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Approximate Cell Doubling Time?

Approximate Cell Doubling Time?


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If I know it takes 12 min to occur in mitotic fibroblast cells and I count there are 32 dividing cells in the microscope slide out of a total population of 32034 cells, how do I calculate the approximate doubling time in hours?

My colleague says it should be around 200 hours. But I don't understand how they reached to this conclusion, can anyone explain this to me please?

Thanks


I figured it out, 32034/32 = ~1000 1000 / 5 = 200 hours

Because 12*5=60 min = 1 hour


6.6.5: Generation Time

  • Contributed by Boundless
  • General Microbiology at Boundless

Examine microbial generation times

Bacterial growth is the division of one bacterium into two daughter cells in a process called binary fission. Providing no mutational event occurs the resulting daughter cells are genetically identical to the original cell. Therefore, &ldquolocal doubling&rdquo of the bacterial population occurs. Both daughter cells from the division do not necessarily survive. The doubling time is the generation time of the bacteria. If the number surviving exceeds unity on average, the bacterial population undergoes exponential growth.

The measurement of an exponential bacterial growth curve in batch culture was traditionally a part of the training of all microbiologists. The basic means requires bacterial enumeration (cell counting) by direct and individual (microscopic, flow cytometry), direct and bulk (biomass), indirect and individual (colony counting), or indirect and bulk (most probable number, turbidity, nutrient uptake) methods. In autecological studies, bacterial growth in batch culture can be modeled with four different phases: lag phase, exponential or log phase, stationary phase, and death phase.

Figure: Bacterial Growth Curve: This chart shows the logarithmic growth of bacteria. Note the Y-axis scale is logarithmic meaning that the number represents doubling. The phases of growth are labelled on top.

During lag phase, bacteria adapt themselves to growth conditions. It is the period where the individual bacteria are maturing and not yet able to divide. During this phase of the bacterial growth cycle, synthesis of RNA, enzymes, and other molecules occurs. The exponential phase (sometimes called the log phase or the logarithmic phase) is a period characterized by cell doubling. The number of new bacteria appearing per unit time is proportional to the present population. If growth is not limited, doubling will continue at a constant rate so both the number of cells and the rate of population increase doubles with each consecutive time period. For this type of exponential growth, plotting the natural logarithm of cell number against time produces a straight line. The slope of this line is the specific growth rate of the organism, which is a measure of the number of divisions per cell per unit time.


Effects of DNA replication on mRNA noise

There are several sources of fluctuations in gene expression. Here we study the effects of time-dependent DNA replication, itself a tightly controlled process, on noise in mRNA levels. Stochastic simulations of constitutive and regulated gene expression are used to analyze the time-averaged mean and variation in each case. The simulations demonstrate that to capture mRNA distributions correctly, chromosome replication must be realistically modeled. Slow relaxation of mRNA from the low copy number steady state before gene replication to the high steady state after replication is set by the transcript's half-life and contributes significantly to the shape of the mRNA distribution. Consequently both the intrinsic kinetics and the gene location play an important role in accounting for the mRNA average and variance. Exact analytic expressions for moments of the mRNA distributions that depend on the DNA copy number, gene location, cell doubling time, and the rates of transcription and degradation are derived for the case of constitutive expression and subsequently extended to provide approximate corrections for regulated expression and RNA polymerase variability. Comparisons of the simulated models and analytical expressions to experimentally measured mRNA distributions show that they better capture the physics of the system than previous theories.

Keywords: analytical solutions chromosome replication master equation stochastic gene expression stochastic simulation.

Conflict of interest statement

The authors declare no conflict of interest.

Figures

Simulation schematic composed of 200…

Simulation schematic composed of 200 simulation replicates shows the progress of the average…

Comparison of the time-dependent (TD)…

Comparison of the time-dependent (TD) and time-independent (TI) theories for constitutively expressed genes.…

Comparison with experiments. ( A–D…

Comparison with experiments. ( A–D ) A comparison of predicted distributions computed assuming…

( A and B ) Deviation of the time-dependent from time-independent theory of…


Results and Discussion

Development of an automated scalable solid-phase doubling time estimation platform

An ideal solid-phase, time-resolved, growth analysis platform would allow for high sample density and be amenable to automated data acquisition and processing. The optimized method, which we have termed ODELAY, is depicted schematically (Figure 1) and consists of four stages: spotting arrays of live yeast onto thin beds of growth substrate on a glass slide support (Figure 1A) periodic bright field image acquisition over a user-specified time course (Figure 1B) processing of raw bright field data to extract microcolony cross-sectional area data (Figure 1C) and postprocessing calculation of growth parameters for each individual microcolony within each spot (Figure 1D). ODELAY is applicable to a wide range of growth substrates and incubation temperatures and is highly scalable, as it can analyze between 10 5 and 10 6 individual microcolonies per experiment.

One-cell Doubling Evaluation by Living Arrays of Yeast (ODELAY). Solid-phase growth parameters are extracted by collecting time course image of growing colonies (A and B). Colony areas are measured from thresholded and binarized images from the time course image series (C). Colonies seeded from single yeast cells are tracked over time and the log2(Area) is used to fit a parameterized version of the Gompertz function (D, left). A control yeast strain (BY4741) was pregrown to saturation with glucose as a carbon source and then assayed on galactose-containing agar. The resulting heterogeneous colonies were clustered based on growth curve characteristics and graphed using colors to represent each cluster (D, right).

ODELAY consists of an automated pipeline that encompasses acquisition and processing of images, identification and measurement of microcolonies at each time point, matching of microcolonies through time, and extrapolation of growth parameters from growth curves. This current platform employs theoretical approximation of ODELAY growth curves using the Gompertz function as an unsupervised method to extract growth parameters (Gompertz 1825 Preibisch et al. 2009). All files required for execution of automated ODELAY analysis, as well as a demonstrative data set, are available as Supplemental Material (File S1 and File S2).

Determination of growth parameters by ODELAY

First, data are acquired, and then growth parameters of doubling time, lag time, and carrying capacity are determined by directly fitting a parameterized version of the Gompertz function (Equation 1). For data acquisition, the first time point would ideally be acquired immediately after spotting onto agar at the desired growth temperature, but for practical purposes, the starting time is when the cells are spotted at room temperature on the solid substrate. The plate is then transferred to an environmentally controlled chamber and growing colonies are tracked until they merge with their neighbors. The time required for colonies to merge is therefore related to the initial cell density and the ultimate carrying capacity of adjacent colonies. While many colonies merge before carrying capacities are observed, ODELAY will still estimate carrying capacity as long as a sufficient number of data points are collected after maximum growth velocity is achieved. This is a feature of the Gompertz function’s symmetry about maximum growth velocity, which permits fair estimation of carrying capacity even when it is not directly measured. Note that caution should be exercised when examining phenotypes associated with increased carrying capacity because the Gompertz function may not accurately estimate all possible outcomes.

Comparison of ODELAY to established methods

We directly compared doubling times and lag times calculated by multiple ODELAY population measurements to liquid culture OD600 measurements made using the BioScreen C instrument for both fast and slow growing strains taken from the MATα yeast deletion library (Winzeler et al. 1999) (Figure 2A). Population doubling times and precision of this measurement across replicates were roughly comparable between the two platforms (Figure 2A). Measured doubling times for 140 yeast strains correlated well between the two platforms with a Pearson coefficient of 0.76 and Spearman coefficient of 0.70 (Figure 2B). These correlation coefficients are similar to growth rate comparisons of growth on solid media versus liquid media reported elsewhere (Zackrisson et al. 2016). Growth rates measured with ODELAY correlate rather modestly with other colony pinned and liquid growth OD600 assays, which is also similarly reported (Zackrisson et al. 2016). These differences likely reflect differences in growth conditions employed in each experiment or method. ODELAY-derived lag times showed less agreement with liquid growth OD600 measurements, likely due to the liquid vs. solid culture medium, and the lack of sensitivity of optical density measurements at low cell concentrations (Figure 2B). In addition, unlike BioScreen, ODELAY identified slow growing outliers because microcolony growth curves are derived from single cells. In contrast, liquid culture OD600 curves measure an aggregate of all cells in a population, and therefore are not sensitive to the contribution of individual cells.

Complex phenotypes observed by ODELAY. Comparisons of doubling times and lag times for repeated measurements (A). Red lines indicate Bioscreen C results. Median ODELAY measurements show good agreement with BioScreen C measurements in doubling time but less so in lag time (B). Population histograms of doubling time from SWR1 complex deletion and GFP-tagged strains (blue) with comparison to the parent strain BY4742 (gray) (C). Heterogeneity in doubling times is observed in strains swc3Δ and ARP6-GFP while arp6Δ and SWC3-GFP appear similar to the parent strain BY4742. GFP, green fluorescent protein ODELAY, One-cell Doubling Evaluation by Living Arrays of Yeast.

Microcolony convergence is the limiting factor of ODELAY’s dynamic range, which can be controlled by altering the initial cell density obtained when spotting yeast cultures. Increased cell density decreases the time it takes for growing colonies to converge. In contrast, the dynamic range of liquid culture measurements is limited by either the nutrient capacity of the media or the linear range of the density sensor. Due to differences in strain doubling times, the dynamic range is best defined by the total number of doublings required to reach the upper limit starting from a single cell. At optimal seed density of ∼200–500 cells per spot (∼25–50 cells/mm 2 ), the dynamic range of ODELAY is 8–12 doublings, from a single cell up to 250 or as many as 4000 cells, which compares favorably to a dynamic range of 3–5 doublings attainable by most currently available technologies.

In traditional pinned colony assays, the size of a colony is dependent on the number of viable individuals contributing to the colony population, the number of doublings these cells have undergone, the amount of nutrients present, and the ability of the colony to transport nutrients to its reproducing members. The contribution of individuals to the overall colony size is not distinguished by traditional methods such as liquid-based or spot-based assays. In contrast, ODELAY tracks individual cells forming into colonies and can quantify population heterogeneity that other methods cannot resolve.

Example applications of ODELAY

Identification of doubling time phenotypes:

To illustrate the ability of ODELAY to compare population heterogeneity of growth phenotypes between strains such that features of the population distributions may be evaluated, we focused on members of the SWR1 complex of chromatin modifiers (Figure 2C). Chromatin modification is one way for the emergence of epigenetic differences that can manifest as heterogeneity within isogenic populations. We observed population heterogeneity in two strains, swc3Δ and ARP6-GFP (Figure 2C), but not in their respective GFP-tagged or deletion mutant. Population heterogeneity has been observed before in SWR1 deletion strains when measuring POT1-GFP expression during a carbon source switch from glucose to oleic acid (Knijnenburg et al. 2011). In that instance, deletion of other members of the SWR1 deletion complex induced bimodal expression of POT1-GFP. Here, the bimodality of growth phenotypes emerged from cells grown strictly on glucose media and without any stimulation from a change in carbon source. This observation demonstrates that ODELAY readily detects subpopulations of cells present in standard culture of deletion and GFP fusion strains.

