Computational Biochemistry and Biophysics Group, Research Unit on Biomedical Informatics, IMIM/Universitat Pompeu Fabra, c/Dr. Aiguader 88, 08003, Barcelona, Catalonia, Spain

Departament de Física i Enginyeria Nuclear, Universitat Politècnica de Catalunya, Edifici GAIA, Rambla de Sant Nebridi s/n 08222, Terrassa, Barcelona, Spain

Institute for Molecular Bioscience, The University of Queensland, Brisbane, QLD, 4072, Australia

COMLAB and OCISB, University of Oxford, Oxford OX1 3QD, UK

Abstract

Background

With increasing computer power, simulating the dynamics of complex systems in chemistry and biology is becoming increasingly routine. The modelling of individual reactions in (bio)chemical systems involves a large number of random events that can be simulated by the stochastic simulation algorithm (SSA). The key quantity is the step size, or waiting time,

Results

In this paper we extend Poisson

Conclusions

The benefit of adapting the extended method to the use of RK frameworks is clear in terms of speed of calculation, as the number of evaluations of the Poisson distribution is still one set per time step, as in the original

Background

It is by now very well known that the biochemical kinetics involving small numbers of molecules can be very different to kinetics described by the law of mass action and differential equations

The basic idea of the SSA is that at each time point a waiting time to the next reaction and the most likely reaction to occur must be sampled from a joint probability density function leading to an appropriate update of the state vector. But if the rate constants and/or the numbers of molecules in the system are large then the waiting time (time step,

Although

Here, we use the Runge-Kutta formulation to construct methods with large stability regions so that efficiencies are gained by allowing larger stepsizes. We note that this is exactly what Abdulle and Cirilli

Thus, in this paper, we explore a series of fully explicit multistage Runge-Kutta methods with extended stability for a fixed

Review of Runge-Kutta methods for SDEs and ODEs

Stability region for RK methods applied to ODEs

Consider the system of initial value ODEs given by

The class of

where

where **b**
^{T }= (_{1},...,_{
s
}), **w **= **Ae **and **e **= (1,...,1)^{T}. Here **A **is the matrix with entries _{
ij
}and **w **is the column vector **w**
^{T }= (_{1},...,_{s})^{T}. A Runge-Kutta method is said to be explicit if the **A **is strictly lower triangular. The method parameters are usually chosen so that a Runge-Kutta method has appropriate efficiency, order and stability characteristics. The **Y**
_{
i
}are considered to be approximations to the solution at the intermediate points _{
n
}+ _{
i
}

In a stability setting an RK method is often applied to the linear, scalar test equation

In which case it is easily seen that (2) gives rise to

Where

Here **y' **= **Λy **in which case it becomes a matrix function of the **Λ**:

where **e **is the unit vector, **I**
_{
s
}is the identity matrix of order **A **⊗ **B **is _{
ij
}
**B**. Notice that, if **Λ **is a scalar value and taking **Λ**,

In the case of an explicit method, as **A **is a strictly lower triangular **A**
^{
s
}= **0**. Therefore, equation (4) can be expanded into a finite power series for **A**:

where _{
j
}= **b**
^{T}
**A**
^{
j-1}
**e**,

Since (3) is asymptotically stable for all Re [

Stability region for RK methods applied to SDEs

In the case of stochastic differential equations (SDEs), we consider the general

where **W**(_{1}(_{
d
}(^{T }is a vector of

and non-overlapping Wiener increments are independent of one another. A sample of a Wiener increment

Equation (8) can arise as the limit of a discrete process through the concept of a diffusion process in which case **f **(**y**) will represent the mean of this process and **g**(**y**) is the **gg**
^{T }is the covariance. Equation (8) can be interpreted in several ways (see

where Δ**W**
_{
n
}= (Δ_{1},....Δ_{
d
})^{T }and Δ_{
i
}:= _{
i
}(_{
n
}+ _{
i
}(_{
n
}),

As with the deterministic case, the quality of a numerical method can be partly characterised by its stability region associated with the scalar, linear test equation

The solutions of (10) in the Itô and Stratonovich cases are, respectively,

In the later case, the solution is mean square stable ^{2}] ≤ 0.

A very general class of stochastic Runge-Kutta methods

where

In the case of the Euler-Maruyama method

and in the (

Results

The

As stated in the Background section, the SSA describes the time evolution of a vector of integer numbers of molecules in the presence of intrinsic noise. More formally, suppose that there are _{1},...,_{N }undergoing _{
i
}(_{
i
}and **X**(_{1}(_{
N
}(^{T}. Now any set of chemical reactions is uniquely characterised by two sets of quantities. These are the update (stoichiometric) vectors _{1},...,_{
m
}for each of the _{1}(**X**(_{
m
}(**X**(

then **X**(^{T}, _{1 }= (-1, -1, 1)^{T }and _{1}(**X**(

Given **X**(_{1}(**
X
**(

and the algorithm repeats.

