Stochastic Simulation of Biological Systems

Chemical Reactions

• Reactants Products

• m1R1 + m2R2 + ··· + mrRr –! n1P1 + n2P2 + ··· + npPp

Example

• 3 reactions, 2 species:

Y1 ! 2Y1 (1)

Y1+Y2 ! 2Y2 (2)

Y2 ! (3)

• Lotka-Volterra

Representations (I)

• Y1 Y2

1 1 0 = A

2 -1 1

3 0 -1

• A = Net Effect Matrix

Example

• x(t0) = (10 5)

• Reaction (2) fires at time t1 > t0

• Then x(t1) = x(t0) + r.A

= x(t0) + (0 1 0).A

= (10 5) + (-1 1)

= (9 6)

Representations (II)

• SBML

• Electronic XML-based system

• Represent models in a standard format

• Sharing models

• Transferring models between different software systems

<reaction name="Protein dimerisation" reversible="false">

<listOfReactants>

<specieReference specie="P" stoichiometry="2"/>

</listOfReactants>

<listOfProducts>

<specieReference specie="P2" stoichiometry="1"/>

</listOfProducts>

<kineticLaw formula="k4*0.5*P*(P-1)"/>

<listOfParameters>

<parameter name="k4" value="1"/>

</listOfParameters>

</reaction>

Simulating the LV model

• ODEs

• Use 2 assumptions

(i) Continuous quantities (not integers)

(ii) System behaves deterministically

• Assumptions are often appropriate…..

• ….but not always.

Y1 ! 2Y1 (1)

Y1+Y2 ! 2Y2 (2)

Y2 ! (3)

• c1, c2, and c3 are the rate constants for reactions 1, 2, and 3 respectively.

• Start at time t0 and state x0

• Simulate time of first reaction event

• ‘Hazards’ of the three reactions are:

h1=c1y1 h2=c2y1y2 h3=c3y2

• Total hazard function is h0=h1+h2+h3

• h0 is constant during inter-event time periods

• T ~ Exponential(h0)

• Now select which of the 3 reactions has occurred.

• Probability of choosing reaction i is given by:

hi / h0

• Update system time:

t := t0 + T

• Update state vector:

x := x0 + ri.A

• Continue moving forwards in time

• Gillespie’s Stochastic Simulation Algorithm (the SSA)

• A single run generates a time vector (tvec) and a state matrix (xmat)

• Algorithm must be run many times

• Calculate statistical quantities

• E.g. mean and standard deviation of Y1 species over the first 50 seconds

• SSA is an exact algorithm

• Each reaction event simulated individually

• Algorithm must run thousands of times

• Very time consuming for large, complex systems

• How to speed things up?

Time Discretization

• Divide time interval into discrete chunks of width Δt.

• Go from t to t+Δt in one “leap”

• More than one reaction may occur in the interval [t, t+Δt)

• Simulate which reactions have occurred in this interval.

• Time interval [0, t]• Let X be the number of times reaction 1 fires in

[0, t]• Divide interval into N discrete chunks of width δt• N large, δt small

• Prob(Reaction 1 fires once in time δt) h1.δt

• Prob(Reaction 1 fires more than once in time δt) is negligible (relative to δt)

• Then X ~ Bin(N, h1.δt) = Bin(N, h1.(t/N))

• E(X) = h1.t

• As N , δt 0 , X~Poisson(h1.t)

“Tau-Leap” Algorithm

• Start at time t0, state x0

• Select discrete step size, • Move forwards to time t0+ • Simulate the number of firings of reaction i during

the interval as a Poisson(hi.) (i=1,…n)

• Update the system state according to the reactions which have occurred.

• Now advance to t0+ 2, t0+ 3, ……

• Tau-Leap is faster than the SSA

• But some accuracy is sacrificed

• Relies on a sensible choice of

Hybrid Algorithms

• Divide the system species and reactions into discrete and continuous regimes

• The discrete regime contains species with low concentrations and reactions which fire infrequently

• The continuous regime contains species with high concentrations and reactions which fire frequently

• Use different methods for simulating the two regimes.

• Simulate the continuous regime using deterministic or stochastic differential equations

• Simulate the discrete regime using the SSA

• The two regimes must be syncronized

• Some species will be involved in both regimes

Measuring Accuracy

• The accuracy of an approximate algorithm (leaping or hybrid) can be assessed by comparing results with SSA

• Run SSA and approximate algorithm 1,000 times for the same system, using the same rate constants and initial concentrations

• Do the results from the approximate algorithm closely resemble the results from the SSA?

The Test Suite for Stochastic Simulators

• Available at www.calibayes.ncl.ac.uk• Contains stochastic simulation results for

some simple biological systems• Results generated using analytic

expressions for species means and standard deviations.

• Can be used to measure the correctness of a stochastic simulator

• Testing stochastic simulators is not

straightforward.• Even if a simulator is working perfectly, it

will not produce exactly the results contained in the test suite.

• Need statistical method to compare the simulator results with the test suite results.

• E.g. if the concentration of species X after 10 seconds varies by 1% from the test suite value, is this evidence of a problem with the simulator?