Aspects of multiscale modelling in a process algebra for biological systems

We propose a variant of the CCS process algebra with new features aiming at allowing multiscale modelling of biological systems. In the usual semantics of process algebras for modelling biological systems actions are instantaneous. When different scale levels of biological systems are considered in a single model, one should take into account that actions at a level may take much more time than actions at a lower level. Moreover, it might happen that while a component is involved in one long lasting high level action, it is involved also in several faster lower level actions. Hence, we propose a process algebra with operations and with a semantics aimed at dealing with these aspects of multiscale modelling. We study behavioural equivalences for such an algebra and give some examples.

💡 Research Summary

The paper addresses a fundamental limitation of existing process‑algebraic formalisms for systems biology: the assumption that all actions are instantaneous. In real biological systems, processes occur on vastly different time scales—slow, high‑level events such as cell division or tissue morphogenesis may last minutes to days, while fast, low‑level events such as enzyme catalysis or ion channel opening happen in milliseconds or less. When a model tries to capture both levels simultaneously, the traditional CCS‑style semantics cannot represent the coexistence of a long‑lasting high‑level action with many rapid low‑level actions involving the same component.

To overcome this, the authors propose an extension of CCS that introduces two new syntactic constructs. The first, a “duration operator” (written as τ·P·τ in the paper), explicitly annotates a prefix with a time interval during which the action persists. The second is an enriched parallel composition that allows actions with different durations to overlap in time. These operators are combined with the usual choice (+), parallel (|), restriction (\) and relabeling operators, yielding a language capable of expressing nested, multiscale behaviours.

Semantically, the authors move from the classic labelled transition system (LTS) to a timed LTS where each transition carries a pair (action, time‑interval). Transition rules are carefully designed to respect three principles: (1) a durational action generates a start transition, a series of “time‑pass” self‑loops, and a termination transition; (2) when two actions overlap, the system produces a set of joint transitions that are enabled exactly over the intersection of their time intervals; (3) a component engaged in a long‑lasting high‑level action may still synchronise with any number of short‑lived low‑level actions, which are inserted as interleaved transitions within the high‑level interval. This formalism makes it possible to reason about “concurrent time‑overlap” in a mathematically precise way.

Recognising that standard behavioural equivalences (weak bisimulation, observational equivalence) ignore timing information, the authors define a new equivalence relation called Time‑Sensitive Bisimilarity. Two processes are bisimilar under this relation if, for every time interval, they can match each other’s sets of observable actions while preserving the exact timing constraints. The paper proves that this relation is an equivalence, is a congruence with respect to all operators of the language, and induces meaningful quotient structures for model reduction.

The theoretical development is illustrated with two biologically motivated case studies. The first models the eukaryotic cell‑cycle. The four phases (G1, S, G2, M) are represented as durational actions lasting several hours, while within each phase a cascade of signalling events (e.g., cyclin‑CDK activation, checkpoint checks) are modelled as rapid actions of millisecond duration. The extended semantics correctly captures that, for example, while the cell is in G1, multiple checkpoint‑related phosphorylations can occur without terminating the G1 phase. The second case study concerns antigen presentation and T‑cell activation. Antigen presentation on an APC surface is a long‑lasting context (hours), whereas T‑cell receptor binding/unbinding events are fast (sub‑second). The model demonstrates that many binding events can be interleaved within a single presentation interval, reproducing experimentally observed kinetic patterns. In both examples, the authors show that a naïve CCS encoding would either conflate the time scales or be forced to introduce artificial synchronisation points, whereas the proposed algebra handles them naturally.

Finally, the authors discuss future work. They suggest extending the deterministic duration annotations to stochastic distributions (exponential, Gaussian) to capture intrinsic biological variability, integrating the language into simulation tools (e.g., a timed stochastic simulator) and model‑checking frameworks, and exploring translations to other formal models such as timed Petri nets or hybrid automata. They also note the potential for compositional verification: because Time‑Sensitive Bisimilarity is a congruence, large models can be decomposed, analysed, and then recomposed without losing multiscale timing guarantees.

In summary, the paper delivers a rigorous, compositional framework for multiscale modelling in systems biology. By enriching CCS with durational operators, a timed transition semantics, and a novel timing‑aware behavioural equivalence, it enables the faithful representation of simultaneous slow and fast biological processes. The approach opens the door to more accurate quantitative analyses, formal verification, and tool support for complex, hierarchically organised biological systems.