Since a typical stepsize (waiting time) is of the size 1/_{0}(**X**(

Gillespie _{
j
}reactions per step, _{
j
}, as coming from a Poisson distribution with mean _{
j
}(**X**
_{
n
}), that is

Using the so-called compensated process given by

which satisfies ^{2}] =

Where

As noted by Gillespie

This is precisely the Euler-Maruyama method applied to the SDE

Thus in the continuous limit the Poisson

These relationships naturally lead to the introduction of the class of Runge-Kutta

where **A **is strictly lower diagonal. We note that (17) requires the same number of samples of Poisson random variables per step as the Poisson

The Poisson

Indeed any Runge-Kutta method for solving an ODE can be incorporated into this framework. We also note that other methods proposed in the literature can be put into this framework. For example, the midpoint method of Gillespie **b**
^{T }= (0, 1), **w **= (0, 0.5)^{T }and where the row-wise entries of **A **are 0, 0, 0.5, 0.

The linear case

As in the case of stability settings in the ODE and SDE regimes, we analyse (17) when applied to linear kinetics, which in this case are described by sets of unimolecular reactions. A general set of

where **x **is the state vector of dimension **W**

so that now the drift or expected step-change can be represented as

If the Runge-Kutta method for ODEs underlying a Runge-Kutta

**Supplementary material for "Simulation methods with extended stability for stiff biochemical kinetics"**. Technical results, coefficients for the optimal stability polynomials and notes on the mitogen-activated protein kinase (MAPK) cascade simulation results.

Click here for file

where **y' **= **Λy **giving **X**
_{
n
}= **Λ**)**X**
_{
n-1}. Thus with fixed stepsize

Therefore, boundedness in the mean requires that the spectral radius, **W**) satisfies

In order to analyse the framework (17) from the perspective of both mean and variance behaviour we consider the reversible isomerisation reaction with fixed total number of molecules given by

as the linear scalar test equation. It is easy to see that this system is a analogous to (3) for ODEs and (10) to SDEs with constant nonzero term. The system is chosen to have constant nonzero term in order to compare its variance, which otherwise would fade to zero, to the variance given by the framework methods (17). In this case

For this set of reactions, the Chemical Master Equation (which describes the probability density function associated with the evolving Markov process **X**) can be solved analytically

Where

and _{1}(_{2}(**e **= (1, 1)^{T}

In the case of non-negative coefficients in the underlying RK method and for constant _{1 }+ _{2}), then in the limit as n → ∞ the mean vector converges to the theoretical mean, that is

Note that with the constraint |_{1 }+ _{2}) then the spectral radius of **W**) is less than or equal to 1, and as there is only one eigenvalue equal to one hence we have boundedness of the mean.

Furthermore, if Var [**X**
_{∞}] denotes the variance of the new method at steady state (_{1 }and _{2 }have the same variance) and if ^{2}(_{1 }+ _{2}), then (see details in Additional File

where

We call this the relative variance at the stationary state associated to

Let us consider some particular cases of this result:

**Poisson τ-leap **For this method

**Two stage methods with **
_{21 }≠ 0 For the family of explicit two-stage methods with _{21 }≠ 0

the stability function is ^{2}, where _{2}
_{21 }and the variance behaviour is determined by

In this case we have one free parameter of the method,

Constraining

and with a

For instance, for 0.5 <

This is the methodology we propose in the following section for the derivation of particular Runge-Kutta methods with

**Implicit midpoint rule **For the implicit midpoint rule

This was first shown by Cao

Methods with bounded variance and extended stability domain

For the general case of

and optimise the value of _{
s, ϵ }such that the range for which this holds is (-_{
s, ϵ}, 0].

Noticing from (24) that

inequality (26) can be restated in terms of

with _{
s ϵ}, 0]. Hence, we can translate constraints in the relative variance into constraints in the stability function. Since we are interested in constructing explicit methods we can ask how we can make

Thus, similar to the case _{1 }= **b**
^{T}
**e **= 1 then we have _{2},...,_{
s
}, we can optimise. In this case, though, the search of the optimal set of parameters has to be performed with numerical optimisation methods rather than analytically. The problem of finding optimal sets of parameters can be stated as a nonlinear program, NLP, and thus its solution approximated numerically (see details in Additional file

Figure

Stability regions for methods with bounded relative variance and optimal stability

**Bound (ϵ)**

**Stages ( s)**

**Stability ( l**

**Factor vs. τ-leap**

**Norm. factor τ-leap**

0.10

3

3.94566

19.73

6.58

5

10.1813

50.9

10.18

0.25

3

5.89563

14.74

4.91

5

11.0001

27.5

5.5

0.50

3

8.12004

12.18

4.06

5

15.5997

23.4

4.68

Factor vs. _{s, ϵ}/_{τ-leap}). Normalised factor vs. _{s, ϵ}/(_{τ-leap})).

