Tuesday 7 February 2023

Stochastic Survival of the Densest: defective mitochondria could be seen as altruistic to understand their expansion

With age, our skeletal muscles (e.g. muscle of our legs and arms) work less well. In some people, there is a substantial loss of strength and functionality called sarcopenia. This has a knock-on effect on the health of elderly people. If we want to delay skeletal muscle ageing, so that we can stay active and healthy for longer, we need to understand the mechanism behind it.

The spread of a type of mitochondrial mutants, deletions, has been causally implicated in the ageing of skeletal muscle. But how do these mutations expand, replacing normal (wildtype) mitochondrial DNA? One could think that they have a replicative fitness advantage, therefore they outcompete wildtypes. However, no definitive explanation of this supposed faster replication is known for non-replicating cells, despite numerous proposals. We might even expect that these mutants are actively eliminated since they are disadvantageous to the cell.

In a recent PNAS paper, (free version here) we  unveil a new evolutionary mechanism, termed stochastic survival of the densest, that accounts for the expansion of mitochondrial deletions in skeletal muscle. We do not assume a replicative advantage for deletions, and even allow the possibility of a higher degradation rate for mutants compared to non-mutants. Our stochastic model predicts a noise-driven clonal wave of advance of mitochondrial mutants, recapitulating experimental observation qualitatively  and quantitatively. 

Stochastic survival of the densest accounts for the expansion of mitochondrial DNA mutants in aged skeletal muscle through a noise-driven wave of advance of denser mutants.

A) Dysfunctional mtDNA mutants expand in muscle fibres with age, occupying macroscopic regions where strength and functionality are lost. B) A stochastic model of a spatially extended system predicts a travelling wave of denser mutants (e.g. the mitochondrial deletions in skeletal muscles), according to a novel evolutionary mechanism termed stochastic survival of the densest.

A key reason behind mutants’ expansion is that they can live at higher densities (or carrying capacity) in the muscle fibres. This, together with the stochastic nature of biological processes, gives rise to the surprising wave-like expansion. Remarkably, if you take the noise away, the effect disappears: it is a truly stochastic phenomenon. We are very excited about this new mechanistic understanding: the clonal expansion of deletions has been the subject of intense research for over 30 years. Moreover, this progress allows principled suggestions of therapies that might slow down skeletal muscles ageing. 

We are possibly even more excited about the evolutionary implications of our work. We believe that stochastic survival of the densest is a previously unrecognised mechanism of evolution that can account for other counterintuitive phenomena. We essentially showed that, in the presence of noise, a species can take over a system because the more of it there is the faster all species in the system replicate. This conforms to one of the strict definitions of altruism. Therefore, the model can also account for the spread of altruism: a species can win a competition because it is altruistic. The model we use to show the effect is one of the simplest models for populations (generalised stochastic Lotka-Volterra) and we think that the effect is robust, and will be reproduced by a variety of similar models and in a range of geometries/topologies

Mitochondrial deletion mutants are bad for our muscles and our health, but in order to understand (and counteract) their expansion we might consider them an altruistic species that, driven by noise, outcompetes wildtype mitochondria. Ferdinando and Nick

Community detection in graphons

Graphs are one way to represent data sets as they arise in the social sciences, biology, and many other research domains. In these graphs, the nodes represent entities (e.g., people) and edges represent connections between them (e.g., friendships). For the analysis of graphs, a vast selection of mathematical and computational techniques has been developed. For example, Google uses eigenvectors to quantify the importance of webpages in a graph that represents the word-wide web.

Community detection


One field of interest in the analysis of graphs is the detection of so-called communities. These are groups of nodes that are strongly connected internally but only sparsely to nodes in other groups. A common example in the social sciences are friendship groups. Community-detection methods can be used to cluster large graph-structured data sets and so provide insights into many complex systems. For example, one can use the community structure to investigate how brain function is organised. In these biological graphs, the different communities represent different parts of the brain, each of which fulfilling different functions. Many algorithms to identify communities exist, the most popular of which is modularity maximisation.




As the scale of these data sets increase, the graphs that represent them get larger, too. This makes the development of computationally-efficient tools for their analysis a pressing issue. One way to investigate these large, finite objects is to abstract them with idealized infinite objects. In this context, graphons have emerged as one way of looking of infinite-sized graphs. 


One way to motivate graphons is to look at graphs that grow in size (i.e., the number of nodes) by iteratively adding nodes and edges. In Fig. 1, we show graphs of varying size in a so-called pixel image. Each pixel image visualises the edges in a graph with n nodes. The n^2 pixels in each image represent a single pair of nodes and are RED if an edge is connecting the pair of nodes and WHITE if not. With growing network size n, we observe that the pixels get smaller, making the discrete image resemble a continuous density plot. The graphon, as shown on the very right, is the continuous limiting object of such growing graphs as n goes to infinity. This intuitive notion of convergence can be concretised and proven by looking at subgraph counts.


Defining community detection for graphons


In our paper, which was just published in SIAM Applied Mathematics "Modularity maximization for graphons" (free version here) joint with Michael Schaub, we explore how one can define and compute these communities for graphons. To achieve this, we define a modularity function for graphons, which is an infinite-sized limit for the well-established modularity function for graphons. The modularity function for graphs is defined as a function of a double-sum over all pairs of edges. The core idea is, that in the infinite-size limit (i.e, for graphons) this double sum approaches a double integral, which may simplify the computational burden of community detection.


In our paper, we explore a selection of synthetic graphons and compute their community structure. Surprisingly (at least for us), the continuous form of graphons allows us to analytically compute the optimal community structure for some of these --- something that is usually not possible for graphs. Lastly, we outline a computational pipeline that would allow us to detect community structure in a privacy-preserving way (see Fig. 2). Florian and Nick.





Stochastic Survival of the Densest: defective mitochondria could be seen as altruistic to understand their expansion

With age, our skeletal muscles (e.g. muscle of our legs and arms) work less well. In some people, there is a substantial loss of strength an...