Cellware is capable of simulating a homogeneous system that exhibits activities at different timescales. The simulation engine includes two different classes of algorithms, namely the Stochastic Simulation Algorithms and Deterministic Algorithms.
7.2.1 Deterministic Algorithms
Deterministic algorithms deal with the aggregated quantity of concentration and rate of change of concentration. It formulates the system based on chemical reaction rate laws governing the species concentration. The substance-reaction relationships are usually expressed as a system non-linear differential equation.
Over the past many years, powerful algorithms with sound mathematical basis have been developed for solving these equations. The tools based on these algorithms have seen wide spread applications especially in the field of metabolic engineering. A couple of explicit and implicit methods have been implemented in this class of algorithms to solve the stiff and non-stiff Ordinary Differential Equations (ODE). The methods include:
The Cellware users can choose the required method from the above options depends on the scale size, expected accuracy and robust of the problem. For example, trapezoidal method is a stable and fast method to solve the stiff ODE problem; however, fourth order Rosenbrock method is a little bit slower but offers more accurate results.
- Euler Forward method
Euler forward method is a fastest solver for most non-stiff ODE problems, but it gives lower order of accuracy and small time step size.
- Euler Backward method
Euler backward method is almost a fastest solver for most stiff ODE problems, but like Euler forward method, it only offers first order accuracy.
- Trapezoidal method
Trapezoidal is a fast solver for stiff ODE problems and it also gives second order accuracy.
- Explicit 4th order Runge-Kutta method
Explicit 4th order Runge-Kutta method is the popular one-step solver for non-stiff ODE problem. It provides high order accuracy and large time step size also.
- Rosenbrock method (Generalized fourth order Runge-Kutta method)
The famous Rosenbrock method is special case of the generalized Runge-Kutta method. It provides high order of accuracy and economic computational cost for stiff ODE problem.
- Advanced ODE Solver (Adams-Bashforth)
Advanced ODE solver is an automated explicit Adams-Bashforth four step method. It can automatically detect whether the problem is stiff or non-stiff and use small time step size.
7.2.2 Stochastic Algorithms
In a deterministic simulation, the system evolves along a predetermined path from its initial state. However, this is not always true for intra-cellular biological processes like gene regulation which are inherently stochastic. Also it is not possible to capture certain emergent phenomena which are inherently random. Deterministic techniques cannot be used for modelling such stochastic or random events.
Another limitation of deterministic approach is that they assume the concentration changes continuously over time.
This assumption is valid as long as the concentrations are high but in living cells the concentrations could be very low (e.g. single DNA molecule, molecules of certain RNA and Protein ranging from tens to a few hundred). As a result, at low concentration the discrete nature of change needs to be considered.
A series of Gillespie stochastic simulation algorithms are implemented in the simulation engine of Cellware.
- Gillespie's Direct method
Gillespie direct method is exact stochastic simulation algorithm for homogeneous, well-mixed chemical reaction systems. Due to the direct proportionality to the number of reactions in the system, the algorithm requires substantial amount of computational effort to simulate a complex system.
- Gibson Next Reaction method
In view of the inherent weaknesses in the Gillepie's direct method, Gibson introduced the next reaction method, which omits the repetitive activities in the original algorithms. In this context, two specific data structures namely a dependency graph and indexed priority queue are used. Numerical experiments prove it is much faster than the Gillespie's direct method, specially for large scale problems.
- Explicit Tau-Leap method
Gillespie introduced the explicit Tau-Leap method to speed up the Gillespie direct algorithms. This method aims to use a larger time step than Gillespie direct algorithm while bounding the local error within the tolerance. Although this method performs faster simulation than Gillespie direct algorithm, the accuracy is highly sensitive to the error control mechanisms and the tolerance.
7.2.3 Hybrid Algorithms
Gibson's next reaction method and Stochsim have been proposed to reduce the computational requirement of stochastic algorithm as the complexity of the system increases. However, none of these algorithms have actually solved the multi-scale problem efficiently. It is necessary to devise a hybrid method which can handle the fast reactions and slow reactions.
A hybrid method will be an integrative platform, which combines different numerical algorithm to simulate a biological system. The underlying idea of a hybrid method will be to integrate the numerical algorithms in such a way that the advantages of one algorithm will help to overcome the disadvantages of another algorithm. The key ideas of the algorithm are to simulate the biological system using ODE and stochastic algorithm in the same simulation platform.
Currently, one hybrid simulation method is included in Cellware.
- StochODE method
StochODE method is developed by Cellware Team to provide preliminary studies on large-scale system that involves multiple time-scale reactions. The method integrates differential equations with stochasticity by introducing an external noise term to the ODE. The noise term is generated based on Poisson distribution of the quantity of the species. This method is fast in a large-scale system but accuracy of the results is sacrificed.