Mitochondrial networks and Ageing in the Variance

Mitochondrial DNA (mtDNA) populations within our cells encode vital energetic machinery. MtDNA is housed within mitochondria, cellular compartments lined by two membranes, that lead a very dynamic life. Individual mitochondria can fuse when they meet, and fused mitochondria can fragment to become individual smaller mitochondria, all the while moving throughout the cell. The reasons for this dynamic activity remain unclear (we’ve compared hypotheses about them before here and here, with blog articles here). But what influence do these physical mitochondrial dynamics have on the genetic composition of mtDNA populations?

MtDNA populations can, naturally or as a result of gene therapies, consist of a mixture of different mtDNA types. Typically, different cells will have different proportions of, say, type A and type B. For example, one cell may be 20% type A, another cell may be 40% type A, and a third may be 70% type A. This variability matters because when a certain threshold (often around 60%) is crossed for some mtDNA types, we get devastating diseases.

We previously showed mathematically (blog) and experimentally (blog) that this cell-to-cell variability in mtDNA proportions (often called “heteroplasmy variance” and sometimes referred to via the “mtDNA bottleneck”) is expected to increase linearly over time. However, this analysis pictured mtDNAs as individual molecules, outside of their mitochondrial compartments. When mitochondria fuse to form larger compartments, their mtDNA is more protected: smaller mitochondria (and their internal mtDNA) are subject to greater degradation. More degradation means more replication, and more opportunities for the fraction of a particular type of mtDNA to change per unit time. In a new paper here in Genetics, we show (using a mathematical tour de force by Juvid) that this protection can dramatically influence cell-to-cell mtDNA variability. Specifically, the rate of heteroplasmy variance increase is scaled by the proportion of mitochondria that exist in a fragmented state. (It turns out that it's the proportion of itochondria that are fragmented that's important -- not whether the rate of fission-fusion is fast or slow).

This has knock-on effects for how the cell can best get rid of low-quality mutant mtDNA. In particular, if mitochondria are allowed to fuse based on their quality (“selective fusion”), we show that intermediate rates of fusion are best for removing mutants. Too much fusion, and all mtDNA is protected; too little, and good mtDNA cannot be sorted from bad mtDNA using the mitochondrial network. This mechanism could help explain why we see different levels of mitochondrial fusion in different conditions. More broadly, this link between mitochondrial physics and genetics (which we’ve also speculated about here (blog) and here) suggests one way that selective pressures and tradeoffs could influence mitochondrial dynamics, giving rise to the wide variety of behaviours that remain unexplained. Juvid, Nick, and Iain

The cell's power station policy

Our cells are filled with populations of mitochondrial DNA (mtDNA) molecules, which encode vital cellular machinery that supports our energy requirements. The cell invests energy in maintaining its mtDNA population, like us using electricity-powered tools to help maintain our power stations. Our cellular power stations can vary in quality (for example, mutations can damage mtDNA), and are subject to random influences. How should the cell best invest energy in controlling and maintaining its power stations? And can we use this answer to design better therapies to address damaged mtDNA?

In a new paper "Energetic costs of cellular and therapeutic control of stochastic mitochondrial DNA populations" free here in PLoS Computational Biology, we attempt to answer this question using mathematical modelling, linking with genetic experiments done by our excellent collaborators at Cambridge (Payam Gammage, Lindsey Van Haute and Michal Minczuk). We first expand a mathematical model for how diverse mtDNA populations within cells change over time – building new power stations and decommissioning old ones, under the “governance” of the cell. We then produce an “energy budget” for the cellular “society” – describing the costs of building, decommissioning, and maintaining different power stations, and the corresponding profits of energy generation.

We find some surprising results. First, it can get harder to maintain a good energy budget in a tissue (a collection of individual cellular “societies”) over time, even if demands stay the same and average mtDNA quality doesn’t change. This is because the cell-to-cell variability in mtDNA quality does increase, carrying with it an added energetic challenge. This increased challenge could be a contributing factor to the collection of problems involved in ageing.

An overview of our approach. A mathematical model for the processes and "budget" involved in controlling mtDNA populations makes a general set of biological predictions and explains gene-therapy observations

Next, we found that cells with only low-quality mtDNA can perform worse than cells with a mix of low- and high-quality mtDNA. This is because low-quality mtDNA may consume less cellular resource, although global efficiency is decreased. Linked to this, removal of low-quality mtDNA (decommissioning bad power stations) alone is not always the best strategy to improve performance. Instead, jointly elevating low- and high-quality mtDNA levels, avoiding this detrimental mixed regime, is the best strategy for some situations. These insights may help explain some of the negative effects recently observed in cells with mixed mtDNA populations.

Our theory suggests that mixed mtDNA populations may do worse than pure ones, even if the pure population is a low-functionality mutant. Image from Hanne's post here 

We identified how best to control cellular mtDNA populations across the full range of possible populations, and used this insight to link with exciting gene therapies where low-quality mtDNA is preferentially removed through an experimental intervention (using so-called “endonucleases” to cut particular mtDNA molecules). We found that strong, single treatments will be outperformed by weaker, longer-term treatments, and identified how the mtDNA variability we know is present can practically effect the outcome of these therapies. We hope that the principles found in this work both add to our basic understanding of ageing and mixed (“heteroplasmic”) mitochondrial populations, and may inform more efficient therapeutic approaches in the future. Iain, Hanne, Nick

(Hanne's also written a post about this paper, you can read it here)

Guessing the spreading time of rumours

Social scientists are fascinated by social influence. That is, how people's beliefs, opinions and actions are influenced by others. This is relevant for understanding voting, health behaviour or opinions on issues like vaccination and climate change (topics our group is interested in). Mathematically inclined social scientists often interpret social influence using network theory.  Networks or graphs are used to represent systems consisting of many individual units, known as nodes, and the interactions between them, which are referred to as edges or links. In social networks the nodes represent people and the links represent social ties such as friendships.

