Random Walk Approximation for Stochastic Processes on Graphs.

We introduce the Random Walk Approximation (RWA), a new method to approximate the stationary solution of master equations describing stochastic processes taking place on graphs. Our approximation can be used for all processes governed by non-linear master equations without long-range interactions and with a conserved number of entities, which are typical in biological systems, such as gene regulatory or chemical reaction networks, where no exact solution exists. For linear systems, the RWA becomes the exact result obtained from the maximum entropy principle. The RWA allows having a simple analytical, even though approximated, form of the solution, which is global and easier to deal with than the standard System Size Expansion (SSE). Here, we give some theoretically sufficient conditions for the validity of the RWA and estimate the order of error calculated by the approximation with respect to the number of particles. We compare RWA with SSE for two examples, a toy model and the more realistic dual phosphorylation cycle, governed by the same underlying process. Both approximations are compared with the exact integration of the master equation, showing for the RWA good performances of the same order or better than the SSE, even in regions where sufficient conditions are not met.

The role of non-equilibrium fluxes in the relaxation processes of the linear chemical master equation.

We propose a non-equilibrium thermodynamical description in terms of the Chemical Master Equation (CME) to characterize the dynamics of a chemical cycle chain reaction among m different species. These systems can be closed or open for energy and molecules exchange with the environment, which determines how they relax to the stationary state. Closed systems reach an equilibrium state (characterized by the detailed balance condition (D.B.)), while open systems will reach a non-equilibrium steady state (NESS). The principal difference between D.B. and NESS is due to the presence of chemical fluxes. In the D.B. condition the fluxes are absent while for the NESS case, the chemical fluxes are necessary for the state maintaining. All the biological systems are characterized by their "far from equilibrium behavior," hence the NESS is a good candidate for a realistic description of the dynamical and thermodynamical properties of living organisms. In this work we consider a CME written in terms of a discrete Kolmogorov forward equation, which lead us to write explicitly the non-equilibrium chemical fluxes. For systems in NESS, we show that there is a non-conservative "external vector field" whose is linearly proportional to the chemical fluxes. We also demonstrate that the modulation of these external fields does not change their stationary distributions, which ensure us to study the same system and outline the differences in the system's behavior when it switches from the D.B. regime to NESS. We were interested to see how the non-equilibrium fluxes influence the relaxation process during the reaching of the stationary distribution. By performing analytical and numerical analysis, our central result is that the presence of the non-equilibrium chemical fluxes reduces the characteristic relaxation time with respect to the D.B. condition. Within a biochemical and biological perspective, this result can be related to the "plasticity property" of biological systems and to their capabilities to switch from one state to another as is observed during synaptic plasticity, cell fate determination, and differentiation.

Analysis of double-resonance crossing in adiabatic trapping phenomena for quasi-integrable area-preserving maps with time-dependent exciters.

In this paper we analyze the adiabatic crossing of a resonance for Hamiltonian systems when a double-resonance condition is satisfied by the linear frequency at an elliptic fixed point. We discuss in detail the phase-space structure on a class of Hamiltonians and area-preserving maps with an elliptic fixed point in the presence of a time-dependent exciter. Various regimes have been identified and carefully studied. This study extends results obtained recently for the trapping and transport phenomena for periodically perturbed Hamiltonian systems, and it could have relevant applications in the adiabatic beam splitting in accelerator physics.

Bistability in the chemical master equation for dual phosphorylation cycles.

Dual phospho/dephosphorylation cycles, as well as covalent enzymatic-catalyzed modifications of substrates are widely diffused within cellular systems and are crucial for the control of complex responses such as learning, memory, and cellular fate determination. Despite the large body of deterministic studies and the increasing work aimed at elucidating the effect of noise in such systems, some aspects remain unclear. Here we study the stationary distribution provided by the two-dimensional chemical master equation for a well-known model of a two step phospho/dephosphorylation cycle using the quasi-steady state approximation of enzymatic kinetics. Our aim is to analyze the role of fluctuations and the molecules distribution properties in the transition to a bistable regime. When detailed balance conditions are satisfied it is possible to compute equilibrium distributions in a closed and explicit form. When detailed balance is not satisfied, the stationary non-equilibrium state is strongly influenced by the chemical fluxes. In the last case, we show how the external field derived from the generation and recombination transition rates, can be decomposed by the Helmholtz theorem, into a conservative and a rotational (irreversible) part. Moreover, this decomposition allows to compute the stationary distribution via a perturbative approach. For a finite number of molecules there exists diffusion dynamics in a macroscopic region of the state space where a relevant transition rate between the two critical points is observed. Further, the stationary distribution function can be approximated by the solution of a Fokker-Planck equation. We illustrate the theoretical results using several numerical simulations.