Identification of lag time phenotype:

Through ODELAY analysis, outliers with highly variable, expanded, or contracted lag periods can be identified by assessing the distribution of lag times for microcolonies of a given strain (lag time variability), as well as relative lag between tested strains. To demonstrate the quantification of lag time by ODELAY, we exploited the well-studied and highly regulated response of yeast to a carbon source shift from its preferred source, glucose, to an alternative source, galactose (Guarente et al. 1982). This shift is characterized by a lag phase, during which the normally repressed galactose utilization machinery, including the galactose transporter, is induced. Exponentially growing yeast preconditioned in either glucose or galactose liquid medium were spotted onto solid media containing galactose and analyzed by ODELAY (Figure 4A). Cells preconditioned in galactose media exhibited a highly synchronized response characterized by short lag times. In contrast, more pronounced and variable lag times were observed for cells that were not primed for growth in galactose. Once glucose-grown cells acclimated to the shift to galactose and entered exponential phase, they doubled at rates similar to those observed for galactose preconditioned cells.

As with heterogeneity of doubling times, ODELAY enables the detection of heterogeneity in lag times. To demonstrate the utility of ODELAY in assessing population heterogeneity of lag times, we compared growth parameters of a control yeast strain (BY4742) in galactose-containing medium after pregrowth in glucose media for differing amounts of time (Figure 3A). We staggered seeding of cultures such that cells were pregrown in glucose media for 3, 6, 24, and 48 hr (Figure 3B). The resulting cultures were then spotted on galactose media and their growth phenotypes observed (Figure 3C). Not only did lag time correlate with the length of time that yeast was cultured in glucose but also colony-to-colony variation in lag times increased for the longer incubation times. This example demonstrates ODELAY’s ability to capture the effects of environmental perturbations on population heterogeneity, a feature which is difficult to distinguish using other solid media growth assays.

Observing lag time after a carbon source switch: Growth curves and histograms depicting lag time (TLag) and doubling time (Td) for control yeast strain (BY4742) pregrown in Gal (left) or Glu (right) and then spotted on Gal media (A). Differing growth phenotypes are observed from samples of a control yeast strain taken from the same culture at multiple times after seeding a source culture (B and C). Histograms of the Td and TLag are depicted below each set of growth curves. Note the changes in the lag time distributions as the source culture is aged. This demonstrates ODELAY’s utility and sensitivity to culture conditions of yeast. Gal, galactose Glu, glucose ODELAY, One-cell Doubling Evaluation by Living Arrays of Yeast.

Large-scale multiparameter analyses with ODELAY

A strength of the ODELAY platform is to extract doubling times and lag times for populations of cells growing on solid media in a high-throughput manner. To demonstrate multiparameter growth rate analysis by ODELAY, we assayed a collection of 140 strains that contained gene deletions of transcription factors, transcriptional regulators, and nuclear transport factors including nucleoporins and karyopherins. The genes selected were previously associated with regulating the response to a carbon source shift (Winzeler et al. 1999 Aitchison and Rout 2012 Knijnenburg et al. 2011 Van de Vosse et al. 2011).

For the deletion strains, we quantified colony doubling times and lag times, and estimated carrying capacities during a carbon source switch from glucose to galactose using galactose-to-galactose transition as a control. This rich multivariate dataset underscores how ODELAY can reveal complex and heterogeneous growth phenotypes of populations of individual cells growing into colonies (Figure 4). Strains with noticeably strong increases in doubling time include dot1Δ, htl1Δ, eaf5Δ, eaf7Δ, and spt20Δ. Of these five examples, only spt20Δ had been reported to have reduced growth rate on galactose media (Roberts and Winston 1996).

Comparison of 140 deletion strains: doubling times, lag times, and estimated carrying capacities of 140 deletion strains that underwent a Glu to Gal switch vs. those that were maintained on Gal as a carbon source. Strains are grouped into categories according to annotated gene function (yeastgenome.org) including chromatin modifiers (Chrom), acetyltransferase enzymes (Acetyl), protein complexes (HDA1, INO80, ISWI NatB/C, NuA4, Rpd3S/L, SET3, SIR, SWI/SNF, SWR1, SAS, and SetC), transcriptional regulators (TRs), nucleoporins and karyopherins (Nup/Kap), and other genes associated with carbon source switching (other). Gal, galactose Glu, glucose.

In general, reporting absolute values of growth parameters is rare in the literature. Here, we present a second large-scale application of ODELAY to compare doubling times of yeast mutants to the parent strain. A commonly overlooked class of mutant includes the C-terminal tagging with GFP, which is often assumed to have negligible effects on growth when compared with the more dramatic growth defects observed in deletion strains. We tested this assumption by comparing the doubling time of the previously mentioned deletion strains and the corresponding GFP fusion strains against their parent strain, BY4742. All measurements were repeated in triplicate on rich glucose media with the most frequently observed doubling time, the population mode, of each replicate compared to the parent strain using the Student’s t-test. ODELAY was able to resolve 12 GFP fusions with doubling times significantly decreased compared to BY4742 and 71 strains that have significantly increased doubling times (Table 1). The deletion strains group had 11 strains with significantly decreased doubling times while 72 had significantly increased doubling times (Table 2). While the majority of the doubling time differences for the GFP strains were <5 min, the presence of the GFP tag does appear to have a widespread and significant impact on growth rates on rich media.

Comparison of ODELAY to other phenotypic analysis methods

ODELAY differs from colony pinning assays and liquid culture assays primarily in its ability to observe heterogeneity in lag time, doubling time, and carrying capacity of colonies forming from individual cells. A summary of features of various growth assays are described (Table S2). While ODELAY is similar to previously published methods that observe colonies forming from single cells in liquid media (Levy et al. 2012), ODELAY is distinguished from these techniques by observing larger areas with image stitching and improved analysis of the resulting growth curves. In its current configuration, ODELAY can resolve individual cells with a pixel resolution of 0.65 μm and can evaluate 96 strains per assay. For each strain, up to 10 3 individual cells are observed growing into colonies with between 10 5 and 10 6 colonies observed per experiment. This feature allows clear observation of population heterogeneity, to which most other methods are insensitive, and allows new avenues for characterizing variations cellular growth phenotypes, even for well-studied organisms such as baker’s yeast.

Periodically imaged colony pinning assays, such as ultrahigh-density omics (Bean et al. 2014), Scan-o-matic (Zackrisson et al. 2016), and ScanLag (Levin-Reisman et al. 2010), have very high throughput in terms observing from 96 to 6144 colonies per plate, with 1536 colonies being typical (Table S2). Each colony has the potential to be a unique strain, allowing complete screening of yeast libraries within a short period of time. However, arraying colonies at higher densities limits the total observation time from when a colony can be clearly resolved by the imaging methods to when the colonies merge (Bean et al. 2014). Herein lies the tradeoff between ODELAY and pinned assays ODELAY may observe heterogeneity at the cost of total number of strains observed. Though it should be noted that, as with colony pinning assays, ODLEAY’s strain throughput can be increased with improved tooling.

Similar to other growth assays, there are caveats associated with ODELAY. Extraction of cell doubling time by ODELAY relies on the assumption that microcolony cross-sectional area is directly proportional to the volume of cells in a given colony and that this relationship between volume and area is unaffected by changes in growth condition and/or genetic background. There will certainly be exceptions to this assumption in yeast and other colony forming micro-organisms however, similar to the limitations in liquid culture OD600 analysis when applied to flocculent mutant strains, such exceptions may yield informative phenotypic information. Furthermore, ODELAY could be adapted to analyze the 3D volume of the growing microcolonies however, this would trade off time for collecting images or limit the total area interrogated. Lag time measurements were also observed to have local variations, which are also commonly observed in other solid phase growth assays (Baryshnikova et al. 2010 Levin-Reisman et al. 2010). Although all initial experiments have utilized haploid baker’s yeast, this methodology can be applied in other colony-forming organisms including medically relevant bacteria such as Mycobacterium tuberculosis, Pseudomonas aeruginosa, Staphylococcus aureus, and others.

In summary, ODELAY is a quantitative tool capable of multiparameter growth analysis based on time resolved microcolony expansion on solid media. The unique features of ODELAY include its relatively large dynamic range, when compared to other available methods, which enables quantitative measurement of doubling time, lag time, and carrying capacity in a single experiment. Additionally, ODELAY has the ability to assess population heterogeneity, including viability, through the analysis of single microcolonies.


Where is it useful? [ edit ]

A constant relative growth rate means simply that the increase per unit time is proportional to the current quantity, i.e. the addition rate per unit amount is constant. It naturally occurs when the existing material generates or is the main determinant of new material. For example, population growth in virgin territory, or fractional-reserve banking creating inflation. With unvarying growth the doubling calculation may be applied for many doubling periods or generations.

In practice eventually other constraints become important, exponential growth stops and the doubling time changes or becomes inapplicable. Limited food supply or other resources at high population densities will reduce growth, or needing a wheel-barrow full of notes to buy a loaf of bread will reduce the acceptance of paper money. While using doubling times is convenient and simple, we should not apply the idea without considering factors which may affect future growth. In the 1950s Canada's population growth rate was over 3% per year, so extrapolating the current growth rate of 0.9% for many decades (implied by the doubling time) is unjustified unless we have examined the underlying causes of the growth and determined they will not be changing significantly over that period.


Growth Curves: Generating Growth Curves Using Colony Forming Units and Optical Density Measurements

Source: Andrew J. Van Alst 1 , Rhiannon M. LeVeque 1 , Natalia Martin 1 , and Victor J. DiRita 1
1 Department of Microbiology and Molecular Genetics, Michigan State University, East Lansing, Michigan, United States of America

Growth curves provide valuable information on bacterial growth kinetics and cell physiology. They allow us to determine how bacteria respond in variable growth conditions as well as to define optimal growth parameters for a given bacterium. An archetypal growth curve progresses through four stages of growth: lag, exponential, stationary, and death (1).


Figure 1: Bacterial growth curve. Bacteria grown in batch culture progress through four phases of growth: lag, exponential, stationary, and death. Lag phase is the period of time it takes for the bacteria to reach a physiological state capable of rapid cell growth and division. Exponential phase is the stage of fastest cell growth and division during which DNA replication, RNA transcription, and protein production all occur at a constant, rapid rate. Stationary phase is characterized by a slowing down and plateauing of bacterial growth due to nutrient limitation and/or toxic intermediate accumulation. Death phase is the stage during which cell lysis occurs as a result of severe nutrient limitation.

