ARTICLE: Generations of generating functions in dividing cells

Closed-form stochastic solutions for non-equilibrium dynamics and inheritance of cellular components over many cell divisions

• Populations of important machines in our cells behave quite randomly: we build a mathematical framework to better understand these populations (which has already helped us understand the inheritance of mtDNA disease)

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) 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 in the Proceedings of the Royal Society A. Iain and Nick [blog article also here]

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

Stochastic modelling, Bayesian inference, and new in vivo measurements elucidate the debated mtDNA bottleneck mechanism

• Disease-causing mutant mtDNA is inherited through a complicated process: we use maths and statistics to shed light on this process and suggest possible therapeutic strategies to address disease inheritance and onset

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, lead 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 here. Iain and Nick [blog article also here]

ARTICLE: Great technological power, great statistical responsibility

Multiple hypothesis correction is vital and undermines reported mtDNA links to diseases including AIDS, cancer, and Huntingdon’s

• Several papers perform incorrect and misleading statistical analyses in seeking links between mtDNA and cancer: these statistical issues must be corrected before scientific and policy progress can be made from these investigations

Biologists often report a result as a "significant" sign of exciting new science if there is less than a 1-in-20 chance that the result they observe could have emerged by chance from boring old science. This is silly (although we do it too!) -- by contrast, for example, physicists require less than a 1-in-3,500,000 chance. But this post won't discuss too many problems with this state of affairs -- that is done admirably elsewhere.

The problem can be compounded when scientists take lots of measurements. Say we take 50 measurements of a boring old system, and every time we see something that has less than a 1-in-20 chance of appearing in a boring old system, we call it "significant". We're playing the odds 50 times, so we expect to see 1-in-20 results appear around 2 or 3 times; just as if we roll a dice 50 times, we'd expect to roll a good few sixes. If we call every 1-in-20 result "significant" without accounting for the fact that we've looked at lots of measurements (and are thus more likely to see 1-in-20s by chance), we are in danger of reporting exciting new science when in fact the boring old science has been true all along.

There are lots of ways of doing this accounting, but a series of papers that have been recently published linking mtDNA to diseases have made no attempt to do it. Generally, these papers look at the mtDNA of people without the disease and the mtDNA of people with the disease. If any mtDNA features appear more in the people with the disease, the paper calculates the chance of that difference occurring in the boring old picture (in which there is no link between the mtDNA feature and the disease). If they drop below the 1-in-20 mark, they report an exciting new link between that feature and the disease. But they test dozens of features and never account for this multiple testing -- so, as above, we'd expect them to see "significant" results emerging just by chance. In a paper in Mitochondrial DNA here (free here) I show, by creating artificial data, that this problem is rife, that most of these reported links are spurious, and that scientists really need to be more responsible, before their flawed analysis starts to misguide health policy and medicine.

The top graph shows how the probability of seeing a 1-in-20 occurrence (p < 0.05 in the jargon), when in fact there is nothing new and exciting to report, increases as a scientist investigates more things. If an experiment consists of one test, then a 1-in-20 occurrence indeed has a 1-in-20 probability (0.05). But as soon as we do more tests, the chance of seeing at least one 1-in-20 occurrence starts to increase, as we are "playing the game" more times. If we do 6 tests there is a 0.27 probability -- between a 1-in-4 and 1-in-3 chance -- that we will see at least one 1-in-20 event. This is illustrated below, where we have six dice and think some of them may be unfair. We roll each one five times and count the number of 6s. One of them comes up 6 three times -- the chance of this happening for one fair die is less than 1-in-20. But because we've looked at six dice, we should be less surprised to see this rare event, because we've looked at more events in total. We need more evidence to claim that this die is unfair.

This quick note only represents the tip of the iceberg. MtDNA studies are often statistically unsound; statistical misdemeanours in biomedical studies are so common that most published research is wrong; scientists increasingly focus on the 1-in-20 chance as opposed to the size and importance of the effect they're measuring; the majority of hallmark papers in vital fields like cancer science are unreproducible (though this last point may have other causes than statistical problems). The 1-in-20 idea was only ever meant to be a step in identifying interesting scientific avenues, not the final measure of scientific truth. This is a big, and growing, problem! Iain

(For accessibility I have used "exciting", "boring", and "1-in-20" instead of their usual, more technical labels; they of course are usually called the "alternative hypothesis", "null hypothesis", and "p < 0.05" respectively).