Toward a microscopic model of bidirectional synaptic plasticity.

We show that a 2-step phospho/dephosphorylation cycle for the alpha-amino-3-hydroxy-5-methyl-4-isoxazole proprionic acid receptor (AMPAR), as used in in vivo learning experiments to assess long-term potentiation (LTP) induction and establishment, exhibits bistability for a wide range of parameters, consistent with values derived from biological literature. The AMPAR model we propose, hence, is a candidate for memory storage and switching behavior at a molecular-microscopic level. Furthermore, the stochastic formulation of the deterministic model leads to a mesoscopic interpretation by considering the effect of enzymatic fluctuations on the Michelis-Menten average dynamics. Under suitable hypotheses, this leads to a stochastic dynamical system with multiplicative noise whose probability density evolves according to a Fokker-Planck equation in the Stratonovich sense. In this approach, the probability density associated with each AMPAR phosphorylation state allows one to compute the probability of any concentration value, whereas the Michaelis-Menten equations consider the average concentration dynamics. We show that bistable dynamics are robust for multiplicative stochastic perturbations and that the presence of both noise and bistability simulates LTP and long-term depression (LTD) behavior. Interestingly, the LTP part of this model has been experimentally verified as a result of in vivo, one-trial inhibitory avoidance learning protocol in rats, that produced the same changes in hippocampal AMPARs phosphorylation state as observed with in vitro induction of LTP with high-frequency stimulation (HFS). A consequence of this model is the possibility of characterizing a molecular switch with a defined biochemical set of reactions showing bistability and bidirectionality. Thus, this 3-enzymes-based biophysical model can predict LTP as well as LTD and their transition rates. The theoretical results can be, in principle, validated by in vitro and in vivo experiments, such as fluorescence measurements and electrophysiological recordings at multiple scales, from molecules to neurons. A further consequence is that the bistable regime occurs only within certain parametric windows, which may simulate a "history-dependent threshold". This effect might be related to the Bienenstock-Cooper-Munro theory of synaptic plasticity.

Analysis of adiabatic trapping phenomena for quasi-integrable area-preserving maps in the presence of time-dependent exciters.

In this paper, results concerning the phenomenon of adiabatic trapping into resonance for a class of quasi-integrable maps and Hamiltonians with a time-dependent exciter are presented and discussed in detail. The applicability of the results about trapping efficiency for Hamiltonian systems to the maps considered is proven by using perturbation theory. This makes possible to determine explicit scaling laws for the trapping properties. These findings represent a generalization of previous results obtained for the case of quasi-integrable maps with parametric modulation, as well as an extension of the work by Neishtadt et al. [Regul. Chaotic Dyn. 18, 686 (2013)10.1134/S1560354713060087] on a restricted class of quasi-integrable systems with time-dependent exciters.

A Cationic Contrast Agent in X-ray Imaging of Articular Cartilage: Pre-Clinical Evaluation of Diffusion and Attenuation Properties.

The aim of this study was the preliminary assessment of a new cationic contrast agent, the CA4+, via the analysis of spatial distribution in cartilage of ex vivo bovine samples, at micrometer and millimeter scale. Osteochondral plugs (n = 18) extracted from bovine stifle joints (n = 2) were immersed in CA4+ solution up to 26 h. Planar images were acquired at different time points, using a microCT apparatus. The CA4+ distribution in cartilage and saturation time were evaluated. Tibial plates from bovine stifle joints (n = 3) were imaged with CT, before and after 24 h-CA4+ bath immersion, at different concentrations. Afterward, potential CA4+ washout from cartilage was investigated. From microCT acquisitions, the CA4+ distribution differentiated into three distinct layers inside the cartilage, reflecting the spatial distribution of proteoglycans. After 24 h of diffusion, the iodine concentration reached in cartilage was approximately seven times that of the CA4+ bath. The resulting saturation time was 1.9 ± 0.9 h and 2.6 ± 2.9 h for femoral and tibial samples, respectively. Analysis of clinical CT acquisitions confirmed overall contrast enhancement of cartilage after 24 h immersion, observed for each CA4+ concentration. Distinct contrast enhancement was reached in different cartilage regions, depending on tissue's local features. Incomplete but remarkable washout of cartilage was observed. CA4+ significantly improved cartilage visualization and its qualitative analysis.

Application of Wavelet Packet Transform to detect genetic polymorphisms by the analysis of inter-Alu PCR patterns.