Lag phase is the period of time it takes for the bacteria to reach a physiological state capable of rapid cell growth and division. This lag occurs because it takes time for bacteria to adjust to their new environment. Once the necessary cellular components are generated in lag phase, bacteria enter the exponential phase of growth where DNA replication, RNA transcription, and protein production all occur at a constant, rapid rate (2). The rate of rapid cell growth and division during the exponential phase is calculated as the generation time, or doubling time, and is the fastest rate at which the bacteria can replicate under the given conditions (1). The doubling time can be used to compare different growth conditions to determine which is more favorable for bacterial growth. The exponential growth phase is the most reproducible growth condition as bacterial cell physiology is consistent throughout the entire population (3). Stationary phase follows the exponential phase where cell growth plateaus. Stationary phase is brought on due to nutrient depletion and/or accumulation of toxic intermediates. Bacterial cells continue to survive in this stage, although the rate of replication and cell division is drastically reduced. The final phase is death, where severe nutrient depletion leads to the lysing of cells. Features of the growth curve that provide the most information include the duration of lag phase, the doubling time, and the maximum cell density reached.

Quantification of bacteria in batch culture can be determined using both colony forming units and optical density measurements. Enumeration by colony forming units (CFU) provides a direct measurement of bacterial cell counts. The standard unit of measure for CFU is the number of culturable bacteria present per 1 mL of culture (CFU/mL) determined by serial dilution and spread plating techniques. For each timepoint, a 1:10 dilution series of the batch culture is performed and 100 µl of each dilution is spread plated using a cell spreader.


Figure 2. Serial dilution plating schematic. General flowthrough for dilution plating from batch culture. The batch culture is serially diluted 1:10 by transferring 1 mL  of the previous dilution into the subsequent tube containing 9ml PBS. From each dilution tube, 100 µl is spread plated using a plate spreader which is an additional dilution of 1:10 as it is 1/10 th the volume of 1 mL volume when calculating CFU/mL. Plates are incubated and enumerated once clonal colonies grow on the plates.

The plates are then incubated overnight and clonal colonies enumerated. The dilution plate which grew 30-300 colonies is used to calculate the CFU/mL for the given timepoint (4, 5). Stochastic variation in colony counts under 30 are subject to greater error in the calculation of CFU/mL and counting colonies greater than 300 can be underestimated due to colony crowding and overlapping. Using the dilution factor for the given plate, the CFU of the batch culture can be calculated for each timepoint.

Optical density gives an instant approximation of bacterial cell count measured using a spectrophotometer. The optical density is a measure of absorbance of light particles that pass through 1cm of culture and detected by a photodiode sensor (6). The optical density of a culture is measured in relation to a media blank and increases as bacterial density increases. For bacterial cells, a wavelength of 600 nm (OD600) is typically used when measuring optical density (4). By generating a standard curve relating colony forming units and optical density, the optical density measurement can be used to readily approximate the bacterial cell count of a batch culture. However, this relationship begins to deteriorate as early as 0.3 OD600 as cells begins to change shape and accumulate extracellular products in the media, influencing the optical density reading as it relates to CFU (7). This error becomes more pronounced during stationary and death phases.

Here, Escherichia coli is grown in Luria-Bertani (LB) broth at 37°C over the course of 30 hours (7). Both CFU/mL and optical density growth curves have been generated as well as the standard curve relating optical density to CFU. 


Figure 3. Escherichia coli optical density at 600 nm wavelength (OD600) growth curve. Optical density values were taken directly from the spectrophotometer after blanking with sterile LB media. OD600 values greater than 1.0 were diluted 1:10 by combining 100 µl culture with 900 µl fresh LB, again measured, and then multiplied by 10 to obtain the OD600 value. This step is taken as the accuracy in measurement of the spectrophotometer is reduced at high cell density. From the curve, lag phase extends to around 1h of growth, transitions to exponential phase from 2h to 7h, then begins to plateau, entering stationary phase. Death phase is not a stark transition, however, as optical density gradually begins to decline after 15h.


Figure 4. Escherichia coli colony forming unit per milliliter (CFU/mL) growth curve. CFU/mL values for each timepoint were calculated from the dilution plate that contained 30-300 colonies. From the curve, lag phase extends out to around 2h of growth, transitions to exponential phase from 2h to 7h, then begins to plateau, entering stationary phase. Death phase is not a stark transition, however, as CFU/mL gradually begins to decline after 15h from a peak of 2 x 10 9 to approximately 5 x 10 8 at 30 hours.


Figure 5. Standardization curve for CFU/mL versus OD600. A linear regression can be used to relate these units so that optical density may be used to approximate bacterial cell density. Optical density can be used to provide and instant approximation of the CFU/mL of the batch culture. Here, only the first six timepoints are plotted as the relationship between OD600 and CFU/mL is less accurate beyond 1.0 OD600 as cell shape and extracellular products begin to accumulate as the bacteria enter stationary phase, which occurs shortly after reaching 1.0 OD600. Changes in cell shape and extracellular products in the media influence the optical density reading and therefore the relationship between optical density and the number of bacteria in the culture is also impacted.

The doubling time has also been determined to be 15 minutes and 19 seconds. From this data, the capacity for growth in LB for E. coli can be visualized and be used for comparison between different media or bacteria.

Procedure

  1. Required laboratory materials: liquid media, solidified agar media, Erlenmeyer flasks, 15 mL test tubes, phosphate buffered saline (PBS), bacterial cell spreader, 70% ethanol, and a spectrophotometer. All solutions and glassware must be sterilized prior to use.
  2. Prepare the work station by sterilizing with 70% ethanol. Work near a Bunsen burner to prevent contamination of media.
  3. When working with bacteria, proper personal protective equipment and aseptic technique should be used. A lab coat and gloves are required when working with bacterial cultures.
  4. Recipes for buffers, solutions, and reagents
    1. Phosphate buffered saline (PBS) (8).
    2. Luria-Bertani Broth (LB) (9).
    1. Preparation of Media
      1. Identify the growth media with which to grow the bacteria and prepare both liquid broth and solid agar (1.5% w/v agar) media in separate autoclavable bottles. Here, LB broth and LB agar were prepared for the growth of Escherichia coli.
      2. Sterilize the media with a semi-tightened cap in an autoclave set to 121 °C for 35 min.
      3. For agar media, after autoclaving, place in a water bath set to 50 °C for 30 minutes to cool. Once cooled, pour 20-25 mL agar media into 100x15mm circular Petri dishes. Allow plates to set 24 hours at room temperature before use.
      1. From frozen stock, streak bacteria for isolation on selected media agar to obtain single colony isolates. Incubate in growth conditions permissible for the chosen bacteria. Here, E. coli is streaked on LB agar and is incubated at 37 °C overnight (16-18h).
      2. Using a sterile inoculation loop, select a single colony from the streak plate and inoculate 4 mL liquid media in a 15 mL test tube and grow in conditions permissible for the chosen bacteria. Here, E. coli is grown at 37 °C with shaking at 210 rpm overnight (16-18h).
      1. Growth flask preparation
        1. Autoclave appropriately sized Erlenmeyer flasks. Typically, a 1:5 ratio of media to total flask volume is used. Here, 100 mL LB media is used in a 500 mL flask.
        2. Using a serological pipette, transfer sterile media to the Erlenmeyer flask.
        1. Label 15 mL test tubes: -1, -2, -3, -4, -5, -6, -7, -8, and -9, distributing 9mL PBS into each. These numbers correspond to the dilution factor used to calculate CFU/mL. A new set of tubes is needed for each collection timepoint. (Figure 2)
        1. Label plates with time of collection and dilution factor. For each timepoint there will be one plate for each dilution.
        1. Inoculation of media
          1. Using the overnight liquid culture prepared as part of step 2.2.2, inoculate the flask media with 1:1000 volume of culture. Here, 100 µL overnight liquid culture is added to 100 mL LB media.
          2. Swirl the media to evenly distribute the bacteria.
          1. Growth condition setup
            1. Place flask in experimental growth conditions chosen for the given bacteria. Timepoints should be taken frequently for fast-growing bacteria and can be taken in longer intervals for slow-growing bacteria. Here, E. coli is grown at 37°C with shaking at 210 revolutions per minute (rpm) and timepoints taken every 1 hour.
            1. At each timepoint, including the starting timepoint (t = 0), withdraw 1 mL of bacterial culture and dispense into a spectrophotometer cuvette.
            2. Wipe the cuvette clean and record the optical density at 600 nm wavelength. If the optical density reading is greater than 1.0, dilute 100 µL of culture 1:10 with 900 µL fresh media, record the optical density, and multiply this value by 10 for the OD600 measurement.
            1. At each timepoint, withdraw 1 mL of bacterial culture and dispense into the -1 glass test tube containing 9 mL of PBS.
            2. For the dilution series, serially transfer 1 mL from the -1 tube down all dilution tubes to the -9, vortexing after each transfer. (Figure 2)
            3. For each dilution, dispense 100 µL of cell suspension to the correspondingly labelled solid media agar plate. (Figure 2)
            4. Using a cell spreader that has been sterilized in ethanol, passed through a Bunsen burner flame, and cooled by touching the surface of the agar, spread the 100 µL of cell suspension until the surface of the agar plate becomes dry.
            5. Incubate the spread plates upside-down at a temperature that supports growth of the bacteria. Here, E. coli is incubated at 37°C.
            6. After incubation, once visible colonies arise, count the number of bacterial colonies on each plate and record these values along with their associated dilution factor for all plates at each timepoint.

            3. Data Analysis and Results

            1. Optical Density (OD600) Growth Curve Plot
              1. Plot the optical density (OD600) versus time on a semi-log scale. (Figure 3)
              1. For each timepoint, choose the dilution plate where the colony counts fell within the range of 30-300 bacteria. Multiply the colony count number by the dilution factor and then by 10 as the 100 µL spread is considered an additional 1:10 dilution when calculating CFU/mL.
              2. Plot the colony forming units versus time on a semi-log scale. (Figure 4)
              1. Plot the colony forming units versus optical density on a linear scale for OD600 readings less than or equal to 1.0 OD600 as the relationship between OD600 and CFU/mL is less accurate beyond 1.0 OD600. Here, the first six timepoints are plotted. (Figure 5)
              2. Generate a linear regression trendline displaying the equation and R 2 value.
              1. Using the colony forming unit growth curve plot, during exponential phase, identify two points on the graph with the steepest slope between them to calculate the doubling time.
              2. Calculating the doubling time
                1. ΔTime = t2 - t1, where t1 = Timepoint 1 and t2 = Timepoint 2
                2. , where b = number of bacteria at t2, B = number of bacteria at t1, and n = number of generations. Derived from: .
                3. Calculate doubling time using:

                Bacteria reproduce through a process called cell division, which results in two identical daughter cells. If the growth conditions are favorable, bacterial populations will grow exponentially.

