Relevant elments, Magnetization and Dynamical Properties in Kauffman Networks: a Numerical Study

This is the first of two papers about the structure of Kauffman networks. In this paper we define the relevant elements of random networks of automata, following previous work by Flyvbjerg and Flyvbjerg and Kjaer, and we study numerically their probability distribution in the chaotic phase and on the critical line of the model. A simple approximate argument predicts that their number scales as sqrt(N) on the critical line, while it is linear with N in the chaotic phase and independent of system size in the frozen phase. This argument is confirmed by numerical results. The study of the relevant elements gives useful information about the properties of the attractors in critical networks, where the pictures coming from either approximate computation methods or from simulations are not very clear.Comment: 22 pages, Latex, 8 figures, submitted to Physica

Structurally constrained protein evolution: results from a lattice simulation

We simulate the evolution of a protein-like sequence subject to point mutations, imposing conservation of the ground state, thermodynamic stability and fast folding. Our model is aimed at describing neutral evolution of natural proteins. We use a cubic lattice model of the protein structure and test the neutrality conditions by extensive Monte Carlo simulations. We observe that sequence space is traversed by neutral networks, i.e. sets of sequences with the same fold connected by point mutations. Typical pairs of sequences on a neutral network are nearly as different as randomly chosen sequences. The fraction of neutral neighbors has strong sequence to sequence variations, which influence the rate of neutral evolution. In this paper we study the thermodynamic stability of different protein sequences. We relate the high variability of the fraction of neutral mutations to the complex energy landscape within a neutral network, arguing that valleys in this landscape are associated to high values of the neutral mutation rate. We find that when a point mutation produces a sequence with a new ground state, this is likely to have a low stability. Thus we tentatively conjecture that neutral networks of different structures are typically well separated in sequence space. This results indicates that changing significantly a protein structure through a biologically acceptable chain of point mutations is a rare, although possible, event.Comment: added reference, to appear on European Physical Journal

Replica-symmetry breaking in dynamical glasses

Systems of globally coupled logistic maps (GCLM) can display complex collective behaviour characterized by the formation of synchronous clusters. In the dynamical clustering regime, such systems possess a large number of coexisting attractors and might be viewed as dynamical glasses. Glass properties of GCLM in the thermodynamical limit of large system sizes $N$ are investigated. Replicas, representing orbits that start from various initial conditions, are introduced and distributions of their overlaps are numerically determined. We show that for fixed-field ensembles of initial conditions, as used in previous numerical studies, all attractors of the system become identical in the thermodynamical limit up to variations of order $1/\sqrt{N}$ because the initial value of the coupling field is characterized by vanishing fluctuations, and thus replica symmetry is recovered for $N\to \infty$. In contrast to this, when random-field ensembles of initial conditions are chosen, replica symmetry remains broken in the thermodynamical limit.Comment: 19 pages, 18 figure

Biodiversity in model ecosystems, II: Species assembly and food web structure

This is the second of two papers dedicated to the relationship between population models of competition and biodiversity. Here we consider species assembly models where the population dynamics is kept far from fixed points through the continuous introduction of new species, and generalize to such models thecoexistence condition derived for systems at the fixed point. The ecological overlap between species with shared preys, that we define here, provides a quantitative measure of the effective interspecies competition and of the trophic network topology. We obtain distributions of the overlap from simulations of a new model based both on immigration and speciation, and show that they are in good agreement with those measured for three large natural food webs. As discussed in the first paper, rapid environmental fluctuations, interacting with the condition for coexistence of competing species, limit the maximal biodiversity that a trophic level can host. This horizontal limitation to biodiversity is here combined with either dissipation of energy or growth of fluctuations, which in our model limit the length of food webs in the vertical direction. These ingredients yield an effective model of food webs that produce a biodiversity profile with a maximum at an intermediate trophic level, in agreement with field studies

Mathematical model of SARS-Cov-2 propagation versus ACE2 fits COVID-19 lethality across age and sex and predicts that of SARS, supporting possible therapy

The fatality rate of Covid-19 escalates with age and is larger in men than women. I show that these variations correlate strongly with the level of the viral receptor protein ACE2 in rat lungs, which is consistent with the still limited and apparently contradictory data on human ACE2. Surprisingly, lower levels of the receptor correlate with higher fatality. However, a previous mathematical model predicts that the speed of viral progression in the organism has a maximum and then declines with the receptor level. Moreover, many manifestations of severe CoViD-19, such as severe lung injury, exacerbated inflammatory response and thrombotic problems may derive from increased Angiotensin II (Ang-II) level that results from degradation of ACE2 by the virus. I present here a mathematical model based on the influence of ACE2 on viral propagation and disease severity. The model fits Covid-19 fatality rate across age and sex with high accuracy ($r^2>0.9$) under the hypothesis that SARS-CoV-2 infections are in the dynamical regimes in which increased receptor slows down viral propagation. Moreover, rescaling the model parameters by the ratio of the binding rates of the spike proteins of SARS-CoV and SARS-CoV-2 allows predicting the fatality rate of SARS-CoV across age and sex, thus linking the molecular and epidemiological levels. The presented model opposes the fear that angiotensin receptor blockers (ARB), suggested as a therapy against the most adverse effects of CoViD-19, may favour viral propagation, and suggests that Ang-II and ACE2 are candidate prognostic factors for detecting population that needs stronger protection.Comment: 1 figur