Formal Modeling and Simulation of Biological Systems with Delays

Delays in biological systems may be used to model events for which the underlying dynamics cannot be precisely observed, or to provide abstraction of some behavior of the system resulting in more compact models. Historically, deterministic modeling of biological systems with delays is based on Delay Differential Equations (DDes), an extension of ordinary ones where the derivative of the unknown function depends on past-states of the system. More recently, stochastic modeling was addressed by using non-Markovian stochastic processes, i.e. processes whose sojourn time in a state and the transition probabilities are not necessarily exponentially distributed. To generate a statistically correct trajectory of such processes Delay Stochastic Simulation Algorithms (DSSas), based on a well-known Stochastic Simulation Algorithm (SSa) for delay-free systems, have been proposed. In the first part of the thesis we study different DSSas. The first two algorithms treat delays according to the “delays as durations” (DDa) and the “purely delayed” (PDa) scheduling strategies. Numerical simulations of a model of the cell cycle with a delay suggest that the PDa, even in its naive definition, is the more suitable approach to model systems in which species involved in a delayed interaction can be involved, at the same time, in other interactions. We provide a strong mathematical foundation for both the approaches by deriving two different Delay Chemical Master Equations (Dcmes), the analogous of the Chemical Master Equation (Cme) for delay-free systems. This rigorously proves the difference between the algorithms. Improvements of the naive PDa result in the PDa with markings, a more precise algorithm finally combined with the DDa to have a unique DSSa with both approaches. In the last algorithm we propose, the DSSa with Delayed Propensity Functions (Dpf), delays are treated as in DDes: changes in the current state of the system are affected by the history of the system, rather than by any scheduling strategy. We prove this algorithm to be correct, and we show that the Dcme for the PDa holds also for the Dpf, thus proving the equivalence between the algorithms. In the second part of the thesis we study formal languages for modeling biological systems with delays. Initially, we start adding delays to actions in the Calculus of Concurrent Systems (CCS) process algebra. This leads to two new qualitative algebras, CCSd and CCSp, the former where actions follow the DDa, the latter where actions follow the PDa. We provide both the algebras a Structural Operational Semantics (SOS) in the Starting-Terminating (ST) style, meaning that the start and the completion of an action are observed as two separate events, as desired when actions with delays require scheduling. To compare processes, we define bisimulation relations in both CCSd and CCSp, and we prove them to be congruences, thus extending standard results on CCS. Finally, we enrich the bio-inspired stochastic process algebra Bio-PEPA with actions following the DDa, yielding a new non-Markovian stochastic process algebra: Bio-PEPAd. This is a conservative extension meaning that the original syntax of Bio-PEPA is retained, hence adding delays to existing Bio-PEPA models is straightforward. We provide Bio-PEPAd with a SOS, again in the ST-style. To analyze Bio-PEPAd models, we formally define their encoding in Generalized semi-Markov Processes (GSMPs), a class of non-Markovian processes, directly at the semantic level. Moreover, we define translation of models in input for the DDa and in sets of DDes. Finally, we investigate the relation between Bio-PEPA and Bio-PEPAd models, outlining a clear connection at their semantic level, as it happens for the corresponding stochastic processes.