                Bacterial growth curves plot the amount of bacteria in a culture as a function of time. A typical growth curve progresses through four stages: lag phase, exponential phase, stationary phase, and death phase. The lag phase is the time it takes for bacteria to reach a state where they can grow and divide quickly. After this, the bacteria transition to the exponential phase, characterized by rapid cell growth and division. The rate of exponential growth of the bacterial culture during this phase can be expressed as the doubling time, the fastest rate at which bacteria can reproduce under specific conditions. The stationary phase comes next, where bacterial cell growth plateaus and the growth and death rates even out due to environmental nutrient depletion. Finally, the bacteria enter the death phase. This is where bacterial growth declines sharply and severe nutrient depletion leads to the lysing of cells.

                Two techniques can be used to quantify the amount of bacteria present in a culture and plot a growth curve. The first of these is via colony forming units, or CFUs. To obtain CFUs a one to ten series of nine dilutions is performed at regular time points. The first of these dilutions, negative one in this example, contains 9mL of PBS and 1mL of the bacterial culture. Resulting in a 1:10 dilution factor. Then, 1mL of this solution is transferred to the next tube, negative two, resulting in a 1:100 dilution factor. This process continues through the last tube, negative nine, resulting in a final dilution factor of 1:1 billion. After this, 100 microliters of each dilution is plated. The plates are then incubated and the clonal colonies are counted. The dilution plate for a given time point that grows between 30 and 300 colonies is used to calculate the CFUs per milliliter for that time point.

                The second common method of measuring bacterial concentration is the optical density. The optical density of a culture can be measured instantly, in relation a media blank, with a spectrophotometer. Typically a wave length of 600 nanometers, also referred to as OD600, is used for these measurements, which increase as cell density increases. While optical density is less precise than CFUs, it is convenient because it can be obtained instantaneously and requires relatively few reagents. Both techniques can be used together to create a standard curve that more accurately approximates the bacterial cell count of a culture. In this video, you will learn how to obtain CFUs and OD600 measurements from timed serial dilutions of E. coli. Then, two growth curves using the CFU and OD600 measurements, respectively, will be plotted before being related by a standard curve.

                When working with bacteria, it is important to use the appropriate personal protective equipment such as a lab coat and gloves and to observe proper aseptic technique.

                After this, sterilize the work station with 70% ethanol. First, prepare the LB broth and LB solid agar media in separate autoclaveable bottles. After partially closing the caps of the bottles, sterilize the media in an autoclave set to 121 degrees Celsius for 35 minutes. Next, allow the agar media to cool in a water bath set to 50 degrees Celsius for 30 minutes. Once cooled, pour 20 to 25 mL into each Petri dish. After this, allow the plates to set for 24 hours at room temperature.

                To prepare the single colony isolates that will later be used to produce a liquid bacterial culture, use previously frozen stock and proper streak plating technique to streak E. coli for isolation on LB agar. Incubate the dish at 37 degree Celsius overnight. After this, cool a flame sterilized inoculation loop on the agar before selecting a single colony from the streaked plate. Inoculate 4 mL of liquid media in a 15 mL test tube. Then, grow the E. coli at 37 degrees Celsius overnight with shaking at 210 rpm.

                To set up the 1:1000 volume of bacterial culture that will be used in the growth curve, first obtain an autoclaved 500 mL Erlenmeyer flask. Then, use a 50 mL serological pipette to transfer 100 mL of sterile media to the flask. Next, label nine 15 ml test tubes consecutively as one through nine. These numbers correspond to the dilution factor that will be used to calculate the colony forming unit, or CFU. Then, add 9 mL of 1X PBS to each tube. After this, label the prepared agar plates with the corresponding time points and dilution factors that will be grown. In this example with E. coli, after the starting time point, time points are taken once every hour. Using the previously prepared overnight liquid E. coli culture, inoculate the media in the autoclave 500 mL Erlenmeyer flask with 1:1000 volume of culture. Swirl the media to evenly distribute the bacteria.

                After blanking a spectrophotometer, clean the cuvette with a lint-free wipe. Next, dispense 1 mL of the culture into the cuvette and place it into the spectrophotometer to obtain the optical density of the culture at time point zero. Then, grow the E. coli at 37 degrees Celsius with shaking at 210 rpm. At each time point after time point zero, withdraw another 1 mL of bacterial culture from the flask and repeat the optical density measurement. If the optical density reading is greater than 1.0, dilute 100 microliters of bacterial culture with 900 microliters of fresh media and then measure the optical density once more. This value can be multiplied by 10 for the OD 600 measurement.

                To obtain the colony forming unit measurement for each time point, withdraw an additional 1 mL of bacterial culture from the flask at each time point. Dispense the bacterial culture into the negative one test tube and vortex to mix. Then, perform the dilution series by first transferring 1 mL from the negative one tube into the negative two tube and vortex to mix. Transfer 1 mL from the negative two tube into the negative three tube and vortex to mix. Continue this serial transfer down all the dilution tubes to the negative nine tube. Dispense 100 microliters of cell suspension onto the correspondingly labeled plate for each dilution. For every dilution, sterilize a cell spreader in ethanol, pass it through a Bunsen burner flame, and cool it by touching the surface of the agar away from the inoculate. Then, use the cell spreader to spread the cell suspension until the surface of the agar plate becomes dry. Incubate the plates upside down at 37 degrees Celsius. Once visible colonies arise, count the number of bacterial colonies on each plate. Record these values and their associated dilution factors for each plate at each time point.

                To create an OD 600 growth curve, after ensuring all the data points are entered correctly into a table, select all of the time points and their corresponding data. To generate a colony forming unit growth curve plot, choose the dilution plate where the colony counts fell within the range 30 to 300 bacteria for each time point. Multiply the colony count number by the dilution factor, and then by ten. This is because the 100 microliters spread is considered an additional 1:10 dilution when calculating colony forming units per milliliter. After this, plot the colony forming units versus time on a semi-log scale.

                These plots produced with OD 600 and CFU measurements, respectively, can provide valuable information on E. coli growth kinetics. The optical density and colony forming units can be related, so that CFUs per milliliter can be estimated from OD 600 measurements, saving time and materials in future experiments.

                To do this, plot the colony forming units against the optical density on a linear scale for OD 600 readings less than or equal to 1. 0. After this, generate a linear regression trend line in Y = MX + B format, where M is the slope and B is the y-intercept. Right click on the data points and select add trend line and linear. Then, check the box to display the equation on the chart and display the R squared value on the chart. The R squared value is the statistical measurement of how closely the data matched the fitted regression line. In this example, the first 6 time points are plotted with OD 600 on the x axis and CFUs per milliliter on the y axis. In future experiments with the same growth conditions, these slope and y-intercept values can be plugged into this equation to estimate CFUs from OD 600 readings. Next, look at the colony forming unit growth curve plot. During the exponential phase, identify two time points with the steepest slope between them. To calculate the doubling time, first calculate the change in time between the selected time points. Then, calculate the change in generations using the equation shown here. Here, lower case b is the number of bacteria at time point three and upper case B is the number of bacteria at time point two. Finally, divide the change in time by the change in generations. In this example, the doubling time is 0. 26 hours or 15 minutes and 19 seconds. Comparing doubling times across different experimental treatment allows us to identify the best growth conditions for a certain bacterial species. Therefore, the treatment with the lowest doubling time will be most optimal of the conditions tested.

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                Results

                Plots of colony forming units and optical density are two ways to visualize growth kinetics. By determining the relationship between CFU/mL and OD600, the optical density plot also provides an estimate of CFU/mL over time. Conditions that result in the shortest doubling time are considered optimal for growth of the given bacteria.

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                Applications and Summary

                Growth curves are valuable for understanding the growth kinetics and physiology of bacteria. They allow us to determine how bacteria respond in variable growth conditions as well as define the optimal growth parameters for a given bacterium. Colony forming unit and optical density plots both contain valuable information depicting the duration of lag phase, maximum cell density reached, and allowing for the calculation of bacterial doubling time. Growth curves also allow for comparison between different bacteria under the same growth conditions. Additionally, optical density provides a means of standardizing initial inoculums, enhancing consistency in other experiments.

                Determining which approach to use when designing a growth curve experiment requires consideration. As the preferred method for generating growth curves, colony forming unit plots more accurately reflect the viable cell counts in batch culture. Colony forming unit plots also allow for measuring bacterial growth in conditions that would otherwise interfere with an optical density measurement. However, it is a more time consuming process, requiring extensive use of reagents, and must be performed manually. Optical density plots are less accurate and provide only an estimate of the colony forming units, requiring a standard curve to be generated for each unique bacteria. Optical density is primarily used for its convenience as it is far less time consuming and does not require many reagents to perform. What is most attractive to optical density, is that spectrophotometric incubators can automatically generate growth curves, vastly increasing the number of culture conditions that can be tested at once and eliminating the need to constantly attend the culture.

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                References

                1. R. E. Buchanan. 1918. Life Phases in a Bacterial Culture. J Infect Dis 23:109-125.
                2. CAMPBELL A. 1957. Synchronization of cell division. Bacteriol Rev 21:263-72.
                3. Wang P, Robert L, Pelletier J, Dang WL, Taddei F, Wright A, Jun S. 2010. Robust growth of Escherichia coli. Curr Biol 20:1099-103.
                4. Goldman E, Green LH. 2015. Practical Handbook of Microbiology, Third Edition. CRC Press.
                5. Ben-David A, Davidson CE. 2014. Estimation method for serial dilution experiments. J Microbiol Methods 107:214-221.
                6. Koch AL. 1968. Theory of the angular dependence of light scattered by bacteria and similar-sized biological objects. J Theor Biol 18:133-156.
                7. Sezonov G, Joseleau-Petit D, D'Ari R. 2007. Escherichia coli physiology in Luria-Bertani broth. J Bacteriol 189:8746-9.

                Transcript

                Bacteria reproduce through a process called cell division, which results in two identical daughter cells. If the growth conditions are favorable, bacterial populations will grow exponentially.

                Bacterial growth curves plot the amount of bacteria in a culture as a function of time. A typical growth curve progresses through four stages: lag phase, exponential phase, stationary phase, and death phase. The lag phase is the time it takes for bacteria to reach a state where they can grow and divide quickly. After this, the bacteria transition to the exponential phase, characterized by rapid cell growth and division. The rate of exponential growth of the bacterial culture during this phase can be expressed as the doubling time, the fastest rate at which bacteria can reproduce under specific conditions. The stationary phase comes next, where bacterial cell growth plateaus and the growth and death rates even out due to environmental nutrient depletion. Finally, the bacteria enter the death phase. This is where bacterial growth declines sharply and severe nutrient depletion leads to the lysing of cells.

