Team:Potsdam Bioware/Project/Modeling

Modeling

Introduction

Modeling is a good tool to understand complex interactions or processes. We created a model to illustrate our selection system and to get a better understanding of its behavior under different conditions. Thereby, we wanted to answer two main questions:

1. At which time point selection is successful?

1. How many antibody presenting cells and how many virus particles do we need at the beginning?

To answer these questions we did both deterministic and stochastic modeling using MATLAB. Furthermore, we did a huge parameter analysis for each model to check what kind of influence each parameter has on our selection system.

Simplest Model

Reaction Scheme

We build a simple model of our in vivo selection system, which contains no more reactions we required. The following scheme (Fig. 1) shows the reactions occurring in our model.

Our system starts with the antibody presenting cell exhibiting a cell division rate (kdiv) and a cell death rate (kdeath). Due to expression of AID the antibody gets mutated and with a particular rate (kmut) we get a mutated cell presenting a more affine antibody. Thereby the virus infection is changed and is able to trigger the selection of mutated cells. We can distinguish between two different cases.

Within the first case the infection rate of the virus to the mutated cell (kintM) is improved compared to the infection rate to WT cell (kintC). In this case through the virus infection a positive stimulus can be mediated, resulting in both a higher cell division rate (kdivAinf) and/or a lower cell death rate (kdeathAinf).

In case two the infection rate of the virus to the mutated cell is decreased compared to the infection rate to WT cell C (kintM < kintC). Hence, through the virus infection, a negative stimulus can be mediated, resulting in a lower cell division rate (kdivAinf) or a higher cell death rate (kdeathAinf) or even both. The stimulus due to virus infection gets lost during the proceeding process (kloss). This means that the virus infection is steadily required for those stimuli.

(Due to the experimental observation which show that without an antibody no infection can be seen, we assume that with a better antibody on the cell surface the infection rate by the virus is improved compared to the WT cell. This means we should apply a positive selection system. For this we use antibiotics as selection pressure and antibiotic resistance as positive stimulus through virus infection. Therefore for modeling we used low cell division rate compared to cell death rate and a high cell division rate after virus infection.)

Deterministic Model

For deterministic modeling we write the reactions as ordinary differential equations (ODE´s) (see Fig. 2) and solved them using MATLAB. Because we had only marginal information about parameter values we estimate them and did a huge parameter analysis (see below).

ODE´s

Figure 2: Ordinary differential equations for our selection system.

Results

For calculation in MATLAB we used the following parameter values as reference state. In figure 3 time dependent concentrations of each species are shown.

Figure 3: Solution of differential equations for cell or virus particle concentrations.

Reference state: initial state: V0=10000 C0=1000 CV0=0 M0=0 MV0=0 parameters: par.k_div=0.02 par.k_divAinf=0.1 par.k_death=0.03 par.k_deathAinf=par.k_death par.k_mut=1e-5 par.k_intC=1e-6 par.k_intM=1e-4 par.k_loss=0.2

We start with a 10fold higher concentration of virus (shown in dark blue) compared to WT cell (shown in green). Concentration of WT cells decreases because of cell death due to antibiotic pressure. Virus concentration decreases because of infection of cells. WT cells were only marginal infected resulting in low concentration of infected WT cells (shown in red). After about 100 h mutated cells arise and were directly infected, because of high infection rate and high virus concentration. This leads to increased concentration of infected mutated cells and mutated cells respectively, due to stimulus of virus infection (antibiotic resistance). From that time point concentration of virus decreases rapidly and gets down to zero at about 200 h. Because of the lack of virus, cells were not infected and will get no positive stimulus. Due to continuous antibiotic pressure cell concentration of both mutated and WT cells decreases.

To act against these decreasing we performed two time steps shown in figure 4. In the first time step we applied high selection pressure due to antibiotics. After selection was successful the antibiotic pressure is removed (e.g. by medium exchange) and so cell concentration increases due to higher cell division rate than cell death rate.

Figure 4: Solution of differential equations for cell or virus particle concentrations.

Parameter Analysis

Because we had only marginal information about parameter values we did a huge parameter analysis. We wanted to know how each parameter influence the selection system. So we looked at the influence on success of selection and the influence on time scale. For this analysis we observed the ratio of mutated cells to WT cells (ratio of M to C). For a successful selection the ratio of mutated cells to WT cells should be higher than one and as much as possible.

one dimensional

For one dimensional analysis we changed each parameter holding all other parameters constant at reference state and observed the ratio of mutated cells to WT cells. Following results we obtain:

two dimensional

For two dimensional analysis we changed two parameter holding all other parameters constant at reference state and observed the ratio of mutated cells to WT cells. Following results we obtain:

Time dependencies

To figure out which parameter is time-dependent, we observed the ratio of mutated cells to WT cells over the time changing only one parameter. If the parameter is time-dependent changes in the ratio of mutated cells to WT cells should be observed earlier or later. Following results we obtain:

Stochastic Model

Stochastic modeling is necessary if one or more species have only low amount. In this case reactions are random events leads to fluctuating concentrations of species, which is characterized by different results in several experiments or runs, respectively. In our selection system we get only low amount of mutated cells due to low mutation rate. Because of this and because of some strange behavior of the determinist model we did stochastic modeling. For this we used the “simbiology”-tool of MATLAB. We build our model in the “simbiology”-tool and calculated stochastic kinetics using the stochastic simulation algorithm (SSA). This algorithm efficiently generates individual simulations that are consistent with the chemical master equation (CME). The advantages of this algorithm is the exactness. [1]

We used the same parameters like in deterministic modeling.

Results

In figure x the concentration of each species is plotted against the time. We see nearly the same results compared to deterministic modeling. The concentrations of virus and WT cells decrease and from a time point infected mutated cells and mutated cells increase while virus is available. After virus concentration get zero the concentrations of infected mutated cells and mutated cells decrease.

The difference between deterministic and stochastic model get obvious when we run the stochastic calculation several times (see figure x). Figure x show that the time point when mutated cells were selected and increase in concentration, differ about huge scales. Furthermore in some runs mutated cells did not increase, means sometimes selection was not successful. This is a typical result and can be explained by the stochastic behavior. Because of low concentration of mutated cells sometimes their become extinct by random events and so selection cannot act.

Parameter Analysis

For stochastic modeling we did parameter analysis both one dimensional and two dimensional, too. We change one or two parameters while holding the other parameters constant and run the calculation 3x 100 times counting the successful selections. As a successful selection we defined a ratio of mutated cells to WT cells higher than 1.

one dimensional

two dimensional

Summary