Draft:Stochastic survival of the densest

Stochastic Survival of the Densest, often referred to as SSD is an evolutionary mechanism that describes the outcomes of the competition between two biological species, under specific hypotheses. Coined by researchers Insalata, Hoitzing, Aryaman, and Jones in a scientific article published in the Proceedings of the National Academy of Sciences (PNAS) in 2022, SSD provides insights into the clonal expansion of deletion mutant mitochondrial DNA in skeletal muscle fibers (see Section Biological background), and more widely it contributes to our understanding of evolutionary dynamics in biological systems.

The term's foundation lies in the evolutionary principle of the survival of the fittest, used to describe the process by which organisms that are better adapted to their environment ("fittest") are more likely to survive and reproduce, passing their advantageous traits to future generations. The addition of the adjective "stochastic" reflects the integration of noise into the proposed model, a characteristic intrinsic to biological processes. The concept of densest is defined so as to characterize the species whose population size sustained by the environment (skeletal muscle fibers, in this case), given food, habitat, and available resources, is maximum (see also carrying capacity).

Biological background
Mitochondria are organelles contained in our cells and essential for their lives. They have specific genetic material, termed mtDNA, which carries all the information required for new cells to proliferate and maintain their essential functions. However, when cells replicate, something might go wrong and the genetic material transmitted to progeny may differ from the parental one: this event is called mutation. In this context, the standard mtDNA is referred to as "wild-type", while the one that underwent changes is referred to as "mutant". Between several types of mutations, some imply the removal of an entire chunk of the genetic material and are therefore called deletions.

Notably, in skeletal muscle fibers, to minimize damage accumulation due to mutations and to support cellular functionality even under stress, mitochondria form a dynamic network through fusion and fission between them. Yet, deletion mutants can overcome this protection mechanism and undergo clonal expansion, i.e., one single type of mutant prevails randomly and expands in a wave-like manner (see Section Comparison to experimental data), contributing to cellular dysfunction. It is also observed that, upon clonal expansion, the regions of muscle fibers dominated by mutants have a higher number of mtDNA copies than regions where wild-type mtDNA prevails.

The spread of these mitochondrial deletion mutants, which adversely affect the functionality of skeletal muscles, is intricately related to the aging process. In particular,  they are involved in skeletal muscle aging, a phenomenon known as sarcopenia, which poses challenges to maintaining an active and healthy lifestyle in older individuals.

Earlier explanations attributed a replicative advantage to deletions,  but there is evidence suggesting that cells have the ability to selectively eliminate mitochondrial DNA that has undergone deleterious mutations. Thus, the appropriate framework appeared to be a model where deletion mutants do not have any explicit advantage over non-mutants, termed neutral model. Several such models have been proposed;  however, the parameters used in these do not correspond to values observed in nature. Thus, despite extensive research, understanding how mtDNA mutants outcompete wild-type ones has remained elusive for years.

Unlike previous theories, SSD does not assume a replicative advantage for mutants (i.e., a more effective replication, leading to a faster replication, or a higher number of offspring, for instance) and even considers the possibility of a higher degradation rate for mutants (i.e., mutants are degraded and removed from the cell faster than wild-types), compared to non-mutants, thus modeling also their preferential elimination. Further, the study's stochastic model predicts a noise-driven clonal wave of mitochondrial mutant expansion, aligning with both qualitative and quantitative experimental observations (see Section Comparison to experimental data). In particular, the fact that regions of muscle fibers dominated by mutants have a higher number of mtDNA copies    is accounted for in the fact that, in the model, the mutant mtDNA is the densest species.

The model and its key mechanism
The model is based on a generalization of the stochastic Lotka-Volterra equations, describing the dynamics of the interactions between two populations in a biological system. Mutants and wild-types have the same degradation and growth rate, implying there is a neutral competition between the two. The degradation rate is a constant, $$\mu$$, while the growth rate is $$\lambda = \mu + c(N-w-\delta m)$$. $$N$$ is the target population, i.e., the maximum number of wild-type individuals that can live in the skeletal muscle fiber. In the system, we have $$w$$ wild-types and $$m$$ mutants. $$c$$ is a parameter introduced to mirror a biological mechanism involved in replication (i.e., to quantify the control of the nucleus in the replication process), while $$\delta$$ is a parameter that takes values between 0 and 1 and it is the one giving rise to the mutants being the densest species (for details, see Ref. ). The model also includes spatial structure, simulating individuals living in nearby regions of muscle fiber and allowing migrations between them.

The crux of the model lies in the mutants' ability to thrive at higher densities within muscle fibers. Coupled with the stochastic nature of biological processes, which is intrinsic to the model, SSD predicts a wave-like expansion of mutants. Remarkably, the effect is purely stochastic; removing noise eliminates the observed expansion. Even in a neutral model, accounting for the three key characteristics of: noise, spatial structure, and mutant density, the stochastic model aligns with experimental data.

Comparison to experimental data
Experimental data reveal a wave-like profile in the spatial expansion of deletion mutant mtDNA. The study observes distinct regions with a high number of mutants, separated by transition zones to regions where the mutant count quickly diminishes to zero.

The profile mentioned above is mathematically modeled by a specific function known as asigmoid, representing the characteristic shape of a traveling wave. In simulation results from the SSD model, the transition region moves to the right as time passes, showing that there is a wave-like expansion in space over time. The parameters derived from the model align closely with experimental data from previous studies investigating the expansion of mtDNA mutants in rats and humans.

Evolutionary implications
In traditional competition scenarios between different species, the prevailing and intuitive notion is that the winning species reproduces more frequently or the losing species faces higher mortality rates. This idea sometimes led to a negative portrayal of the survival of the fittest, contributing to the linkage of evolutionary theory with social Darwinism and eugenics.

In the SSD model, a species can emerge victorious by utilizing fewer resources and thriving at higher population densities within the system. Notably, a species exhibiting resource efficiency can be perceived as altruistic in the biological sense of the term - altruism denotes actions or behaviors exhibited by an individual that enhance the fitness of another individual, even at the cost of reducing the actor's own fitness -, broadening the model's applicability to explain the propagation of altruistic traits in competitive environments.