Taking randomness into account

In this post Dr. Ulrich Dobramysl, a Postdoctoral Research Assistant working at the WCMB, discusses stochastic modelling and randomness in biological systems.


My work focusses on the modelling of biological systems involving some sort of randomness. I try to understand and quantify the relevance of random components in specific model systems. This randomness might stem from intrinsic reaction noise in biochemical systems or random influences from a complex environment in ecosystems. I will discuss some examples below.

Traditionally, mathematical modelling of biological systems is done using deterministic techniques, which works quite well in the majority of cases. We use differential equations to provide predictions for, e.g. molecular concentrations in a given part of a cell, or the average number of animals in an ecosystem. In this case we are guaranteed to always get the same results if we start with the same initial conditions. However, we do not always know the initial conditions exactly. In laboratory conditions or in vivo, there is always some external noise the sources of which would be impossible to incorporate into our model. Hence, when we use deterministic modelling techniques, we need to make sure that these influences won't affect our predictions in a profound way which would render our model useless.

Stochastic modelling is a way to incorporate randomness into mathematical models. We employ this kind of modelling when we are interested in the behaviour of only some components of a real system. This is possible, e.g. when there is a separation in time scales between the interesting parts and the rest. Take for example the movement of large biomolecules suspended in water: H2O molecules are moving extremely fast compared to the slower speeds of the biomolecules. They will collide at random with with the larger molecules, transferring momentum and energy. In this case, we can approximate the movement of the small water molecules by an effective random force [1]. Consequently, we do not have to track the H2O molecules and thereby can reduce the needed computational effort tremendously. Another example is the modelling of biochemical reaction, where we assume that the actual binding between reaction partners happens on much faster time scales than we are interested in. Then we are able to use effective binding rates in order to describe reactions such as A+B -> C [1].

There is a profound shift in the underlying modelling philosophy when going from a deterministic to a stochastic description, namely that now we don't know anything for certain any more. We can only make statements about probabilities of outcomes. For example, in a stochastic model for the diffusion of molecules, we can predict the probability of having a certain number of molecules in a given region of space. We can make statements about the average number and its variance. In principle we can derive these from the solution of the so-called master equation, which describes the time evolution of the probability distribution for the system to be in a particular state. However, this equation is generally not solvable by any analytical or direct numerical means, and we need to carry out numerical simulations of our model. A single run of such a simulation is called a realization (of the noise history, and sometimes also of the initial conditions) and we usually need to perform many such realizations in order to approximate information about the probability distribution of the quantities we are interested in.

After this introduction, here are a two of my projects that can serve as an example for stochastic modelling:


Intracellular Calcium signalling: Cells can regulate their cytoplasmic concentration of calcium ions via receptor channels that are located in clusters on calcium reservoirs (in this case, the endoplasmic reticulum). These channels are opened by calcium ions binding to activating sites, and releases ions from the reservoir (this process is called CICR - Calcium-Induced Calcium Release). This leads to a local increase in the number of calcium ions and consequently to the opening of other channels in the cluster and to a further dramatic increase of freed calcium in the vicinity of the channel site. This excess of ions then causes them to bind to the channel's inhibitory binding sites which closes the channels in the cluster. To sum up, this process shows a rapid increase of the local concentration of calcium followed by an exponential decrease after the channels closed and the ions diffuse away, which we call a calcium "puff". Here is a video visualizing calcium puffs (and waves), where they imaged calcium ions in heart muscle cells:

It is quite clear from this video and from other experimental data, that the appearance of puffs is a random process and the cell can regulate the rate at which this process occurs. So we use a stochastic model that takes into account the randomness of the channel activation and inhibition process (i.e. a stochastic channel state model) [2,3]. We also include a way for ions to diffuse away from the channel site [3]. I am comparing various ways of including stochastic ion diffusion to "tune" the randomness and thereby assess its importance.

The effect of rate randomness on predator-prey ecosystems: The Lotka-Volterra model is a very prominent and well-studied model for the interaction between a predator (A) and a prey (B) species, where they behave according to the rules A -> 0 (mortality), B -> 2B (reproduction) and A+B -> 2A (predation) [4]. It is a wonderful example for a system that behaves differently in a a stochastic versus a deterministic modelling approach. The deterministic equations that describe this system predict perpetual oscillations in this cycle: (i) The number of prey rises because of reproduction, (ii) the number of predators rises because they feed off the prey, (iii) the predator population falls due to mortality and starvation, and (iv) the prey population recovers. But in stochastic versions of this model the oscillations are damped, and both species eventually reach co-existence population levels where they don't show oscillations any more. The frequency of the damped oscillations is also quite different from characteristic frequency in the deterministic system because of the inherent reaction randomness [5]. When we include random reaction rates, either because they vary spatially (environmental randomness) or individually (demographic randomness), both the prey and the predator population profit and their numbers increase [6,7]. This quite unexpected result lead us to eventually study the co-evolution between predators and prey in this very simple setting [7].

In addition to these projects, I am also interested in modelling cellular protrusions, called filopodia; the collective behaviour of animal swarms and crowds; as well as the development of efficient algorithms and software for the numerical simulation of these models.

I hope that I could convince you that stochastic modelling is quite interesting, and that it is important to study the influence of randomness on biological systems. 

References

1. R. Erban, J. Chapman and P. Maini, "A practical guide to stochastic simulations of reaction-diffusion processes", Lecture Notes, available as http://arxiv.org/abs/0704.1908, 35 pages (2007) 
2. G. Dupont, L. Combettes, L. Leybaert, "Calcium Dynamics: Spatio‐Temporal Organization from the Subcellular to the Organ Level", International Review of Cytology 261, 193-245 (2007)
3. M. Flegg, S. Ruediger and R. Erban, "Diffusive spatio-temporal noise in a first-passage time model for intracellular calcium release", Journal of Chemical Physics 138, 154103 (2013)
4. J. D. Murray, "Mathematical Biology" Volume I, 3rd edition, pages 437-449, Springer-Verlag Berlin Heidelberg (2003)
5. U. C. Täuber, "Population oscillations in spatial stochastic Lotka–Volterra models: a field-theoretic perturbational analysis", Journal of Physics A: Mathematical and Theoretical 45, 405002 (2012)
6. U. Dobramysl and U. C. Täuber, "Spatial Variability Enhances Species Fitness in Stochastic Predator-Prey Interactions, Physical Review Letters 101, 258102 (2008)
7. U. Dobramysl and U. C. Täuber, Environmental Versus Demographic Variability in Two-Species Predator-Prey Models, Phys. Rev. Lett. 110, 048105 (2013)