3. Material and Methods
3.4. Simulation
+
C Con
S Sstop
rC
sC dCon
rS
sS
colicin diffusion
ø
Figure 3.9
Two-Strain Interaction Parameters adapted from [10]. rX replication rate of strain X (C or S), sCswitching rate of C into the stateCon, anddCondegradation rate ofConwith subsequent cell lysis and toxin release. sS switching of S into the Sstop state when encountering a colicin in the environment, making switching intoSstop dependent on the concentration of colicin ([col]) in the environment. Figure adapted from [51].
∂tC =rC ·C−sC·C
∂tCon =sC ·C−dCon·Con
∂tS =rS·S−sS·[col]·S
(3.2)
Where each strain S or C, can reproduce with a raterX. The C strain has the ability to switch into the ON state with a constant switching rate sC depending on the stress level. When this happens, it can release toxin via cell lysis with the rate dCon. This rate is dependent on the expression of the gene cel in the cell and is the inverse of the lysis time Tlysis measurable in single cell time-lapse microscopy. When colicin is released into the environment it diffuses in the surrounding, which is implemented into the simulation with an exponential gradient from the point where C cell lysis occurred [46]. A S strain encountering colicin in the environment can switch into the Sstop state with a switching rate of sC. Switching thus depends on the concentration of colicin in the environment ([col]).
The two strain interaction parameters were implemented into a simulation by von Bronk et al. [10, 11, 67]. In the following the basic principles of this model will be explained.
For expansion in 2D a 250x250 lattice was chosen. A space in the lattice can be empty or contain any of the states described in Figure 3.9 and shown in Figure 3.10 A. A spot in the lattice has 8 neighbors that are considered in each simulation step and influence the occupation of an empty site (see Figure 3.10 B). Interaction is possible for the 8 neighbors (Moore neighborhood) [68] and is weighed depending on distances as shown in Equation (3.3), which is comparable to the neighbors shown in Figure
3.4. Simulation
A
lattice configuration
empty lattice C Con S Sstop
possible lattice occupations
possible transitions
B C
Figure 3.10
Lattice configuration and transitions for simulation. A) Definition of possible lattice occupations corresponding to states shown in Figure 3.9 and Chapter 2. B) Lattice configuration for considered lattice space (black) and its neighbors (dark grey: Neumann neighbors, lightgrey: Diagonal neighbors, light and dark grey: Moore neighbors) in a Moore neighborhood used for simulations (Equation (3.3)). C) Possible transitions for all lattice occupations described in A) depending on the environment. Probabilities are given in Equations (3.4, 3.5, 3.6). Simulation adapted from [10].
3.10 B. Here direct neighbors (Neumann neighborhood) [68] are marked in dark gray while diagonal neighbors are shown in light gray.
NM oore =NN eumann +NDiagonal
= 4·1 +4· 1
√2
(3.3)
Each lattice site has a defined probability for switching into another state or staying in its current one, depending on occupation of their Moore neighborhood. Possible transitions are shown in Figure 3.10 and depend on the lattice site under consideration.
Probabilities pfor all transitions are listed in Equations 3.4-3.6.
pempty,S = NS ·rS ·∆t pempty,C = NC ·rC ·∆t
pempty,stay = 1− (NS·rS+NC·rC)·∆t
(3.4)
With NX: number of neighbors of type X in the surrounding (with distance factor shown in Equation (3.3)) and rX: replication rate of X strain. Switching of an empty state thus depends on the number of S or C cells in the environment. Each state can
either switch into another state according to the transitions shown in Figure 3.9 or stay in its current state.
pS,switch = [col]·sS·∆t
pS,stay = 1− [col]·sS·∆t (3.5)
With [col] being the concentration of colicin in the surrounding, sS switching rate of the S strain into the Sstop state and ∆t the time step size in the simulation. S cells switch into Sstop depending on the colicin concentration in their environment. Which, as described above, is simulated with an exponential decay from its point of origin.
pC,switch = sC ·∆t pC,stay = 1− sC·∆t pCon,switch = dCon·∆t
pC,stay = 1− dCon·∆t
(3.6)
Where sC describes the switching rate of the C strain into the ON state (Con) and dConthe rate of degradation of a Con cell and thus emptying a lattice space. The rate dConwith which the Con cells release toxin is limited by the minimal time to cell lysis that is set for the system to react to encountered stresses.
Initial positioning of cells on the lattice is random in a circular pattern (resembling experimental conditions) but kept at a ratio of approximately 1 C cell for 100 S cells.
Furthermore, in order to decide which switching takes place, all probabilities for a space are summed up and a random number between 0 and 1 is chosen. Depending on this number and its position in the probability sum, the new state of this lattice space is determined. This is repeated for all lattice sites occupied or neighboring an occupied space for each time-point.
When a filled lattice space ’touches’ a border of the defined lattice, rescaling of the current lattice occupation as well as all rates (growth, switching etc.) takes place.
Detailed parameters that are fixed for all simulations are given in Table 3.3.
The switching rate sC was varied between 1 % and 99 % representing induction levels from low to very high external stress. Analytical solution of the ODE system (Equa-tion (3.2)) was performed by von Bronk and steady state solu(Equa-tions were obtained for the fractions of switching C cells [10, 67]:
3.4. Simulation
Parameter Definition Size
N lattice sites in x and y dimension 250
∆x initial lattice spacing 2µm
Z factor used for rescaling of lattice 5
∆t time step size 1.5 min
tend complete simulation time 2790 min C:S mean initial ratio of C to S strain 1:100
Table 3.3
Fixed parameters for competition model were adapted from von Bronk [10, 67]
F rac(t) =h Con(t) Con(t) +Cof f(t)
i
t→∞ = sC
rC +dCon (3.7)
Other parameters, such as lysis time, growth rate, toxin effectivity and toxin amount have to be adjusted by extraction from measurements as follows:
GRs: Growth rates for all strains S and CX were extrapolated from linear fits of control measurements as described above. The conversion factor from area growth rate to simulation growth rate was chosen as in Bronk et al. [10, 11].
dCon: Degradation rates for all CX strains were calculated from their median time to cell lysis (Tlysis) determined in single-cell time-lapse microscopy.
ntox: Toxin amounts were determined in experiments as described in Section 3.3 and inserted as factors depending on the lowest toxin amount, which was set to factor 1. More detailed information is given in Chapters 4 and 5.
σS: Toxin sensitivity of the competitor (S) is a factor that was determined by adjusting simulation to experimental results over broad ranges, as it cannot be measured easily. In previous studies by von Bronket al. [10, 11], sS was chosen as 1500. Here, this also fits for the CREP1 strain over a broad range of stresses.
sS: Toxin effectivity fo toxin to the S strain is calculated as sS = σS ·ntox, making it dependent on both the S strain sensitivity to toxin and the amount of toxin being released by CX.
[col]: The colicin concentration is determined by the position of the observed lattice space compared to that of C strains releasing toxin. For each cell lysis of a C cell, an exponential decaying profile of toxin in the lattice is calculated [10, 46]. This leads to a decrease in [col] the farther the distance between the observed lattice cite and a lysed C lattice site.
In summary, this model facilitates theoretical analysis of bacterial interaction over a broad time-scale from single cell interaction to macroscopic competition outcome.
For detailed description of the model and all its components, please see von Bronk 2018 [67]. The key components driving ColicinE2 driven interaction are implemented including stochasticity in gene expression as well as positioning, toxin production and sensitivity, growth rates and switching of the C strain into the toxin producing strain.
This theoretical approach enables investigation of the different components separately, even if they are biologically linked. Selective variation of parameters is supported by the framework.