Delay Stochastic Simulation of Biological Systems: A Purely Delayed Approach

Abstract

Abstract. Delays in biological systems may be used to model events for which the underlying dynamics cannot be precisely observed. Mathematical modeling of biological systems with delays is usually based on Delay Differential Equations (DDEs), a kind of differential equations in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times. In the literature, delay stochastic simulation algorithms have been proposed. These algorithms follow a "delay as duration" approach, which is not suitable for biological systems in which species involved in a delayed interaction can be involved at the same time in other interactions. We show on a DDE model of tumor growth that the delay as duration approach for stochastic simulation is not precise, and we propose a simulation algorithm based on a "purely delayed" interpretation of delays which provides better results on the considered model. Moreover, we give a formal definition of a stochastic simulation algorithm which combines both the delay as duration approach and the purely delayed approach.

Mean Field for Performance Models with Generally-Distributed Timed Transitions

“…Systems of DDEs have been derived to approximate stochastic models of reaction networks where deterministic delays are possible after reactions occur [3,6,21]. However, these models differ from those considered here in a number of critical ways; most importantly, the presentation in this paper lacks the severe rigidity of models encountered in biology and chemistry, making it suitable for a much larger class of population models.…”

Hybrid semantics for Bio-PEPA

“…In each prefix term (α i , κ i ) op i C, α i is an action name from a set A and gives the name or label of a reaction, κ i is the stoichiometric coefficient 3 of the species and the prefix combinator op i represents the role of the element in the reaction. If a species is a reactant in the reaction then ↓ is used, if a product then ↑, if an activator then ⊕, if an inhibitor then , and is used for a generic modifier 4 . The operator + expresses the choice between two sequential components.…”

“…In the reaction A + 3B → 2C + D, three times as much B as A is consumed to produce as much D as A and twice as much C as A. 4 The generic modifier is used when a species plays a role in a reaction that is different to any of the other roles. This has been useful for constructing Bio-PEPA models that abstract from a reaction-based view and also applying Bio-PEPA in non-biological contexts.…”

“…In this approach, values of variables and their derivatives from earlier time points may be used to define the differential equations; whereas in ordinary differential equations, values are used only from the current time. Stochastic approaches are based on modifications of Gillespie's algorithm [16,5,4]. The approach taken in Bio-PEPAd is delay-as-duration where reactants are removed at the start of the reaction and products added at the end.…”

“…The approach taken in Bio-PEPAd is delay-as-duration where reactants are removed at the start of the reaction and products added at the end. There are other approaches to modelling delays such as purely-delayed and random-delayed [4,21,23,65].…”