BACKGROUND: The analysis of Inter-Alu PCR patterns obtained from human genomic DNA samples is a promising technique for a simultaneous analysis of many genomic loci flanked by Alu repetitive sequences in order to detect the presence of genetic polymorphisms. Inter-Alu PCR products may be separated and analyzed by capillary electrophoresis using an automatic sequencer that generates a complex pattern of peaks. We propose an algorithmic method based on the Haar-Walsh Wavelet Packet Transformation (WPT) for an efficient detection of fingerprint-type patterns generated by PCR-based methodologies. We have tested our algorithmic approach on inter-Alu patterns obtained from the genomic DNA of three couples of monozygotic twins, expecting that the inter-Alu patterns of each twins couple will show differences due to unavoidable experimental variability. On the contrary the differences among samples of different twins are supposed to originate from genetic variability. Our goal is to automatically detect regions in the inter-Alu pattern likely associated to the presence of genetic polymorphisms. RESULTS: We show that the WPT algorithm provides a reliable tool to identify sample to sample differences in complex peak patterns, reducing the possible errors and limits associated to a subjective evaluation. The redundant decomposition of the WPT algorithm allows for a procedure of best basis selection which maximizes the pattern differences at the lowest possible scale. Our analysis points out few classifying signal regions that could indicate the presence of possible genetic polymorphisms. CONCLUSIONS: The WPT algorithm based on the Haar-Walsh wavelet is an efficient tool for a non-supervised pattern classification of inter-ALU signals provided by a genetic analyzer, even if it was not possible to estimate the power and false positive rate due to the lacking of a suitable data base. The identification of non-reproducible peaks is usually accomplished comparing different experimental replicates of each sample. Moreover, we remark that, albeit we developed and optimized an algorithm able to analyze patterns obtained through inter-Alu PCR, the method is theoretically applicable to whatever fingerprint-type pattern obtained analyzing anonymous DNA fragments through capillary electrophoresis, and it could be usefully applied on a wide range of fingerprint-type methodologies.

A stochastic compartmental model to simulate the Covid-19 epidemic spread on a simple network.

The recent Covid-19 epidemic has pointed out the inadequacy of the plans applied by industrial countries to limit the epidemic spread and frailty of the global economy to cope with a pandemic. Many countries were forced to a global lockdown with a great socio-economic impact. In Italy, one of the problems was the complex mobility network structure of the Northern regions that made ineffective the attempts to isolate the initial hotspots. In the paper we study a simple model that simulates the epidemic spread on a community network that may exchange population according to a daily mobility rate. In each community the epidemic evolution is provided by a stochastic compartmental model whose parameters are tuned to reproduce the Covid-19 evolution observed in Italy before the global lockdown policies. We initially study the delay in the epidemic spread due to the finite local mobility by proposing a power law relation for the increasing of the infection peak time in each node and the network distance from the initial node where the epidemic starts. We consider two scenarios to study the effectiveness of local lockdown policies: the presence of two clusters weakly connected by the mobility or a homogeneous chain of communities that exchange the population at a fixed rate. In both cases we show the existence of a threshold effect, in a probabilistic sense, for the effectiveness of lockdown policies as a function of the delay time at which such policies are applied, or of the network distance from the outbreak node.

Frailness and resilience of gene networks predicted by detection of co-occurring mutations via a stochastic perturbative approach.

In recent years complex networks have been identified as powerful mathematical frameworks for the adequate modeling of many applied problems in disparate research fields. Assuming a Master Equation (ME) modeling the exchange of information within the network, we set up a perturbative approach in order to investigate how node alterations impact on the network information flow. The main assumption of the perturbed ME (pME) model is that the simultaneous presence of multiple node alterations causes more or less intense network frailties depending on the specific features of the perturbation. In this perspective the collective behavior of a set of molecular alterations on a gene network is a particularly adapt scenario for a first application of the proposed method, since most diseases are neither related to a single mutation nor to an established set of molecular alterations. Therefore, after characterizing the method numerically, we applied as a proof of principle the pME approach to breast cancer (BC) somatic mutation data downloaded from Cancer Genome Atlas (TCGA) database. For each patient we measured the network frailness of over 90 significant subnetworks of the protein-protein interaction network, where each perturbation was defined by patient-specific somatic mutations. Interestingly the frailness measures depend on the position of the alterations on the gene network more than on their amount, unlike most traditional enrichment scores. In particular low-degree mutations play an important role in causing high frailness measures. The potential applicability of the proposed method is wide and suggests future development in the control theory context.

Gut microbiota ecology: Biodiversity estimated from hybrid neutral-niche model increases with health status and aging.