ARTICLE: The function of mitochondrial networks

What is the function of mitochondrial networks? A theoretical assessment of hypotheses and proposal for future research

• Mitochondria in our cells sometimes form large networks and sometimes remain independent, with changes between these structure often linked to disease: we use physics and maths to explore why these networks may form and be valuable to the cell, and to suggest ways to find out more

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) 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 [blog article also here]

ARTICLE: Turbocharging the back of the envelope

Explicit tracking of uncertainty increases the power of quantitative rule-of-thumb reasoning in cell biology

• Estimated numbers in biology (and life) are often uncertain: we've made a calculator to work with this uncertainty and help make calculations more interpretable (with a particular focus on understanding how the cell works)

The numbers that we use to describe the world are rarely exact. How long will it take you to drive to work? Perhaps "between 20 and 30 minutes". It would be unwise (and unnecessary) to say "exactly 23.4 minutes".

This uncertainty means that "back-of-the-envelope" calculations are very valuable in estimating and reasoning about numerical problems, particularly in the sciences. The idea here is to perform a calculation using rough guesses of the quantities involved, to get an "order of magnitude" estimate of the answer you're after. Made famous in physics as "Fermi problems", attributed to Enrico Fermi (who used rough reasoning to deduce quantities from the power of an atomic bomb to the number of piano tuners in Chicago), this approach is integral in many current applications of maths and science. Cool books like "Street-fighting Mathematics", "Guesstimation", "Back of the envelope physics", the excellent "What If?" section of xkcd, and the lateral interview questions facing some job candidates: "how much of the world's water is contained in a cow?" are all examples.
 

Calculations in biology, such as the time it takes for a protein (foreground) to diffuse through an E. coli cell (background), are often subject to large uncertainties. Our approach and web tool allows us to track this uncertainty and obtain a probability distribution over possible answers (plotted).

We've built a free online calculator (Caladis -- calculate a distribution) that complements this approach by allowing one to take the uncertainty in one's estimates into account throughout a calculation. For example, what volume of CO2 is produced by our yearly driving? We could say that we cover 8000 miles per year "give or take" 1000 miles, and find that our car's CO2 emissions are between 100 and 150 grams per kilometre. Our calculator allows us to do the necessary conversions and sums while taking this possible variability into account -- doing maths with "probability distributions" describing our uncertainty. We no longer obtain a single (possibly inaccurate) answer, but a distribution telling us how likely any particular answer is -- in this case a rather concerning bell-shaped distribution between 1 and 2 tonnes which can be viewed here.

In the sciences, particularly in biology, measurements often have substantial uncertainties -- due to experimental error, natural variability in the system of interest, or both -- and so using distributions rather than single numbers in calculations allows us to understand and process more about the question of interest. "Back-of-the-envelope" calculations are certainly useful in biology but, owing to the uncertainties involved, one can trust one's estimates better if one has a smart envelope that takes that uncertainty into account.  We've written an accompanying paper (free here in Biophysical Journal) showing how to use our calculator -- in conjunction with the excellent Bionumbers online database, a collection of (often uncertain) experimental measurements in biology -- to make real biological calculations more powerful. Do have a go at using our calculator at www.caladis.org: it's user-friendly and there are lots of examples showing how it works! Iain and Nick [blog article also here]

ARTICLE: Therapies for mtDNA disease: models and implications

Mitochondrial DNA disease and developmental implications for reproductive strategies

• The inheritance of mutant mtDNA can cause devastating diseases: we review how this inheritance occurs and the ways modern medicine can help (and how some therapies may be improved)

Mitochondrial DNA (mtDNA) is a molecule in our cells that contains information about how to build important cellular machines that provide us with the energy required for life. Mutations in mtDNA can prevent our cells from producing these machines correctly, causing serious diseases. Mutant mtDNA can be passed from a carrier mother to her children, and as the amount of mutated mtDNA inherited can vary, children's symptoms can be much more severe (often deadly) than those in the mother.

