Markov Jump Model

class pints.toy.stochastic.MarkovJumpModel(x0, V, propensities)

A general purpose Markov Jump model used for any systems of reactions that proceed through jumps.

A population of N different species is simulated, reacting through M different reaction equations.

Simulations are performed using Gillespie’s algorithm [1], [2]:

1. Sample values \(r_0\), \(r_1\), from a uniform distribution

\[r_0, r_1 \sim U(0,1)\]

2. Calculate the time \(\tau\) until the next single reaction as

\[\tau = \frac{-\ln(r)}{a_0}\]

where \(a_0\) is the sum of the propensities at the current time.

3. Decide which reaction, i, takes place using \(r_1 * a_0\) and iterating through propensities. Since \(r_1\) is a a value between 0 and 1 and :math`a_0` is the sum of all propensities, we can find \(k\) for which \(s_k / a_0 <= r_2 < s_(k+1) / a_0\) where \(s_j\) is the sum of the first \(j\) propensities at time \(t\). We then choose \(i\) as the reaction corresponding to propensity k.

4. Update the state \(x\) at time \(t + \tau\) as:

\[x(t + \tau) = x(t) + V[i]\]

4. Return to step (1) until no reaction can take place or the process has gone past the maximum time.

Extends pints.ForwardModel, pints.toy.ToyModel.

Parameters:

• x_0 – An N-vector specifying the initial population of each of the N species.

• V – An NxM matrix consisting of stochiometric vectors \(v_i\) specifying the changes to the state, x, from reaction i taking place.

• propensities – A function from the current state, x, and reaction rates, k, to a vector of the rates of each reaction taking place.

References

[1]

A Practical Guide to Stochastic Simulations of Reaction Diffusion Processes. Erban, Chapman, Maini (2007). arXiv:0704.1908v2 [q-bio.SC] https://arxiv.org/abs/0704.1908

[2]

A general method for numerically simulating the stochastic time evolution of coupled chemical reactions. Gillespie (1976). Journal of Computational Physics https://doi.org/10.1016/0021-9991(76)90041-3

interpolate_mol_counts(time, mol_count, output_times)

Takes raw times and inputs and mol counts and outputs interpolated values at output_times

n_outputs()

Returns the number of outputs this model has. The default is 1.

n_parameters()

See pints.ForwardModel.n_parameters().

simulate(parameters, times)

See pints.ForwardModel.simulate().

simulate_raw(rates, max_time)

Returns raw times, mol counts when reactions occur.

suggested_parameters()

Returns an NumPy array of the parameter values that are representative of the model.

For example, these parameters might reproduce a particular result that the model is famous for.

suggested_times()

Returns an NumPy array of time points that is representative of the model