In this work we propose an index to estimate the gut microbiota biodiversity using a modeling approach with the aim of describing its relationship with health and aging. The gut microbiota, a complex ecosystem that links nutrition and metabolism, has a pervasive effect on all body organs and systems, undergoes profound changes with age and life-style, and substantially contributes to the pathogenesis of age-related diseases. For these reasons, the gut microbiota is a suitable candidate for assessing and quantifying healthy aging, i.e. the capability of individuals to reach an advanced age, avoiding or postponing major age-related diseases. The importance of the gut microbiota in health and aging has been proven to be related not only to its taxonomic composition, but also to its ecological properties, namely its biodiversity. Following an ecological approach, here we intended to characterize the relationship between the gut microbiota biodiversity and healthy aging through the development a parsimonious model of gut microbiota from which biodiversity can be estimated. We analysed publicly available metagenomic data relative to subjects of different ages, countries, nutritional habits and health status and we showed that a hybrid niche-neutral model well describes the observed patterns of bacterial relative abundance. Moreover, starting from such ecological modeling, we derived an estimate of the gut microbiota biodiversity that is consistent with classical indices, while having a higher statistical power. This allowed us to unveil an increase of the gut microbiota biodiversity during aging and to provide a good predictor of health status in old age, dependent on life-style and aging disorders.

Dynamical model for sympatric speciation in an ecological niche.

The speciation phenomenon is the process used by the evolution to allow populations to become distinct species. The speciation is the primary cause of the complexity of the ecological network. Sympatric speciation concerns the rise of a new species from a surviving ancestral species while both continue to inhabit the same ecological niche or geographical region. In sympatric speciation, reproductive isolation evolves within a population in an ecological niche without the aid of geographic barriers. Different models have been proposed for alternative modes of sympatric speciation. The most popular was first put forward by John Maynard Smith in 1966 who suggested that in a given population homozygous individuals may, under particular environmental conditions, have a greater fitness than those with alleles heterozygous for a certain trait, eventually leading to speciation in the population. In this framework we assume an effective description of the speciation process based on a dynamical model for the populations in an ecological system. Our basic assumption is the existence of an ancestral population in an ecological niche that can express two phenotypes. In presence of certain environmental conditions one of the phenotypes has the propensity to separate from the original population in the reproduction process. Then new individuals may give rise to a new species in the ecosystem realizing a sympatric speciation. Due to the finite resources in the niche the populations are continuously competing each other's, and their numerousness fluctuates according to the changes of the environmental conditions. The effect of natural selection is introduced in the model by stochastic perturbations, that decrease the reproduction rate of the populations in the niche. We show some the dynamical properties of the system and we prove the existence of a threshold values in the environmental stress in order to observe the speciation process. We also discuss some biological implications of the model and the validation problem using empirical data.

Master equation and relative species abundance distribution for Lotka-Volterra models of interacting ecological communities.

DESCRIPTION: Understanding the factors that control the dynamics of interacting species is a fundamental problem in ecology. The nature of the interactions among different species is usually not completely understood, but it is assumed that the species interaction plays an important role in the ecosystem properties as predicted by the niches models for an ecosystem. However, recent studies point out as the neutral hypothesis proposed by Hubbell of non-interacting species with an external source from the surrounding environment, allows to explain the relative species abundance distribution when the ecosystem has reached a stationary situation. In this paper we use the concept of fitness landscape to introduce a random dynamical model that describes the evolution of different communities near a stationary situation. The average dynamics can be associated to a system of Lotka-Volterra equations with mutualistic interactions. Then we derive a Master equation that satisfies the detailed balance condition of thermodynamical equilibria and allows to analytically compute the relative species abundance distribution near the stationary state as a multinomial negative distribution. These results suggest a possible approach to a synthetic theory that joins the niche theories and the Hubbell's neutral theory for RSA distribution.

A stochastic model of randomly accelerated walkers for human mobility.

Recent studies of human mobility largely focus on displacements patterns and power law fits of empirical long-tailed distributions of distances are usually associated to scale-free superdiffusive random walks called Lévy flights. However, drawing conclusions about a complex system from a fit, without any further knowledge of the underlying dynamics, might lead to erroneous interpretations. Here we show, on the basis of a data set describing the trajectories of 780,000 private vehicles in Italy, that the Lévy flight model cannot explain the behaviour of travel times and speeds. We therefore introduce a class of accelerated random walks, validated by empirical observations, where the velocity changes due to acceleration kicks at random times. Combining this mechanism with an exponentially decaying distribution of travel times leads to a short-tailed distribution of distances which could indeed be mistaken with a truncated power law. These results illustrate the limits of purely descriptive models and provide a mechanistic view of mobility.