                Two techniques can be used to quantify the amount of bacteria present in a culture and plot a growth curve. The first of these is via colony forming units, or CFUs. To obtain CFUs a one to ten series of nine dilutions is performed at regular time points. The first of these dilutions, negative one in this example, contains 9mL of PBS and 1mL of the bacterial culture. Resulting in a 1:10 dilution factor. Then, 1mL of this solution is transferred to the next tube, negative two, resulting in a 1:100 dilution factor. This process continues through the last tube, negative nine, resulting in a final dilution factor of 1:1 billion. After this, 100 microliters of each dilution is plated. The plates are then incubated and the clonal colonies are counted. The dilution plate for a given time point that grows between 30 and 300 colonies is used to calculate the CFUs per milliliter for that time point.

                The second common method of measuring bacterial concentration is the optical density. The optical density of a culture can be measured instantly, in relation a media blank, with a spectrophotometer. Typically a wave length of 600 nanometers, also referred to as OD600, is used for these measurements, which increase as cell density increases. While optical density is less precise than CFUs, it is convenient because it can be obtained instantaneously and requires relatively few reagents. Both techniques can be used together to create a standard curve that more accurately approximates the bacterial cell count of a culture. In this video, you will learn how to obtain CFUs and OD600 measurements from timed serial dilutions of E. coli. Then, two growth curves using the CFU and OD600 measurements, respectively, will be plotted before being related by a standard curve.

                When working with bacteria, it is important to use the appropriate personal protective equipment such as a lab coat and gloves and to observe proper aseptic technique.

                After this, sterilize the work station with 70% ethanol. First, prepare the LB broth and LB solid agar media in separate autoclaveable bottles. After partially closing the caps of the bottles, sterilize the media in an autoclave set to 121 degrees Celsius for 35 minutes. Next, allow the agar media to cool in a water bath set to 50 degrees Celsius for 30 minutes. Once cooled, pour 20 to 25 mL into each Petri dish. After this, allow the plates to set for 24 hours at room temperature.

                To prepare the single colony isolates that will later be used to produce a liquid bacterial culture, use previously frozen stock and proper streak plating technique to streak E. coli for isolation on LB agar. Incubate the dish at 37 degree Celsius overnight. After this, cool a flame sterilized inoculation loop on the agar before selecting a single colony from the streaked plate. Inoculate 4 mL of liquid media in a 15 mL test tube. Then, grow the E. coli at 37 degrees Celsius overnight with shaking at 210 rpm.

                To set up the 1:1000 volume of bacterial culture that will be used in the growth curve, first obtain an autoclaved 500 mL Erlenmeyer flask. Then, use a 50 mL serological pipette to transfer 100 mL of sterile media to the flask. Next, label nine 15 ml test tubes consecutively as one through nine. These numbers correspond to the dilution factor that will be used to calculate the colony forming unit, or CFU. Then, add 9 mL of 1X PBS to each tube. After this, label the prepared agar plates with the corresponding time points and dilution factors that will be grown. In this example with E. coli, after the starting time point, time points are taken once every hour. Using the previously prepared overnight liquid E. coli culture, inoculate the media in the autoclave 500 mL Erlenmeyer flask with 1:1000 volume of culture. Swirl the media to evenly distribute the bacteria.

                After blanking a spectrophotometer, clean the cuvette with a lint-free wipe. Next, dispense 1 mL of the culture into the cuvette and place it into the spectrophotometer to obtain the optical density of the culture at time point zero. Then, grow the E. coli at 37 degrees Celsius with shaking at 210 rpm. At each time point after time point zero, withdraw another 1 mL of bacterial culture from the flask and repeat the optical density measurement. If the optical density reading is greater than 1.0, dilute 100 microliters of bacterial culture with 900 microliters of fresh media and then measure the optical density once more. This value can be multiplied by 10 for the OD 600 measurement.

                To obtain the colony forming unit measurement for each time point, withdraw an additional 1 mL of bacterial culture from the flask at each time point. Dispense the bacterial culture into the negative one test tube and vortex to mix. Then, perform the dilution series by first transferring 1 mL from the negative one tube into the negative two tube and vortex to mix. Transfer 1 mL from the negative two tube into the negative three tube and vortex to mix. Continue this serial transfer down all the dilution tubes to the negative nine tube. Dispense 100 microliters of cell suspension onto the correspondingly labeled plate for each dilution. For every dilution, sterilize a cell spreader in ethanol, pass it through a Bunsen burner flame, and cool it by touching the surface of the agar away from the inoculate. Then, use the cell spreader to spread the cell suspension until the surface of the agar plate becomes dry. Incubate the plates upside down at 37 degrees Celsius. Once visible colonies arise, count the number of bacterial colonies on each plate. Record these values and their associated dilution factors for each plate at each time point.

                To create an OD 600 growth curve, after ensuring all the data points are entered correctly into a table, select all of the time points and their corresponding data. To generate a colony forming unit growth curve plot, choose the dilution plate where the colony counts fell within the range 30 to 300 bacteria for each time point. Multiply the colony count number by the dilution factor, and then by ten. This is because the 100 microliters spread is considered an additional 1:10 dilution when calculating colony forming units per milliliter. After this, plot the colony forming units versus time on a semi-log scale.

                These plots produced with OD 600 and CFU measurements, respectively, can provide valuable information on E. coli growth kinetics. The optical density and colony forming units can be related, so that CFUs per milliliter can be estimated from OD 600 measurements, saving time and materials in future experiments.

                To do this, plot the colony forming units against the optical density on a linear scale for OD 600 readings less than or equal to 1. 0. After this, generate a linear regression trend line in Y = MX + B format, where M is the slope and B is the y-intercept. Right click on the data points and select add trend line and linear. Then, check the box to display the equation on the chart and display the R squared value on the chart. The R squared value is the statistical measurement of how closely the data matched the fitted regression line. In this example, the first 6 time points are plotted with OD 600 on the x axis and CFUs per milliliter on the y axis. In future experiments with the same growth conditions, these slope and y-intercept values can be plugged into this equation to estimate CFUs from OD 600 readings. Next, look at the colony forming unit growth curve plot. During the exponential phase, identify two time points with the steepest slope between them. To calculate the doubling time, first calculate the change in time between the selected time points. Then, calculate the change in generations using the equation shown here. Here, lower case b is the number of bacteria at time point three and upper case B is the number of bacteria at time point two. Finally, divide the change in time by the change in generations. In this example, the doubling time is 0. 26 hours or 15 minutes and 19 seconds. Comparing doubling times across different experimental treatment allows us to identify the best growth conditions for a certain bacterial species. Therefore, the treatment with the lowest doubling time will be most optimal of the conditions tested.


                Methods

                IncuCyte ZOOM™ Assay

                We perform a monolayer scratch assay using the IncuCyte ZOOM™ live cell imaging system (Essen BioScience, MI USA). This system measures scratch closure in real time and automatically calculates the relative wound density within the initially-vacant area at each time point. The relative wound density is the ratio of the occupied area to the total area of the initial scratched region. All experiments are performed using the PC-3 prostate cancer cell line [16], which is obtained from the American Type Culture Collection (ATCC, Manassas, USA). Cells are routinely propagated in RPMI 1640 medium (Life Technologies, Australia) in 10 % foetal calf serum (Sigma-Aldrich, Australia), with 110 u/mL penicillin, 100 μg/mL streptomycin (Life Technologies), in plastic flasks (Corning Life Sciences, Asia Pacific) in 5 % CO 2 and air in a Panasonic incubator (VWR International) at 37 °C. Cells are regularly screened for Mycoplasma (ATCC). Cells are removed from the monolayer using TrypLE™(Life Technologies) in phosphate buffered saline, resuspended in medium and seeded at a density of 20,000 cells per well in 96-well ImageLock plates (Essen BioScience). After seeding, cells are grown overnight to form a spatially uniform monolayer. We use a WoundMaker™(Essen BioScience) to create uniform, reproducible scratches in all the wells of a 96-well plate. After creating the scratch, the medium is aspirated and the wells are washed twice with fresh medium to remove any cells from the scratched area. Following the washes, for the control assay, 100 μL of fresh medium is added to each well. We also perform a series of experiments where, following the washes, fresh medium containing different concentrations of EGF (Life Technologies) is added to the wells. The concentrations of EGF we use are: 25, 50, 75, 100 and 125 ng/mL. We will refer to these assays as EGF-25, EGF-50, EGF-75, EGF-100 and EGF-125, respectively. Once the fresh medium is added, the plate is placed into the IncuCyte ZOOM™ apparatus and images of the collective cell spreading are recorded every 2 hours for a total duration of 46 hours. For the control assay and each different EGF concentration we perform three identically prepared experimental replicates (n=3).

                Image analysis

                We use Matlab’s Image Processing Toolbox [23, 24] to estimate the position of the leading edge of the spreading cell population in the IncuCyte ZOOM™ images. The experimental image is imported and converted to greyscale using the imread and rgb2gray commands, respectively. We detect edges in the images using edge with the Canny method [25] and automatically-selected threshold values. Detected edges outside of these threshold values are ignored. Remaining edges are dilated using the imdilate command and a structuring element, defined using strel , with a circular element of size 15. Using the bwareaopen command with a component size of 10,000 pixels, we remove any remaining vacant spaces in the image while preserving the vacant scratch. Edge dilation is reversed using the imerode command with the same structuring element defined previously to erode the image. Finally, edges within the image are smoothed using medfilt2 and the area of the remaining vacant space, A(t), representing the vacant area, is estimated using the regionprops command.

                We calculate the position of the leading edge, which we define to be the distance between the centre of the experimental domain and the position of the leading edge using

                where L x is the horizontal width of the image and L y is the vertical height of the image. For all experiments we have L x=1970μm and L y=1430μm. Eq. (1) allows us to examine the time evolution of the scratched area in terms of L E(t), which is the half-width of the scratch (Fig. 1(a)).

                Mathematical model

                We interpret our experimental results using the Fisher-Kolmogorov equation [26–28], which is a continuum reaction-diffusion model describing the spatiotemporal evolution of cell density in a population of cells where cell migration is driven by random (undirected) cell motility and cell proliferation is driven by carrying capacity limited logistic growth. The Fisher-Kolmogorov equation, and extensions of the Fisher-Kolmogorov equation, have been previously applied to in vitro [29–31] and in vivo [32, 33] data describing collective cell spreading in a range of contexts including wound healing [34, 35], tissue repair [3, 4] and malignant spreading [8–10, 36, 37].

