Computational approaches to cellular rhythms
Goldbeter (2002): Computational approaches to cellular rhythms

The authors suggest that "mathematical models and numerical simulations are needed to fully grasp the molecular mechanisms and functions of biological rhythms" because of "the large number of variables involved and of the complexity of feedback processes that generate oscillations". Also, they write that models are "necessary to comprehend the transition from simple to complex oscillatory behaviour and to delineate the conditions under which they arise".

Historically, "theoretical models for biological rhythms were first used in ecology to study the oscillations resulting from interactions between populations of predators and prey", and "[n]eural rhythms represent another field where such models were used at an early stage: the formalism developed by Hodgkin and Huxley still forms the core of most models for oscillations of the membrane potential in nerve and cardiac cells".

This review focuses on "oscillations of intracellular calcium, pulsatile signalling in intercellular communication, and circadian rhythms", and, in addition, the author describes "how computational biology can help in understanding the transition from simple periodic behaviour to complex oscillations including bursting and chaos".

The author starts his paper by explaining the phenomena of steady state and limit cycle:

In the course of time, open systems that exchange matter and energy with their environment generally reach a stable steady state. However, as shown by Glansdorff and Prigogine, once the system operates sufficiently far from equilibrium and when its kinetics acquire a nonlinear nature, the steady state may become unstable. Feedback processes and cooperativity are two main sources of nonlinearity that favour the occurrence of instabilities in biological systems. When the steady state becomes unstable, the system moves away from it, often bursting into sustained oscillations around the unstable steady state. In the phase space defined by the system’s variables (for example, the concentrations of the biochemical species that are involved in the oscillatory mechanism), sustained oscillations correspond to the evolution towards a closed curve — the limit cycle. [...] Limit-cycle oscillations thus represent an example of non-equilibrium self-organization and can therefore be viewed as temporal dissipative structures. The oscillations are characterized by their amplitude and by their period.

It is also possible that a system has multiple steady states, and the most common case of this is called "bistability".

When spatial inhomogeneities develop, instabilities may lead to the emergence of spatial or spatiotemporal dissipative structures. These can take the form of propagating concentration waves, which are closely related to oscillations.

What is the step by step approach used by computational biologists to describe the molecular mechanism of a biological rhythm? The author lists five steps:

First, the key variables of the phenomenon are identified, together with the nature of their interactions that form the relevant feedback loops. Second, differential equations describing the time evolution of the system are constructed. In spatially homogeneous conditions, these take the form of ordinary differential equations, whereas in the presence of diffusion, partial differential equations are used to describe the system’s spatiotemporal evolution. Third, the steady state(s) admitted by these equations are determined analytically or by numerical integration. The fourth step probes the stability properties of the steady state(s). This is generally done by using linear stability analysis. [...] Using this approach, the fifth step is to determine the domains of occurrence of sustained oscillations in parameter space.

In the next chapters, the author discusses various deterministic models for cellular rhythms. Among many other things, he writes about circadian rhythms in continuous darkness:

Computational biology can provide surprisingly counterintuitive insights. A case in point is the puzzling observation that circadian rhythms in continuous darkness can sometimes be suppressed by a single pulse of light and restored by a second such pulse. Winfree proposed the first theoretical explanation for this long-term suppression. He hypothesized that the limit cycle in each oscillating cell surrounds an unstable steady state. The light pulse would act as a critical perturbation that would bring the clock to the singularity, that is, the steady state. Because the steady state is unstable, each cell would eventually return to the limit cycle, but the population would be spread out over the entire cycle so that the cells would be desynchronized and no global rhythm would be seen. An alternative explanation is based on the coexistence of sustained oscillations with a stable steady state.

Finally, in the start of the conclusion chapter, the author writes:

Given the rapid accumulation of new data on gene, protein and cellular networks, it is increasingly clear that computational biology will be crucial in making sense of the puzzle of cellular regulatory interactions. Models and simulations are particularly valuable for exploring the dynamic phenomena associated with these regulations. Such an approach has long been applied to the study of biological rhythms, from the periodic activity of nerve and cardiac cells to population oscillations in ecology. [...] At the genetic level, models show that regulatory interactions between genes can result in multiple steady states or oscillations.

Moreover, he describes "two main approaches followed in computational biology":

The first is based on minimal models — a complex system is decomposed into simpler modules, each of which can be modelled by simple equations. Once these are understood, they are assembled into increasingly complex networks that can exhibit collective properties not apparent in the modules’ behaviour. The second relies on large-scale models that aim at incorporating from the outset all known details about the variables and processes of interest. This approach may someday lead to the construction of an electronic cell in silico, although that day remains far off. With models as with maps, I believe that an intermediate scale will often prove most fruitful.

In the end, he writes:

Models for cellular rhythms illustrate the roles and advantages of computational biology. First and foremost, modelling takes over when pure intuition reaches its limits. This situation commonly arises when studying cellular processes that involve a large number of variables coupled through multiple regulatory interactions. Here one cannot make reliable predictions on the basis of verbal reasoning. But mathematical models can show the precise parameter ranges that give rise to sustained oscillations. Models also help clarify the molecular mechanisms of these oscillations. Indeed, simulations allow rapid determination of the qualitative and quantitative effects of each parameter, and thereby can help to identify key parameters that have the most profound effect on the system’s dynamics. Testing various models permits swift exploration of different mechanisms over a large range of conditions. One of the main roles of models will be to provide a unified conceptual framework to account for experimental observations and to generate testable predictions.