Evolution: geometrical and dynamical aspects.

We develop a possible axiomatic approach to the evolution theory that has been previously discussed in Freguglia [2002]. The axioms synthesize the fundamental ideas of evolution theory and allow a geometrical and dynamical interpretation of the generation law. Using the axioms we derive a simple reaction-diffusion model which introduces the species as self-organized stationary distribution of a finite population and simulates the evolution of a phenotypic character under the effect of an external perturbing action. The dynamical properties of the model are briefly presented using numerical simulations.

Role of connectivity in immune and neural network models: memory development and aging.

In this paper we analyzed how connectivity (defined as number of connections between network elements) can affect the memory capacity of a network-based model of the Immune System (IS) and of a model of the Nervous System (NS) synaptic plasticity (BCM model). The key point is the concept of competition between the characteristic variables that represent the response of such systems to environmental stimuli: the clonal concentrations for the IS, and the neuron responses for the BCM model. The memory states of both systems are characterized by a high selectivity to specific input patterns, reflecting a similar behaviour of their development rules. This selectivity property of memory states can be controlled by changing the degree of the internal connectivity in each system. We can explain the changes occurring in IS memory states during lifespan as due to a reshaping of its internal connectivity. This assumption is in agreement with experimental observations, reporting an increase of IS memory cells during lifespan. The change of connectivity in the BCM model leads to the introduction of quasilocal variables governing the plasticity of groups of synaptic junctions. This could be interpreted as the result of a refinement of neuron internal mechanisms during development, or it could be seen as a different learning rule deriving from the original BCM theory. We argue that connectivity seems to play an important role in a large class of biological systems controlled by competition mechanisms. Moreover, changes in connectivity may lead to changes in memory properties during development and aging.

Dynamics on genes network structures. An ago-antagonist approach.

This paper is a carrying on of the theme examined in (Bazzani, Freguglia 2013) where we discuss a proposal of studying of essential structural aspects of Darwinian Evolution Theory. Also in this case we apply a mathematical ago-antagonist theory inspired by Y. Cherruault's ideas (Cherruault 1998). In the present paper we consider the network structure of genes activity and its dynamics.

Analysis of adiabatic trapping for quasi-integrable area-preserving maps.

Trapping phenomena involving nonlinear resonances have been considered in the past in the framework of adiabatic theory. Several results are known for continuous-time dynamical systems generated by Hamiltonian flows in which the combined effect of nonlinear resonances and slow time variation of some system parameters is considered. The focus of this paper is on discrete-time dynamical systems generated by two-dimensional symplectic maps. The possibility of extending the results of neo-adiabatic theory to quasi-integrable area-preserving maps is discussed. Scaling laws are derived, which describe the adiabatic transport as a function of the system parameters using a probabilistic point of view. These laws can be particularly relevant for physical applications. The outcome of extensive numerical simulations showing the excellent agreement with the analytical estimates and scaling laws is presented and discussed in detail.

Stochastic approach to diffusion inside the chaotic layer of a resonance.

We model chaotic diffusion in a symplectic four-dimensional (4D) map by using the result of a theorem that was developed for stochastically perturbed integrable Hamiltonian systems. We explicitly consider a map defined by a free rotator (FR) coupled to a standard map (SM). We focus on the diffusion process in the action I of the FR, obtaining a seminumerical method to compute the diffusion coefficient. We study two cases corresponding to a thick and a thin chaotic layer in the SM phase space and we discuss a related conjecture stated in the past. In the first case, the numerically computed probability density function for the action I is well interpolated by the solution of a Fokker-Planck (FP) equation, whereas it presents a nonconstant time shift with respect to the concomitant FP solution in the second case suggesting the presence of an anomalous diffusion time scale. The explicit calculation of a diffusion coefficient for a 4D symplectic map can be useful to understand the slow diffusion observed in celestial mechanics and accelerator physics.

Ago-antagonist theory in Darwinian evolution.

In this paper we discuss a proposal on the essential structural aspects of Darwinian Evolution Theory. Using this point of view we apply a mathematical ago-antagonist theory inspired by Y. Cherruault's (1998) ideas, which we have extended. In the ago-antagonist model, the phenotype characters measure the individual propensity to perform an innovative x(t) (agonist) or conservative y(t) (antagonist) action with respect to mutation and to speciation process. We have mathematically introduced the conflict concept and we present a model that takes into account the environmental effects by means of a stochastic multiplicative process. We shortly discuss the properties of the related stochastic differential equations.