                Since our scratch assay takes place on a two-dimensional substrate (Fig. 1(a)–(c)), we start with the two-dimensional analogue of the Fisher-Kolmogorov equation in Cartesian coordinates

                where (overline (x,y,t)) [cells/ μ m 2 ] is the cell density, or average number of cells per unit area, at location (x,y) and time t. For our experiments we have 0≤xL x and 0≤yL y. There are three parameters in the Fisher-Kolmogorov equation: (i) the cell diffusivity, D [ μ m 2 /h], (ii) the cell proliferation rate, λ [/h], and (iii) the carrying capacity density, K [cells/ μ m 2 ]. The proliferation rate, λ, is related to the cell doubling time t d=loge(2)/λ. We note that we make the standard assumption that D, λ and K are constants [1–4, 29, 34]

                Since the initial cell monolayer is spatially uniform and the initial scratch is made perpendicular to the x-direction (Fig. 1(a)), we can simplify the mathematical model by averaging in the y-direction [8–10]. To do this we average the two-dimensional cell density

                which allows us to write Eq. (2) as a one-dimensional partial differential equation

                In general, approximating a two-dimensional nonlinear partial differential equation, such as Eq. (2), by an averaged one-dimensional nonlinear partial differential equation, such as Eq. (4), can introduce an averaging error. However, for initial conditions such as ours where the initial density is independent of the vertical direction, this error vanishes, and a detailed analysis of this error is presented elsewhere [38, 39]. The initial condition for Eq. (4) is given by the width of the scratch (Fig. 1(a))

                where C 0 is the initial density of cells in the monolayer and L E(0) is the initial position of the leading edge. Since we use a WoundMaker™tool to create uniform scratches in all experimental replicates, the initial condition, given by Eq. (5), applies to all experiments and cannot be varied.

                The physical distribution of cells in each experiment extends well-beyond the L x×L y rectangular region imaged by the IncuCyte ZOOM™ apparatus. Therefore, since the cells are spatially uniform except for the scratched region, there will be no net flux of cells across the vertical boundaries along the lines x=0 and x=L x. We model this by using zero-flux boundary conditions

                These boundary conditions do not imply that cells are stationary at x=0 and x=L x. Instead, these boundary conditions imply that the cell density profile is spatially uniform, C(x,t)/ x=0, so that there is no net flux of cells across the boundaries at x=0 and x=L x.

                We solve Eq. (4) using a finite difference numerical method [40]. The spatial domain, 0≤xL x, is discretised uniformly with grid spacing δ x, and the spatial derivatives are approximated using a central-difference approximation [40]. This leads to a system of coupled nonlinear ordinary differential equations that are integrated through time using a backward-Euler approximation with constant time steps of duration δ t [40]. The resulting system of coupled nonlinear algebraic equations are linearised using Picard (fixed-point) iteration, with absolute convergence tolerance ε [41]. The associated tridiagonal system of linear equations is solved using the Thomas algorithm [40]. For all results presented here we always chose δ x, δ t and ε so that our numerical algorithm produces grid-independent results.

                We also apply Eq. (4) to some simplified situations where we focus on the time evolution of the cell density in small subregions, located well-behind the initial position of the scratch, where the cell density is approximately spatially uniform. This implies that C(x,t)≈C(t) within these subregions [18, 21, 22]. Since the cell density is approximately spatially uniform we have C(x,t)/ x=0, and the first term on the right of Eq. (4) vanishes and, subsequently in these subregions, the partial differential equation simplifies to the logistic equation,

                whose solution is given by

                where C(0)=C 0 is the initial density at t=0. The simplification of approximating Eq. (4) by Eq. (7) in subregions well-behind the leading edge where the cell density is spatially uniform does not imply that cells are stationary in these subregions. Instead, Eq. (7) represents the situation where there is no gradient in cell density and cells are free to move amongst the extracellular space within these subregions. The key advantage of applying this approximation is that cell motion in these spatially uniform subregions does not contribute to any temporal changes in cell density. Instead, when the cell density is spatially uniform, any temporal change in cell density is solely associated with the proliferation term in Eq. (4) [18, 21, 22].

                Parameter estimation

                We estimate the three parameters in the Fisher-Kolmogorov model using a sequential approach. First, using cell counting, we estimate the parameters governing cell proliferation: K and λ. Second, using data describing the temporal changes in the position of the leading edge, we estimate the cell diffusivity, D. Although it is possible to use a different approach, based on a multivariate regression technique to estimate D, λ and K simultaneously, we prefer to estimate these parameters sequentially. Estimating the three parameters sequentially, one at a time, emphasises the differences in the interpretation of these parameters, as well as emphasising the differences in the mechanisms of cell proliferation and cell motility. If, instead, a multivariate approach is used to estimate the three parameters simultaneously, we anticipate that the interpretation of the mechanisms associated with these parameters might not be obvious as it is in our approach.

                Carrying capacity density

                To estimate K we focus on experimental images from the latter part of the experiment, t=46 h, where the cell population has grown to confluence. We identify three smaller subregions, located well-behind the initial leading edge, and count the number of cells within each subregion, N. To quantify the variability in our estimate we analyse three different subregions in each image and count N in each replicate subregion. Using this data we estimate the average carrying capacity density as

                where 〈N〉 is the average number of cells within the subregion of area A SR=3.789×10 4 μ m 2 . To examine whether EGF has any impact on the carrying capacity density we estimate K for the control assay and for each experiment treated with a different EGF concentration. Figure 2 shows IncuCyte ZOOM™ images at t=46 h with the location of three subregions superimposed. The images in Fig. 2(a)–(c) show the control, EGF-50 and EGF-100 assays, respectively. We note that the location of all three subregions in each image is located well-behind the initial position of the scratch (Fig. 1(a)) so that after t=46 h the local density of cells within each subregion has grown to confluence. To quantify the variability in our estimate of K, we calculate the sample standard deviation for each EGF concentration, and report results as a mean value for K, with the variation in our estimate given by plus or minus one standard deviation about the mean. Results are summarised in Table 1.

                Final time experimental images (t=46 hours) for three IncuCyte ZOOM™ assays for (a) Control, (b) EGF-50, and (c) EGF-100. The three coloured boxes indicate the location of the three subregions used to estimate K and λ. Each coloured square within the subregions indicates the centre of an individual cell in the cell counting step. Scale bar corresponds to 300 μm

                Proliferation rate

                The logistic equation, given by Eq. (7), describes the time evolution of the cell density where there is, on average, no spatial variation in cell density. To apply the logistic equation to our data we analyse three subregions within each IncuCyte ZOOM™ image at several time points during the assay. Counting the total number of cells in each subregion and dividing by the area of the subregion gives an estimate of the local cell density in that subregion. In all cases the subregions we considered always started off with approximately 20–30 cells at t=0 h. Repeating this procedure for three different subregions, at fixed locations, for each experimental replicate, at five different time points, allows us to calculate the average cell density as a function of time, C(t). With this data, together with our previous estimates of K, we find the value of λ in Eq. (8) that matches our C(t) data across several time points. For consistency, when we estimate λ we always analyse the same three subregions that we used previously to estimate K. The location of these three subregions is shown in Fig. 3. We estimate the initial cell density, C(0), from the first image taken immediately after the scratch is made at t=0 h. Images of the assay in Fig. 3(a)–(e) correspond to 0, 8, 16, 24 and 46 h, respectively. The location of each subregion is chosen to be well-behind the initial position of the leading edge of the population so that the cell density is approximately spatially uniform locally within each subregion. In each subregion C(t) increases with time, and we attribute this increase to cell proliferation. The data in Fig. 3(f) shows the time evolution of the average cell density, C(t), calculated by averaging the three estimates of cell density from each subregion, at each time point. Using our previous estimate of K, we estimate λ by matching the solution of Eq. (7) with the observed C(t) data.

                a-e Time evolution of an EGF-75 IncuCyte ZOOM™ assay. Images taken after (a) 0, (b) 8, (c) 16, (d) 24, and (e) 46 h after the scratch was performed. The three coloured boxes indicate the location of the three subregions used to calculate K and λ. Each coloured square within the subregions indicates the centre of an individual cell in the cell counting step. Scale bar corresponds to 300 μm. f Comparison of the average experimental cell density C(t) (crosses) and the logistic growth curve using our estimates of K and λ (solid)

                For each EGF concentration we have three sets of data describing the temporal variation in average cell density per experimental replicate. For each set of time series data we use Matlab’s lsqnonlin function, a nonlinear least squares minimisation routine [42], to estimate λ. To quantify the average proliferation rate we calculate the average λ value by averaging the three estimates from each experimental replicate. A comparison of the resulting logistic growth curve using our average estimate of K and λ with the observed C(t) data is given in Fig. 3(f) for the EGF-25 experiment, indicating that the solution of Eq. (7) matches the data reasonably well. To quantify the variability in λ we report our mean estimate of λ and the variation as the mean plus or minus one standard deviation. We also report the mean and variability in the C 0 values used to obtain estimates of λ. Results are summarised in Table 1.

                Cell diffusivity

                Several different approaches have been used in previous studies to estimate D from in vitro assays describing collective cell spreading processes. For example, Treloar et al. [22] estimate D in a circular barrier assay by applying two different methods to the same data set. First, they estimate D using a cell labelling and cell counting technique to provide an estimate of the cell density profiles near the leading edge of the spreading population. Treloar et al. [22] calibrate the solution of a partial differential equation to that data to give an estimate of D which matches the position and shape of the spreading cell density profile. Second, using the same data set, Treloar et al. [22] use automated leading edge image analysis [23, 24] to quantify temporal changes in the position of the leading edge of the spreading cell density profile without counting individual cells. Treloar et al. [22] calibrate their model to this leading edge data to obtain a second estimate of D. Given that the two approaches implemented by Treloar et al. [22] produce similar estimates of D, here we choose to estimate D using a similar technique based on leading edge data since this is the most straightforward approach which avoids the need for labelling and counting individual cells within the spreading population.