Stability and relative variance for the different methods

**Stability and relative variance for the different methods**. Stability and relative variance functions for the Poisson ^{6 }simulations each. **(a)**, **(b)**: Relative variance bounded by 0.1. **(c)**, **(d)**: Relative variance bounded by 0.25. **(e)**, **(f)**: Relative variance bounded by 0.5.

Efficient methods with bounded variance and extended stability

Runge-Kutta methods with a given stability polynomial **A **but on **b **and **A **of the Butcher tableau, see details in Additional file _{
s
}= 1, _{j }= 0, **A **has all its elements set to zero except those on the first subdiagonal. These Runge-Kutta schemes obtained in this way are very natural, can be regarded as fixed point iterations and allow the following efficient reformulation of (17)

It is thus clear that these methods are computationally more efficient than the general case as they only require **f**(·) instead of the

Numerical results

Reversible isomerisation

We compare the new Runge-Kutta framework to the Poisson _{1 }= _{2 }= 10 (z = -20**X**(0) = (100, 100)^{T}. We sampled 10^{6 }trajectories for each of the methods and for different fixed _{1 }obtained from the different methods and some of the values of

Histogram of _{1 }in the Reversible isomerisation reaction

**Histogram of X _{1 }in the Reversible isomerisation reaction**. Histogram of

Schlögl reaction

We also consider Schlögl's autocatalytic reaction system ^{6 }simulations for each method and **A **more precise comparison of the plots is given in Figure _{E}) and the PDFs of each of these methods (_{M}), given by:

Details of the Schlögl reaction system

**Reactions**

**Parameters**

**Propensities**

_{1 }= 3·10^{-7}

_{1}

_{2 }= 10^{-4}

_{2}

_{3 }= 3.5

_{3}

_{4 }= 10^{-3}

_{4}

Histogram of

**Histogram of X in the Schlögl reaction**. Histogram of

Kullback-Leibler divergence for the Schlögl reaction

**Kullback-Leibler divergence for the Schlögl reaction**. Kullback-Leibler divergence between the exact stationary distribution of ^{6 }samples solved by SSA) and the approximate stationary distributions obtained with the Poisson ^{-3}.

The MAPK cascade

Finally, we have tested the performance of our methods on a larger system of chemical reactions with stiffness due to different reaction time scales and species amounts ranging over several orders of magnitude. For this purpose we considered the Huang and Ferrell model for the mitogen-activated protein kinase (MAPK) cascade ^{5 }molecules. With the chosen initial conditions the system undergoes a transient change and finally settles down into a stationary state at around

We have compared the methods in two distinct situations. First we have run them with the same time step ^{-5}. In this case, the Poisson

Then we have compared these methods when run at their respective maximum time steps such that the relative variance at the stationary state is bounded to 1.1 (estimated from the simulations). The maximum time steps allowed with this constraint were: ^{-5 }for the Poisson ^{-4 }for the RK ^{-4 }for the Optimal RK

Discussion

Biochemical kinetics typically deals with multiscale problems, in which several scales of time, space and concentrations, simultaneously affect the dynamical behaviour of the system. Thus, the systems biology community is deeply interested in the development of methods that lead to a multiscale view of biochemical systems. As a first step in this workflow, we have presented here a new set of methods that considerably expands the classical

We see from Table

Initially we had hoped that an approach via Chebyshev methods using ideas from ODEs and SDEs applied to the discrete cases would have been fruitful. It turns out that while such methods have good mean behaviour, the variance behaviour is poor. This is because the variance growth function satisfies (24) and an s-stage Chebyshev method would have

Our results on the nonlinear bimodal Schlögl problem show that the RK methods still behave appropriately even on nonlinear problems. For example, from Figure

Finally, we note that we could extend our RK methods to allow more than one set of Poisson random variables to be simulated per step. We imagine that this would allow even bigger stepsizes but at the cost of taking more simulation time in that the additional Poisson sampling is expensive. We emphasise that although our analysis of these new methods has been given for unimolecular reactions, the simulations of the nonlinear Schlögl reaction and the MAPK cascade indicate that these methods have a more general applicability and we will consider nonlinear analysis via Taylor series expansions in future work.

Authors' contributions

KB, PR and JVF designed the research. KB and PR developed the algorithms and PR implemented them and performed and analysed the simulations. KB, PR and JVF wrote the manuscript. All authors have read and approved the final version of the manuscript.

Acknowledgements

PR would like to thank Marta Dies for helpful discussions. PR acknowledges Obra Social "