Collaborations
Our collaborations
Modeling for Toulouse 2016
Introduction
We conceived a model in order to handle questions concerning the following situation: A bacterial growth is carried out in a bioreactor, continually supplied in substrate. Bacteria can possess 2 type of plasmids:
- plasmids 1 carry the toxin A gene and the anti-toxin B gene
- plasmids 2 carry the toxin B gene and the anti-toxin A gene
So during the growth each bacterium can have no plasmids, either one type of plasmids, or both types.
The aim of the model is to assess the evolution of plasmids throughout the culture, to determine which parameters can matter in the loss of those plasmids, and to precise what are the probabilities for a bacteria to loose its plasmids during cell division. As a plasmids can be a disadvantage for growth (energy spent into replicating processes) or a advantage (protection against a toxin) this question is hard to answer. But in this situation, where one type of plasmid can influence on the presence of the other type of plasmid in (and reciprocally) in the same bacteria, the question become too tough to answer and only a mathematical model can resolve such a interrogation!
Equations
EQUATIONS DE 1 A 17
Progamming code
/*
* Program to run a model of bacteria in a fermenter
* with a 2 plasmid contention system.
*/
#include
#include
#include
#include
#include
#include
#include
#include
#include
#define MAX_POPULATION 2E6
// Model parameters
//
// Fermentation variables
double Sf = 10.; // Substrate concentration (g/l) in feed solution
double S = 0.19; // Current substrate concentration in fementer (g/l)
double V = 0.01; // Simulation volume with about 10^6 bacteria in ml
double Vtot = 100.; // Volume of fermenter
double D = 100.; // Dilution rate ml/hr
double alpha = 3.4e-11;// growth yield g of substrate needed for 10^6 cells
double mumax = 3.0; // maximum growth rate on substrate doublings per hour
double KS = 0.1; // Monod constant for substrate (g/l)
//
// Plasmid 1 parameters
double KZ1 =100.0; // Growth inhibition constant
int M1 = 1; // Hill constant for growth inhibition
double k1 = 20.0; // Plasmid replication rate in hr^-1
double K1 = 0.1; // Plasmid replication inhibition constant
double Z1max = 10.0; // Maximum plsmid copy number
//
// Plasmid 2 parameters
double KZ2 =100.0; // Growth inhibition constant
int M2 = 1; // Hill constant for growth inhibition
double k2 = 20.0; // Plasmid replication rate in hr^-1
double K2 = 0.1; // Plasmid replication inhibition constant
double Z2max = 10.0; // Maximum plsmid copy number
//
double sigma = 5.0; // Division rate for large cells in hr^-1
//
// Contention system parameters
double ka1 = 2.0; // Toxicity parameter for toxin on plasmid 1
double ka2 = 1.0; // Toxicity parameter for toxin on plasmid 2
double kb = 2.0; // Ratio of anti-toxin to toxin production rates
//
// Integrator parameters
size_t total = 1e6; // Total number of bacteria at start
double tmax = 100.0; // Number of hours to simulate
double dt = 0.05; // Timestep in hours
double t = 0.0; // Current time
double michaelis(double Vmax, double Km, double S)
{
return Vmax * S /(Km+S);
}
gsl_rng *r;
void setup_seed()
{
const gsl_rng_type * T;
gsl_rng_env_setup();
T = gsl_rng_default;
struct timeval tv;
gettimeofday(&tv,0);
gsl_rng_default_seed = tv.tv_sec + tv.tv_usec;
r = gsl_rng_alloc(T);
}
struct bstate { double Z[3]; } *population = NULL;
size_t pop_size = 0;
void pop_alloc(void)
{
population = calloc(MAX_POPULATION, sizeof(struct bstate));
if (population == NULL) {
fprintf(stderr, "%s: alloc error\n", __func__);
exit(1);
}
}