                Figures 4(a)-(d) show IncuCyte ZOOM™ images at 0, 10, 20 and 30 h, respectively. The position of the two detected leading edges of the spreading population is superimposed on each image. A visual comparison of how the position of the detected leading edges changes with time suggests that the initially-vacant region closes symmetrically with time. The edge detection results allow us to calculate the area of the vacant region, A(t), and with this information Eq. (1) allows us to estimate the half-width, L E(t), which is decreasing function of time. Previously, Treloar et al. [22, 24] showed that the location of the automatically detected leading edge corresponds to a cell density of approximately 2 % of the carrying capacity density.

                a-d Indicate the area of remaining vacant space, A(t), as determined by the edge detection algorithm at a 0, b 10, c 20, and d 30 h for the control assay. The position of the detected leading edge is given in green. The straight vertical lines superimposed on a (white) indicate the average width of the scratch, 2L E(t). Scale bar corresponds to 300 μm. e Average L E(t) data estimated from the control assay experimental images (blue). The error bars correspond to one standard deviation about the mean. Numerical L E(t) data (red), corresponding to the numerical solution of Eq. (4) using the relevant estimates of D, λ and K (Table 1). (f) Evolution of C(x,t) profiles at t=0, 10, 20, 30 h corresponding to the numerical solution of Eq. (4) using the relevant estimates of D, λ and K (Table 1). Arrows indicate the direction of increasing time. Numerical solutions of Eq. (4) correspond to δ x=1μm, δ t=0.1 h and ε=1×10 −6 . The vertical lines show the locations of the subregions where the estimates of λ and K were obtained

                Given our previous estimates of K and λ, and assuming that the position of the detected leading edge corresponds to the location where the density is 2 % of the carrying capacity, we use Matlab’s lsqnonlin function to find an estimate of D that minimises the difference between the observed time series of L E(t) and the time series of L E(t) data from the numerical solution of Eq. (4). We present an example of the match between the experimental measurements of L E(t) and numerical prediction of L E(t) in Fig. 4(c). For all time points, the numerical estimate of L E(t) is always within one standard deviation of the mean of the experimental measurements. Given our estimates of D, λ and K, we can use our numerical solution of Eq. (4) to explore how C(x,t) varies across the entire width of the domain, for the duration of the assay, as illustrated in Fig. 4(f). These profiles show that the cell density remains approximately spatially uniform well-behind the initial location of the scratch. In fact, we have indicated the position of the location of the various subregions used to estimate K and λ on the profiles in Fig. 4(f), and we see that the predicted cell density profile is spatially uniform, C(x,t)/ x=0, at these locations for the duration of the assay, which is visually consistent with the subregions presented in Fig. 2. The cell density in the three subregions well-behind the initial location of the scratch increases with time owing to cell proliferation. These profiles also show the cell density front near the location of the scratch moves inward to close the initially-scratched region with time.

                Using our approach we calculate an average value of D by estimating the cell diffusivity for each experimental replicate and then averaging the results. We note that Treloar et al. found that varying the Matlab edge detection threshold parameters led to a small variation in the position of the detected leading edge corresponding to a cell density in the range of approximately 1–5 % of the carrying capacity density [24]. To quantify the variability in our estimate of D we repeated the edge detection by assuming that the position of the detected leading edge corresponds to both 1 and 5 % of the carrying capacity density. Results are summarised in Table 1. We note that the maximum value of D is obtained by assuming that the detected leading edge corresponds to 5 % of the carrying capacity since this upper bound implies additional spreading. Conversely, the minimum value of D is obtained by assuming that the detected leading edge corresponds to 1 % of the carrying capacity since this lower bound implies less spreading.


                Discussion

                In this work, we developed and used a mechanistic model of B-cell targeted T-cell engaging bispecific drugs to support the translation of preclinical experience with the CD20/CD3 bispecific antibody, mosunetuzumab, to clinical predictions on the relative safety and efficacy of different dose regimens in NHL. The model describes the dynamics of B and T-lymphocytes and their interactions in multiple physiological compartments in the presence of mosunetuzumab or the CD19-targeting BiTE blinatumomab. The model was built and calibrated using in vitro potency data and pharmacodynamic data in cynomolgus monkeys treated with mosunetuzumab. Preclinical and translational validation were performed using additional mosunetuzumab preclinical data and blinatumomab clinical data in r/r ALL, providing confidence in the predictive capabilities of the model for different B-cell targets (CD19 vs. CD20), species (cyno vs. human), drug format and PK (BiTE vs. full-length antibody), and B-cell malignancies (ALL and DLBCL). The dynamics of activated CD8+ T-cells, total CD8+ T-cells, and target cells in circulation and tissues were well-characterized by the model for a wide range of mosunetuzumab doses from 0.001 to 1 mg/kg, which encompasses the projected range of doses currently being tested in a first-in-human Phase I study. The model was used to inform the clinical design by evaluating the proposed dosing strategies to maximize the therapeutic index of mosunetuzumab in patients with r/r NHL, based on prior clinical experience with blinatumomab as well as adoptive T-cell therapies (e.g., CAR T-cells) for the treatment of hematologic malignancies. The results suggested that a single-step or double-step fractionated dosing would mitigate peak cytokine levels with minimal impact on antitumor response. This result provided a strong rationale and quantitative guidance for the dosing schedule in the Phase I clinical trial of mosunetuzumab (ClinicalTrials.gov Identifier NCT02500407). Emerging results from the trial appear to confirm the predictions and have enabled higher dosing with reduced safety events.

                Although clinical data on step-up dosing with blinatumomab provided some insight on how to reduce first cytokine peak, we could not directly extrapolate from blinatumomab experience due to the fundamental differences between blinatumomab and mosunetuzumab in structural format and PK (short half-life of two linked single-chain antibody fragments vs. longer half-life of a full-length antibody), administration (continuous vs. intermittent dosing), and potency (different EC50 values for T-cell activation and B-cell killing). In addition, the model enabled us to explore whether there would be a detrimental effect of step-fractionated dosing on efficacy, whereas data were not available comparing efficacy for step-up vs. constant dose regimens of blinatumomab. This was an important consideration, especially for the treatment of aggressive tumors such as DLBCL.

                A range of different models for T-cell engaging agents have been presented in recent years. These are typically based on preclinical data and include consideration of factors such as target levels and binding interactions along with PK to explore how drug properties considerations can influence tissue distribution and efficacy 36 , to capture and predict preclinical responses for different dose regimens 37 , to propose first-in-human dose 38 , and to project efficacious dose 39 for various different bispecific agents. Each model is tailored to its application and has unique strengths. To our knowledge, ours is the first to explicitly include both blood and lymphoid tissues, address preclinical and clinical settings, include both safety (cytokines) and efficacy (target cell depletion) predictions, and explicitly address population variability. In our work, we have focused on including both efficacy and safety readouts to make more robust predictions to inform dosing strategies in highly variable clinical populations, whereas the prior studies have focused on cell killing and efficacy and not safety readouts that can limit dose levels.

                Predicting the impact of dosing regimen on clinical IL6 levels, as a surrogate biomarker for CRS, was a primary goal in this application of the model. While there is clear evidence for the central role of IL6 in CRS 40,41 , neither has a clear correlation between IL6 levels and severity of CRS been established, nor is there an accepted IL6 threshold above which patients will experience CRS. Nevertheless, higher IL6 levels are generally associated with a higher risk of severe CRS, and anti-IL6R antibody tocilizumab has been approved for the treatment of severe and life-threatening CRS arising following administration of CAR T-cell therapies 42 . Thus, IL6 serves as a reasonable surrogate biomarker for CRS risk, and our goal in comparing different dosing strategies was to identify a dosing regimen that would reduce IL6 levels across the patient population.

                Preclinical studies with mosunetuzumab and blinatumomab and clinical observations with blinatumomab have both shown peak release of IL6 and other cytokines upon first administration of drug, followed by drastically reduced cytokine release upon subsequent doses 34 . These observations might reflect desensitization of cytokine-producing cells. However, we found our model could capture the time-dependent cytokine attenuation observed for both blinatumomab and mosunetuzumab in nonhuman primates and human B-cell malignancies without invoking immune desensitization or similar regulatory mechanisms. We implemented only the established mechanism of target- and drug-dependent activation of T-cells in each compartment and represented cytokine production as a consequence of this activation. We did not explicitly represent different cellular sources of IL6, and instead assumed that activated T-cells through direct or indirect mechanisms drive cytokine production, either producing the IL6 themselves or rapidly stimulating other cells such as macrophages to produce IL6 in a manner proportional to activated T-cell numbers and activation signals 32 . With this implementation, we find that if initial administration of blinatumomab or mosunetuzumab is sufficient to deplete peripheral B-cells, systemic T-cell activation and cytokine production upon subsequent drug administration is greatly attenuated. In fact, recurrent cytokine peaks observed in blinatumomab-treated chimpanzees with partial B-cell recovery suggests that cytokine release need not be a first dose phenomenon and that systemic target abundance may play a role. However, this does not exclude a potential role for immune desensitization or other mechanisms such as exhaustion of cytokine-producing capability, as drivers of cytokine attenuation, and normal and tumor B-cells may differentially influence the PD consequences of T/B-cell engaging agents such as mosunetuzumab and blinatumomab. Clinical studies in newly diagnosed DLBCL patients who have not been exposed to prior treatments and are not B-cell depleted offer an important opportunity to distinguish between normal and tumor B-cell effects and determine how strongly circulating B-cell counts influence cytokine release. Finally, the relative roles of systemic target depletion and other mechanisms influencing cytokine secretion might differ significantly among indications, and especially between hematological malignancies and solid tumors.

                We have compared the model predicted range of IL6 peaks after the first dose with the clinical data. The virtual population used in this work explores the variability in the IL6 module and hence we have quantitative confidence in the predictions for IL6. However, limited variability was explored for the efficacy-related parameters due to the limited patient data that we had from blinatumomab study. Since the focus was comparison of dosing regimens, we have confidence on the relative efficacy outputs comparing across the dosing regimens. Additional clinical data would be required to quantitatively validate the predicted tumor regression rates.

                In the model, we have focused primarily on PD, with a fit-for-purpose approach to PK. Rather than using a PK model for calibration of the virtual cyno, we used the PK measured in cyno studies directly, allowing us to exactly capture each animal’s PK, including the effect of ADAs. ADAs were observed broadly in low- and mid- but not high-dose animal groups. This pattern of dose-dependent ADA effect is frequently attributed to high drug concentrations overcoming the impact of ADA on PK and to ADA assay interference at high drug concentrations. It is however possible that the more thorough depletion of B-cells is responsible for the reduced ADA in the high-dose group. Even so, this should not be a serious concern for the use of low doses in the step-up regimen, because the step-up phase is limited to the first 2 weeks (Days 0–14) during which ADA responses are not yet fully developed. Furthermore, preclinical ADAs are not predictive of clinical ADAs. Thus, our clinical predictions assume no significant ADA impact on PK for blinatumomab, reported immunogenicity rates are <1% 43 , and analysis of mosunetuzumab clinical data collected so far has shown no evidence of ADA 35 .

