Generations of generating functions in dividing cells

Cell biology is a unpredictable world, as we've written about before. The important machines in our cells replicate and degrade in processes that can be described as random; and when cells divide, the partitioning of these machines between the resulting cells also looks random. The number of machines we have in our cells is important, but how can we work with numbers in this unpredictable environment?

In our cells, machines are produced (red), replicate (orange), and degrade (purple) randomly with time, as well as being randomly partitioned when cells split and divide (blue). Our mathematical approach describes how the total number of machines is likely to behave and change with time and as cells divide.

Tools called "generating functions" are useful in this situation. A generating function is a mathematical function (like G(z) = z², but generally more complicated) that encodes all the information about a random system. To find the generating function for a particular system, one needs to consider all the random things that can happen to change the state of that system, write them down in an equation (the "master equation") describing them all together, then use a mathematical trick to push that equation into a different mathematical space, where it is easier to solve. If that "transformed" equation can be solved, the result is the generating function, from which we can then get all the information we could want about a random system: the behaviour of its mean and variance, the probability of making any observation at any time, and so on.

We've gone through this mathematical process for a set of systems where individual cellular machines can be produced, replicated, and degraded randomly, and split at cell divisions in a variety of different ways. The generating functions we obtain allow us to follow this random cellular behaviour in new detail. We can make probabilistic statements about any aspect of the system at any time and after any number of cell divisions, instead of relying on assumptions that the system has somehow reached an equilibrium, or restricting ourselves to a single or small number of divisions. We've applied this tool to questions about the random dynamics of mitochondrial DNA (which we're very interested in! And this work connects explicitly with our recent eLife paper - blog article here) in cells that divide (like our cells) or "bud" (like yeast cells), but the approach is very general and we hope it will allow progress in many more biological situations. You can read about this, free, here under the title "Closed-form stochastic solutions for non-equilibrium dynamics and inheritance of cellular components over many cell divisions" in the Proceedings of the Royal Society A. Iain and Nick

How evolution deals with mitochondrial mutants (and how we can take advantage)

Our mitochondrial DNA (mtDNA) provides instructions for building vital machinery in our cells. MtDNA is inherited from our mothers, but the process of inheritance -- which is important in predicting and dealing with genetic disease -- is poorly understood. This is because mitochondrial behaviour during development (the process through which a fertilised egg becomes an independent organism) is rather complex. If a mother's egg cell begins with a mixed population of mtDNA -- say with some type A and some type B -- we usually observe hard-to-predict mtDNA differences between cells in the daughter. So if the mother's egg cell starts off with 20% type A, egg cells in the daughter could range (for example) from 10%-30% of type A, with each different cell having a different proportion of A. This increase in variability, referred to as the mtDNA bottleneck, is important for the inheritance of disease. It allows cells with higher proportions of mutant mtDNA to be removed; but also means that some cells in the next generation may contain a dangerous amount of mutant mtDNA. Crucially, how this increase in variability comes about during development is debated. Does variability increase because of random partitioning of mtDNAs at cell divisions? Is it due to the decreased number of mtDNAs per cell, increasing the magnitude of genetic drift? Or does something occur during later development to induce the variability? Without knowing this in detail, it is hard to propose therapies or make predictions addressing the inheritance of disease.

We set out to answer this question with maths! Several studies have provided data on this process by measuring the statistics of mixed mtDNA populations during development in mice. The different studies provided different interpretations of these results, proposing several different mechanisms for the bottleneck. We built a mathematical framework that was capable of modelling all the different mechanisms that had been proposed. We then used a statistical approach called approximate Bayesian computation to see which mechanism was most supported by the existing data. We identified a model where a combination of copy number reduction and random mtDNA duplications and deletions is responsible for the bottleneck. Exactly how much variability is due to each of these effects is flexible -- going some way towards explaining the existing debate in the literature.  We were also able to solve the equations describing the most likely model analytically. These solutions allow us to explore the behaviour of the bottleneck in detail, and we use this ability to propose several therapeutic approaches to increase the "power" of the bottleneck, and to increase the accuracy of sampling in IVF approaches.

A "bottleneck" acts to increase mtDNA variability between generations. But how is this bottleneck manifest? Our approach suggests that a combination of copy number reduction (pictured as a "true" copy number bottleneck), and later random turnover of mtDNA (pictured as replication and degradation), is responsible.

Our excellent experimental collaborators, led by Joerg Burgstaller, then tested our theory by taking mtDNA measurements from a model mouse that differed from those used previously and which, could in principle have shown different behaviour. The behaviour they observed agreed very well with the predictions of our theory, providing encouraging validation that we have identified a likely mechanism for the bottleneck. New measurements also showed, interestingly, that the behaviour of the bottleneck looks similar in genetically diverse systems, providing evidence for its generality. You can read about this in the free (open-access) journal eLife under the title "Stochastic modelling, Bayesian inference, and new in vivo measurements elucidate the debated mtDNA bottleneck mechanism"  Iain and Nick

The function of mitochondrial networks

Mitochondria are dynamic energy-producing organelles, and there can be hundreds or even thousands of them in one cell. Mitochondria (as we've blogged about before - e.g. here) do not exist independently of each other: sometimes they form giant fused networks across the cell, sometimes they are fragmented, and sometimes they take on intermediate shapes. Which state is preferred (fragmented, fused or in between) seems to depend on, for example, cell-division stage, age, nutrient availability and stress levels. But what is exactly the reason for the cell preferring one morphology over another?

Nonlinear phenomena -- like some percolation effects -- could help account for the functional advantage of mitochondrial networks

We recently wrote an open-access paper (free here in the journal BioEssays) in which we try to answer the question: what is it about fused mitochondrial networks that could make them preferable to fragmented mitochondria? Our paper differs from previous work in that we attempt to use a range of mathematical tools to gain insight into this complex biological system and we try to hit on the root physiological and physical roles. We use physical models, simulations, and numerical estimations to compare ideas, to reason about existing hypotheses, and to propose some new ones. Among the possibilities we consider are the effects of fusion on mitochondrial quality control, on the spread of important protein machinery throughout the cell, on the chemistry of important ions, and on the production and distribution of energy through the cell. The models we use are quite simple, but we propose ideas for improving them, and experiments that will lead to further progress.

Taking a mathematical perspective leads to a central idea: for fused mitochondria to be 'preferred' by the cell, there must be some nonlinear advantage to fusion. That's what the fuzzy line is representing in the figure above. A big mitochondrion formed by fusing two smaller ones must in some sense be 'better' than the sum of the two smaller ones, or there would be no reason why a fused state is preferred.

Mitochondria can fuse to form large continuous networks across the cell. From a mathematical and physical viewpoint, we evaluate existing and novel possible functions of mitochondrial fusion, and we suggest both experiments and modelling approaches to test hypotheses

What is the source of this nonlinearity? We find several physical and chemical possibilities. Large pieces of fused mitochondria are better at sharing their contents (e.g. proteins, enzymes, and possibly even DNA) than smaller pieces of fused mitochondria. If the 'fusedness' of the mitochondrial population increases by a factor of two, the efficiency with which they share their contents increases by more than two! Also, fusion can reduce damage. If a mitochondrion gets physically or chemically damaged, having some fused non-damaged neighbours can help to reduce the overall harm to the cell. Finally, fusion may increase energy production because of a nonlinear chemical dependence of energy production on mitochondrial membrane potential. Fusing more mitochondria may, under certain circumstances, have the effect of increasing energy production. Hanne, Iain and Nick