void pop_append(struct bstate *p)
{
if (pop_size == MAX_POPULATION) {
fprintf(stderr, "%s: max population reached\n", __func__);
exit(1);
}
population[pop_size++] = *p;
}
/* pop_delete(i)
* overwrite population[i] with last element
* decrement pop_size
*/
void pop_delete(size_t i)
{
if (i >= pop_size) {
fprintf(stderr, "%s: out of range\n", __func__);
exit(1);
}
population[i] = population[--pop_size];
}
void printZ(double Z[])
{
for (int i = 0; i < 3; i++)
printf("%lf ", Z[i]);
printf("\n");
}
int main(void)
{
setup_seed();
/*
* Create a random population of bacteria
* in parameter space Z0(1-2),Z1(0-Z1max),Z2(0-Z2max)
*/
puts("Creating initial population");
pop_alloc();
double AvZ[3] = {0};
double growth = 0.0;
uintmax_t free = 0;
for (size_t i = 0; i < total; i++) {
struct bstate bacteria = {{
1+gsl_rng_uniform(r), // Random size [1-2]
Z1max, // Maximum number of plasmids
Z2max // for both types
}};
pop_append(&bacteria);
AvZ[0] += bacteria.Z[0];
AvZ[1] += bacteria.Z[1];
AvZ[2] += bacteria.Z[2];
}
puts("Starting integrator");
while (t < tmax) {
/*
* Display or output for visualization
* population (density on Z1/Z2 (all Z0 or Z0>2),
*/
printf("%lf %lf %lf %zu %ju %lf %lf %lf\n",
t, V, S, pop_size, free,
AvZ[0]/pop_size, AvZ[1]/pop_size, AvZ[2]/pop_size);
memset(AvZ, 0, sizeof(double[3])); // average values for parameters
growth = 0;
free = 0; // number of bacteria with no plasmids
size_t i = pop_size;
while (i-- != 0) {
double* Z = population[i].Z;
double cntrl = gsl_rng_uniform(r) - (dt * D / Vtot);
if (cntrl < 0) {
pop_delete(i); // Bacterium washout
continue;
}
// Bacteria grow
AvZ[0] += Z[0];
AvZ[1] += Z[1];
AvZ[2] += Z[2];
if (Z[1] == 0.0 && Z[2] == 0.0)
free++;
double dotZ0 = michaelis( mumax, KS, S ); // Growth on substrate
dotZ0 *= michaelis( 1.0, gsl_sf_pow_int(Z[1], M1), KZ1 ); // Inhibition by plasmid 1
dotZ0 *= michaelis( 1.0, gsl_sf_pow_int(Z[2], M2), KZ2 ); // Inhibition by plasmid 2
double tox1 = gsl_sf_exp(-ka1*(Z[1]-kb*Z[2] ));
double tox2 = gsl_sf_exp(-ka2*(Z[2]-kb*Z[1] ));
dotZ0 *= fmin(1.0, tox1); // Inhibition by toxin 1
dotZ0 *= fmin(1.0, tox2); // Inhibition by toxin 2
double dotZ1 = (Z[1] < 1.0)? 0.0 : michaelis( k1, K1, Z[0] ) * (Z1max - Z[1]);
double dotZ2 = (Z[2] < 1.0)? 0.0 : michaelis( k2, K2, Z[0] ) * (Z2max - Z[2]);
Z[0] += dt * dotZ0; // Increment the internal state
Z[1] += dt * dotZ1;
Z[2] += dt * dotZ2;
growth += dt * dotZ0;
if (cntrl < (dt * sigma) && Z[0] > 2.0) {
struct bstate new_bacterium = {{
gsl_ran_gaussian_ziggurat(r, 0.05) + Z[0]/2.0,
gsl_ran_binomial(r, 0.5, Z[1]),
gsl_ran_binomial(r, 0.5, Z[2])
}};
for (int j = 0; j < 2; j++)
Z[j] -= new_bacterium.Z[j];
pop_append(&new_bacterium);
}
}
S += (D*dt*(Sf-S)/1000 - (growth*alpha*Vtot/V))/Vtot;
if (pop_size > 16e5) { // Too many bacteria
// Throw out half of then and reduce the volume
pop_size /= 2;
V /= 2.0;
AvZ[0] /= 2.0;
AvZ[1] /= 2.0;
AvZ[2] /= 2.0;
}
if ((pop_size < 7e5) && (V < Vtot/2.0)) {
// Too few bacteria increase the volume
// and double the bacteria
memcpy(population + pop_size, population, pop_size);
pop_size *= 2;
V *= 2.0;
AvZ[0] *= 2.0;
AvZ[1] *= 2.0;
AvZ[2] *= 2.0;
}
t += dt; // Increment time
}
// Output final population
return 0;
}
Results
METTRE LES COURBE
Data recovery Bordeaux 2016
In order to help the iGEM team Bordeaux, we had to read the paper « Dynamics of plasmid transfer on surfaces » and collect and organize some data about their experiments on plasmids transfer. Thanks to this, the iGEM team Bordeaux can compare those results to their own computaionnal results.
Colaboration made by Pretoria 2016
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