Team:Baltimore BioCrew/Model

2020 Baltimore Biocrew

Modeling

Overview

The goal of our project is to use engineering to improve the growth of cyanobacteria in low iron environments. As part of the math modeling team, we are modeling the growth of un-engineered and engineered cyanobacteria in different concentrations of iron. We based our model on the Monod Model for substrate-limited cell growth. We tested our optimization methods using data from a published paper. Next, we plan to optimize the parameters in our model using our own results from growing cyanobacteria in different iron concentrations at home. We hope to use this model to compare the kinetic parameters of our engineered cyanobacteria and identify how the addition of different iron metabolism genes changes the growth of the cyanobacteria.



About the Monod Model

We are modeling the growth of cells based on how much iron they have using the Monod Model. The Monod Model is a system of differential equations which uses information about the properties of the bacteria and the substrate to generate a growth curve. The graph is composed of an exponential growth curve which, at the point where the substrate becomes limited, levels out with a very small growth. We derived the basis for the Monod Model from the Maier & Pepper chapter on Bacterial Growth (2015).


Parameters

The Monod Model predicts the number of cells growing over time, N(t), as they consume a substrate, S(t). The parameters for the Monod Model are the maximum specific growth rate (Mu_max), the half-saturation constant (K_s), and the cell-yield (Z). They have the following definitions:

  1. Mu_max is the maximum growth rate of the bacteria, which determines the steepness of the exponential growth phase
  2. K_s is a constant that depends upon the bacteria and the substrate and represents the concentration of substrate which make the bacteria grow at a rate of ½*Mu_max
  3. Z is the relationship between how much substrate is lost to produce a number of cells.

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To develop a method for optimizing parameters from the Monod Model, we looked for published experimental data to begin.



Using Published Data to Optimize Parameters

In 2017, Patel et al. published a study characterizing the growth of seven unique species of cyanobacteria. One experiment here, grew Synechocystis PCC 6803 in media for 19 days and measured the optical density each day. Since we have been designing experiments for Synechocystis in our lab, we used Patel’s data as a starting point for optimization.

Optimization is a method we use to identify the best-fit values for parameters (Mu_max, K_s, and Z) for this system. We performed optimization using the lsqnonlin function in Matlab -- this is a non-linear least squares data-fitting solver. We used this function to find parameters that would minimize the difference between our experimental data and simulation data. In order to do that, we need to provide starting estimates and bounds (lower and upper) for each parameter. We also provided initial values for the concentration of cells and substrate at the beginning of the experiment, which is also shown in the table below.

  • Table 1: Optimization settings for cells and substrate, including: initial conditions and rationale
    Variables Descriptor Value for initial condition(units) Rational
    N Cell Concentration 4*10^5 (cells/mL) From Patel 2017, page 4
    S Substrate Concentration 6*10^-6 (g/mL) From Patel 2017, page 3. Because ferric ammonium citrate has one of the lowest concentrations, we are assuming iron is the limiting nutrient.
  • Table 2: Optimization settings for model parameters, including: starting estimate, bounds (upper and lower), and rationale
    Parameters Descriptor Value for starting estimate(units) Bounds (lower, upper) Rational
    mu_max Maximum growth rate 1.0 (1/day) (0.1, 10) From Patel 2017 Table 2
    K_s “half saturation constant”. The substrate concentration at which growth occurs at half of the rate of mu_max 1.0*10^-6 (g/mL) (1*10^-9, 1*10^-3) Estimated from Maier Chapter table of Ks examples
    Z Cell yield, converts substrate lost per cells gained (or vice versa). (e.g. micrograms of substrate per micrograms that are produced) 2.0 *10^-13 (g of substrate / # of cells) (2.5*10^-18, 2.5*10^-8) Estimated from Maier Chapter table for Example Calculation 3.3

Results from Optimization with Patel et al., 2017 Data

We performed optimization using Matlab to identify values for our model parameters, as described above, and got the following values (Table 3). For Mu_max, the maximum growth rate of the cells, the value is 1.14 per day, similar to that of the Patel et al., 2017 paper. Similar to that of the Patel et al., 2017 paper. For the half-saturation constant (K_s), the level of substrate such that the growth rate is half of mu_max, is 6.76*10^-6 grams per milliliter. Finally, the amount of substrate required to make 1 cell of cyanobacteria, Z, is 2.52*10^-13 grams, meaning that the substrate decreases by this amount per every new bacteria cell that is created.

These parameters produce simulations that closely resemble the experimental results from Patel et al., 2017 (Figure 2). The biggest difference we see is that the simulation levels off (zero growth) at a lower cell density than experimental data. But still, our estimate for mu_max matches that of Patel et al., 2017.


  • Table 3: Optimization results using Patel et al., 2017 data
    Parameters Optimized value (units) Value in Patel et al., 2017
    mu_max 1.14 (1/days) 1.033 +/- 0.027 (1/day)
    K_s 6.76*10^-6 (g/mL) (not estimated)
    Z 2.52*10^-13 (g/cells) (not estimated)

Figure 1. Simulated concentration of cells and substrate over time.

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Figure 2. Simulated concentration of cells over time vs. Patel et al., 2017 experimental data.

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Limitations of this Approach

Through analyzing our optimization results, we found that our optimization result greatly depends on our starting estimate. The starting estimates are the initial values we gave the optimization function (lsqnonlin) to start looking for parameter values, and this affects what the end values are. To explore the effect of small changes in our starting estimates, we changed the starting estimate for one parameter at a time. For example, we kept the starting mu_max and Ks value as they are in Table 3 while changing the starting estimate for Z in small increments. Although some variability in final parameters due to the starting estimate is to be expected, we found ours had significant differences -- leading us to believe that we were attempting to estimate too many parameters from too little experimental data. It is also possible that the Monod model generates a highly irregular optimization surface in this parameter space.

To improve our parameter estimates, we plan to grow cyanobacteria at different concentrations of iron over time. We will measure growth at varying concentrations to allow the optimization to be more specific to the cyanobacteria’s reliance on iron. With this, we will be able to better estimate parameters for our model, and thus more accurately predict the growth of native and engineered cyanobacteria.


Comparing Growth Parameters for Our Cyanobacteria Culture Experiments

We aim to gather experimental data from the wetlab groups to conduct an optimization. This will allow us to fit a curve to the data points and let us make conclusions about the effect of the inserted genes on cell growth. .

Our original data comes from our experiments with growing cyanobacteria (sp. CB0101) in semi-controlled conditions in our own homes. We grew the bacteria in solutions with varying concentrations of iron, from 0 up to 10x the recommended amount, to generate a control group that showed how an unmodified strain was able to cope in each environment.

The bacteria density was measured daily with a Secchi stick--a small ruler with a black-and-white disk at the end. The disk was lowered slowly into the solution, and the depth at which it disappeared was recorded in millimeters.

As expected, our cell growth measurements (1/mm from secchi) indicated more growth at later times and with higher iron concentrations. However, our experimental data didn’t follow the logistic curve produced by the Monod Model. The growth of our cyanobacteria was much closer to a linear trend. Further discussion of our experimental outcomes can be found on our Contribution Results Page.

To compare the parameters obtained from the Patel data with our own experiment, we will need to input our spreadsheet into MatLab and continue adjusting our input variables to generate the optimals with the lowest cost and the tightest-fitting curve. We predict that our data will not fit the Patel optimals well; our growth curve was much less exponential in form and more linear than the Patel experiment, possibly due to a much lower mu_max, so significant adjustment may be necessary.