                We also used a simplified representation of pharmacokinetics (a linear two-compartment model) for prospective simulation in human. We acknowledge that the preclinical PK is nonlinear as demonstrated by the population PK model in 44 , however, in the absence of clinical PK data, the linear PK model was used for the purpose of IL6 predictions, as cytokine levels typically peak and drop within 24–48 h, for which the projected PK profiles from the linear and nonlinear PK models are comparable. Future work can include the PK data from clinical studies. Mechanistic representations of the effects of target levels, turnover, and binding affinity can also help capture nonlinear PK and downstream PD. Such mechanisms have been included in previous preclinical modeling efforts 36,37,38,39 . However, this requires either knowledge or calibration of expression and turnover of all targets (CD3, CD19, CD20, including subsequent alteration due to drug action) in all tissues, species, and conditions modeled, which becomes more challenging in clinical application, especially in peripheral tissues for which clinical measurements are not available. Thus, we have used the linear PK with approximate partition coefficient-based distribution to tissues, and we have modeled the direct drug effects as a function of T:B ratio and drug concentration, based on corresponding in vitro data for each drug to implicitly account for the ternary-synapse formation and downstream cellular function. The reasonable behavior and validation of the model over a broad range of drug concentration and T:B ratios supports this approach.

                Beyond the specific application to the candidate dose regimen assessment presented, the model serves to integrate preclinical and clinical biomarker data from related molecules and indications in a unified quantitative explanation of the biology of T-cell engaging agents in B-cell malignancies. This single mathematical description of the mechanisms of drug and target cell-dependent T-cell activation, proliferation, margination/migration, cytokine secretion, cytotoxicity, and target cell depletion quantitatively describes and predicts the diverse blood and tissue biomarker data for the two species (cyno and human), two drug molecules with similar mechanisms of action but very different PK properties (blinatumomab and mosunetuzumab), and three pathological/physiological contexts (healthy cynos, human ALL, and human NHL) considered. This broad fidelity increases our confidence in our understanding of the mechanisms triggered by T-cell recruiting bispecifics. The consistency of the systemic cytokines with systemic target load and drug level for example improves our understanding of mechanisms driving toxicity, whereas various other factors such as tumor proliferation rates and the abundance of T-cells relative to tumor cells influence efficacy.

                Overall, the systems modeling presented here offered a novel and valuable approach for evaluation of clinical dosing strategies for mosunetuzumab and can potentially be extended to other related molecules and/or other B-cell malignancies. Potential future applications of the systems model would be to inform the clinical efficacious dose projections and combination strategies of mosunetuzumab with other therapeutics to achieve a more favorable benefit-risk profile in the patient population of interest.


                Author information

                These authors contributed equally: Abhishekh Gupta, Prson Gautam.

                Affiliations

                Institute for Molecular Medicine Finland (FIMM), University of Helsinki, Helsinki, Finland

                Abhishekh Gupta, Prson Gautam, Krister Wennerberg & Tero Aittokallio

                Center for Quantitative Medicine, University of Connecticut School of Medicine, Farmington, CT, USA

                Biotech Research & Innovation Centre (BRIC) and Novo Nordisk Foundation Center for Stem Cell Biology (DanStem), University of Copenhagen, Copenhagen, Denmark

                Department of Mathematics and Statistics, University of Turku, Turku, Finland

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                Contributions

                A.G. and P.G. conceived this study and wrote the manuscript. A.G. devised the NDR metric and performed the computational analyses. P.G. designed and performed the drug screening experiments. K.W. and T.A. supervised the work and critically reviewed and revised the paper.

                Corresponding authors


                Population doubling time of keratinocyte,etc. - (Mar/01/2005 )

                my preliminary test will be calculating the differences of PDT between the treated and untreated keratinocyte culture.. because my hypothesis will be the treated keratinocyte culture with some kinds of wounf dressing materials will enhance the PDT and thus will serve a better wound healing material.

                i'm wondering to know what is the precise confluency level for me to start putting the test substances and counting the doubling populatiom time (PDT) of human keratinocyte culture?? i gotta do it in triplicates. my time interval will be 24 hours to 96 hours. and what passage would be suitable to put test substances and start doing the PDT??

                anyone here would suggest any better idea,websites or journals that there were people had done it before, at least regarded.

                ur suggestions will be great appreciated!!

                for my experiments, i started with a "cell solution" which i calculated the concentration (cells/ml for example). I put about 500 000 cells on 15 cm plates and calculate the number of cells after 24h 48h and 72h.

                Due to the fact i should replicate assays, my colleague help me with a cell sorter. it's a faster method but available in few labs.

                But my first experiments gave me good results.

                thanks fred!! but i'm wondering also if u put 500 000 cells into a culture plate, those cells which adhered might be less than 500 000 cells, let say 50 000 cells only get adhered, and others might be adhered 2 or 3 days later. how do u deal with it?? once u trypsinize ur cells for number calculation, do u reculture ur calculated cells into the plates??

                if u do not reculture, that means u make some plates for other time intervals,right?? let say u need to calculate the cells at time 72 hours..96hours..and so on. if u had 3 interval time to do, that means u have 3 culture plates and each of them has 500 000 cells, right??

                hi
                first i want to apologize for my long answer, that is probably confused. I'll try to be clear.

                you told : ---"if u do not reculture, that means u make some plates for other time intervals,right?? let say u need to calculate the cells at time 72 hours..96hours..and so on. if u had 3 interval time to do, that means u have 3 culture plates and each of them has 500 000 cells, right??"

                ->you're right.
                I needed to compare three conditions of culture, and wanted to check at 24 48 72h. That's why i seeded three plate for 24 h (1/condition of culture) 3 for the 48h-measure and 1 for 72h-measure.
                As oi wanted more repruductibility, i made my experiments three times.

                You said that i put 500 000 cells on plate but less than 500 000 cells adhered. I'm ok with that. But it's not troubling me for two reasons.
                The first one is that all my polls of cells have been treated in the same conditions. Hence, i assume that if 5-10% of cells in condition 1.24h did not adered, the same percentage of conditions 2.24h 3.24H not adhered. so for a comparison of these conditions it's ok.
                The second reason is that i measure my doubling time on three days. if we admit that the first measure is a little wrong (restart of cell division, and stuff like this) we should consider that these problems are minimized in the assays at day 2 and day 3.

                Finally, when i analyzed my measurements, there was not a big difference of calculations on these three separate days. That's why i took the average doubling time.

                But if you think fir your experiments the 24h calculation is too aproximate, do not take this one and make your experiment either at 48h 60h 72h or 48 72 96H.

                I hope my answer was helpful for you.
                Do not hesitate to mail me for more.

                thanks a lot ,Fred!! i really appreciated ur prompt reply.

                i have done the same procedures as what u clarified to me indeed. so, here..again can i know which passage u started to use or initialize ur population doubling time (PDT) calculation?? in my thought was the second passage, because i might be using some chemical substances to treat on the cultures and compare the PDT between the treated and untreated cultures.

                so, if i started the treatment on cultures at the second passage, and only then i do the PDT. is that any consequences?? pls list ur opinions??

                futher more, if i do the treatment on the culture monolayer using the direct contact method, what do u think of the best or more suitable cultures confluency level before i initialize the treatment, and start the PDT??

                hi
                i use 293 cells for my calculation. That's why i can't really speak about "passage".
                But for my experiments, i take 2 cares (not sure if it's the right word. )

                First, i seed a first plate like a preculture (two or three, depending on the number of further assays you want to do). When cells are 70-80% confluency, i trypsinise cells, centrifuge (1000 rpm 5') and resuspend them in 20ml media. I calculate the cell concentration and then seed the plates for my experiments.

                Second, i do not use cells that have been just defrozed from nitrogen or cels that have stayed at 100% confluency. That's why i do this kind of "preculture".

                That's why i think that second passage is ok for your experiments.

                For the treatment :
                you said the treatment will increase the PDT.You must take in mind the aproximate doubling time of your cells. For my experiments, know that it is feasible for a 293cell to make a full population. For example, IMR90cells are not able to. I don't know for keratinoytes.

                But for a 15cm plate, i think that 500 000 cells are ok, assuming the fact that this plate ma have more than 10millions cells

                The best advice i can tell you for you, is to make a "dummy experiment".
                Try to seed with 250 000, 500 000, 1million and 2 millions of cells the plate you will use for further experiments, and analyze if after 72 or 96h (depending on the longer time for our experiment) if cells are less than 80% confluency.
                Then you will get the max seed quantity of cells for the "untreated" cells.

                Let say your treatment will divide by 2 the PDT.
                Then, you seed all plates by half of the quantity which you get by your "dummy" assay

                so, there i can see ur way to deal with ur PDT, all of them were done under one flask or culture dish. cos u never subcultured ur cells. and ur PDT of ur 293 cell line will be calculated in only one culture dish, but u get it in mean of 3 culture dishes..namely u get the average PDT.

                here again, let say i do it at the 2nd passage. both the flasks were seeded with the similar density. after 2 days, the attached cells were detached by trypsin,and then get counted. so, from here i got the number of previous living cells.

                then, i proceeded the culture. of course the trypsinized cells were thrown. i assumed the similar results happened in another treated cultures. so, here, do i need to wait until the 40% of confluency level and start the treatment or, i start the treatment right away?? in my thought was do the treatment immediately as the cells might have entered log phase. then, i followed the time intervals accordingly to get the PDT between the treated and untreated cells. my hypothesis would be the treated cells might have faster PDT than the untreated cells. what's ur opinion??

                hi
                i think that if you want to get best resutls :
                day 1 : seed plates at 30% confluency
                day 2 : count cell number on (minimum) 2 plates : you'll get the number of cells that did survive after the seeding of the plate and the number of cells that could be used for your experiment.
                Start the treatment at this day.

                Day 3 : count cells
                Day 4 : count cells
                Day 5 : count cells (if it is possible as you say your treatment will decrease the PDT (more divisions in the same interval of time)
                Do the averages and estimate your PDT in the different conditions.

                At the day 5, if you can't count cells of your "treated cells" condition, (take care of confluency at the day 5), that means that you should do more experiments (more plates or more tries, the best is more tries).

                hi! thanks for ur valuable comments & opinions!!

                i have just done the almost similar thing as u mentioned. i seeded the cells at the 2 X 10**5 cells/ml at a culture flask. and counted the living cells (recognized by trypan blue and adherent) after 2 to 3 days as the cells might be started to proliferate and survive. i cant do it after 1day of the seeding cos most of the cells might be taking time to get adhered.

                so, others were similar to urs. i counted the cells everyday until 5th day. but the treated cells might have to undergo the subculture. then i only stopped the PDT. then, everything was re-done as above. do u think any problems if i do it as mentioned above??

                thanks for ur opinions again!!take care!!

                hi
                your experiment should e ak as mentionned above.
                I would add a commentary of my colleague who said that cells need 3-5 hours to become adherent but 18-24 hours are needed if you want to transfect or add treatment. (if i'm wrong, anyone, please correct me)
                Good luck to you and tell me the results !