Several therapies exist to prevent or minimise the inheritance of mutant mtDNA from mother to daughter. These range from simply using a donor mother's eggs (in which case the child inherits no genes from the "mother") to amazing new techniques where a mother's nucleus is transferred into a donor's egg cell which has had its nucleus removed (so that the child inherits nuclear DNA from the mother and father, and healthy mtDNA from the donor). The UK is currently debating whether to allow these new therapies: several potential scientific issues have been identified in their application.

 

If a mother carries an mtDNA mutation, (A) no clinical intervention can lead to her child inheriting that mutation and developing an mtDNA disease. Several "classical" (B-C) and modern (D-E) strategies exist to attempt to prevent the inheritance of mutant mtDNA, which we review (see paper link below)

As experiments with human embryos are heavily restricted, experiments in animals provide the bulk of our knowledge about how these therapies may work. We have previously written about our research in mice, highlighting a possible issue arising from mtDNA "segregation", where one type of mtDNA (possibly carrying a harmful mutation) may proliferate over another: this phenomenon could, in some circumstances, nullify the beneficial effects of mtDNA therapies. Another possible issue involves the effects of "mismatching" between the mother and father's nuclear DNA and the donor's mtDNA: current experimental evidence is conflicted regarding the strength of this effect. Finally, mismatch between donor mtDNA and any leftover mother mtDNA may also lead to biological complications.

We have recently written a paper (free here) explaining and reviewing the current state of knowledge of these effects, summarising the evidence from existing animal experiments. We are positive about implementing these therapies, which have the potential to prevent the inheritance of devastating diseases. However, we note cautions about this implementation, noting that several scientific questions remain debated or unanswered. We particularly highlight that "haplotype matching", a strategy to ensure that donor and mother mtDNA are as similar as possible, will largely remove these concerns. Iain [blog article also here]

ARTICLE: Mitochondrial motion in plants

FRIENDLY regulates mitochondrial distribution, fusion, and quality control in Arabidopsis

• Plant mitochondria play central roles in carbon metabolism, photosynthesis, and plant growth, but the genes controlling their structure are poorly understood: we make progress understanding how one important gene influences mitochondria and plant structure

Mitochondria are often likened to the power stations of the cell, producing energy that fuels life's processes. However, compared to traditional power stations, they're very dynamic: mitochondria move through the cell, and fuse together and break apart (among other things). Interestingly, their ability to move and undergo fusion and fission affects their functionality, and so has powerful implications for understanding disease and cellular energy supplies.

Because of this central role, it is important to understand the fundamental biological mechanisms that govern mitochondrial dynamics. Several important genes controlling mitochondrial dynamics are known in humans (and other organisms), but plant mitochondria (despite the fundamental importance of plant bioenergetics for our society) are less well understood.

Our collaborators, David Logan and his team, working with a plant called Arabidopsis, observed that a particular gene, entertainingly called "FRIENDLY", affected mitochondrial dynamics when it was artificially perturbed. (This approach, artificially interfering with a gene to explore the effects that it has on the cell and the overall organism, is a common one in cell biology.) We've just written a paper with them (free here) exploring these effects. Plants with disrupted FRIENDLY had unusual clusters of mitochondria in their cells, their mitochondria were stressed, and cell death and poor plant growth resulted.

A simulation of mitochondrial dynamics in plant cells under our simple mathematical model, which we compared to observations in real plants.

We used a 3D computational and mathematical model of randomly-moving mitochondria within the cell to show that an increased "association time" (the friendly mitochondria stick around each other for longer) was sufficient to explain the experimental observations of clustered mitochondria. Our paper thus identifies an important genetic player in determining mitochondrial dynamics in plants; and explores in substantial detail the intra-cellular, bioenergetic, and physiological implications of perturbation to this important gene. Iain and Nick [blog article also here]