Given a particular graph there are tools for modelling how opinions and beliefs can spread through a graph. However, in practice we often don’t know the structure of the social network itself. This could be because: i) the data we would like is unavailable ii) privacy concerns about social network data mean we can't share it even if we have it iii) the data exists but is full of errors or omissions. Fortunately, we know a lot about the structure of social networks from decades of past research by social scientists and statisticians. For example, many social networks are known to be homophilous - this means that people who are similar to each other are more likely to share a social connection (e.g many of your friends are probably a similar age to you).

Inspired by this, we consider a simple mathematical model for homophilous networks known as a Random Geometric Graph (RGG). In an RGG the nodes are assigned random positions in a (unit) box. Nodes are connected to all the nodes which are within a set distance (see figure), which we refer to as the connection radius. Positions of nodes may represent the positions of individuals in geographic space or in some “social space” where the coordinate axis might represent attributes such as age, income and education level. Since social networks are homophilous we will expect those who are closer together in “social space” to share a social tie.

Example of a Random Geometric Graph with 100 nodes and a connection radius of 0.2.

One basic question we can ask about a network is: “how long does it take something to spread across it?” We refer to this as the diffusion timescale. The diffusion timescale in a graph is indicative of how well connected the graph is and governs how quickly we might expect a disease, rumour or the adoption of a new behaviour to spread through it (or even how long it will take a zombie apocalypse to take hold). In our recent research we focus on the question:

“If we do not know the network (but perhaps know some of its properties) how precisely can we know the diffusion timescale?”

We show that different RGGs drawn at random with the same number of nodes and connection radius can have very different diffusion timescales. This implies that if we don’t have a good grasp of the graph structure then it could be difficult to predict the outcomes of processes such as the spread of an opinion through a social network. Or alternatively we can gain lots of extra information about diffusion timescales if we happen to know the social co-ordinates of individuals. On the other hand, we do find some classes of RGGs where the diffusion time scale is very predictable given only knowledge of the number of nodes and the connection radius.

Our work helps put limitations on how accurately we can forecast the outcome of processes on networks given the available data (which is always imperfect). Future work may involve asking the same questions for real world datasets. In addition, most of our new results were obtained through computer simulations, meaning that there is also scope for more theory.

You can read about our research in the paper “Large algebraic connectivity fluctuations in spatial network ensembles imply a predictive advantage from node location information” for free here or for not-free here in Physical Review E. Matt and Nick.

How mitochondria can vary, and consequences for human health

Mitochondria are components of the cell which are involved in generating “energy currency” molecules called ATP across much of complex life. Since many mitochondria exist within single cells (often hundreds or thousands), it is possible for the characteristics of individual mitochondria to vary within cells, and within tissues. This variation of mitochondrial characteristics can affect biological function and human health.

Since mitochondria possess their own, small, circular, DNA molecules (mtDNA), we can split mitochondrial characteristics into two categories: genetic and non-genetic. In our review, we discuss a number of aspects in which mitochondria vary, from both genetic and non-genetic perspectives. 

In terms of mitochondrial genetics, the amount of mtDNA per cell is variable. When a cell divides, its daughters receive a share of its parents mtDNA, but the split isn’t precisely 50/50, so cell division can cause variability in the number of mtDNAs per cell. As mtDNAs are replicated and degraded over time, errors in the copying process may give rise to mtDNA mutations, which may spread throughout a cell. Factors such as: the total amount, the rate of degradation/replication, the mean fraction of mutants, and the extent of fragmentation in the mitochondrial network, can all influence how variable the fraction of mutated mtDNAs becomes through time (see here for a preview of some upcoming work on this topic). The total amount, and mutated fraction of mtDNAs, are implicated in diseases such as neurodegeneration, as well as the ageing process.

Apart from genetic variations, there are many non-genetic features of mitochondria which also vary within and between cells. Changes in mtDNA sequence can change the amino-acid sequence of the proteins encoded by mtDNA, causing structural changes in the molecular machines which generate ATP. The shape of the membranes of mitochondria are also highly variable, and respond to mitochondrial activity through quantities such as pH, where mitochondrial activity itself may depend on mtDNA sequence. The previous two examples (mitochondrial protein and membrane structure) demonstrate how the genetic state of mitochondria may influence their non-genetic characteristics. Mitochondrial non-genetic characteristics may also influence the genetic state: for instance, mitochondrial membrane potential can influence the probability of a mitochondria being degraded, along with its mtDNA.

The inter-dependence of genetic and non-genetic characteristics demonstrate the complex feedback loops linking these two aspects of mitochondrial physiology. We suggest here that, since changes in mitochondrial genetics occur more slowly than most physical aspects of mitochondrial physiology, understanding mitochondrial genetics may be especially important in explaining phenomena such as ageing, which appears to be closely related to mitochondrial heterogeneity. You can freely access our work, which has recently been published in Frontiers in Genetics, as “Mitochondrial Heterogeneity” https://www.frontiersin.org/articles/10.3389/fgene.2018.00718/full Juvid, Iain and Nick.