時弘 哲治

Abstract We introduce a three-dimensional mathematical model for the dynamics of vascular endothelial cells during sprouting angiogenesis. Angiogenesis is the biological process by which new blood vessels form from existing ones. It has been the subject of numerous theoretical models. These models have successfully replicated various aspects of angiogenesis. Recent studies using particle-based models have highlighted the significant influence of cell shape on network formation, with elongated cells contributing to the formation of branching structures. While most mathematical models are two-dimensional, we aim to investigate whether ellipsoids also form branch-like structures and how their shape affects the pattern. In our model, the shape of a vascular endothelial cell is represented as a spheroid, and a discrete dynamical system is constructed based on the simple assumption of two-body interactions. Numerical simulations demonstrate that our model reproduces the patterns of elongation and branching observed in the early stages of angiogenesis. We show that the pattern formation of the cell population is strongly dependent on the cell shape. Finally, we demonstrate that our current mathematical model reproduces the cell behaviours, specifically cell-mixing, observed in sprouts.

We discuss the characteristics of the patterns of the vascular networks in a mathematical model for angiogenesis. Based on recent in vitro experiments, this mathematical model assumes that the elongation and bifurcation of blood vessels during angiogenesis are determined by the density of endothelial cells at the tip of the vascular network, and describes the dynamical changes in vascular network formation using a system of simultaneous ordinary differential equations. The pattern of formation strongly depends on the supply rate of endothelial cells by cell division, the branching angle, and also on the connectivity of vessels. By introducing reconnection of blood vessels, the statistical distribution of the size of islands in the network is discussed with respect to bifurcation angles and elongation factor distributions. The characteristics of the obtained patterns are analysed using multifractal dimension and other techniques.

Recent studies have suggested that thaw-aging can improve sensory attributes of freeze-thawed meat. Acceleration of proteolysis is expected to promote tenderisation and improve taste; however, the details of protein degradation, including substrate proteins and cleavage sites, remain unclear. Here, we report a time course overview of the peptidome of beef short plates during thaw-aging. The accelerated degradation of key proteins for meat tenderisation, such as troponin T and desmin, was confirmed. Additionally, 11 cleavage sites in troponin T related to taste-active peptide generation were identified. Terminome analysis showed that the contribution of each protease varies depending on the substrate proteins and the thaw-aging period. Based on our results; proteases, not only calpains, but also others contributed to the degradation of myofibrillar proteins. The techniques employed indicate that meat proteolysis during thaw-aging is not constant but dynamic.

We introduce an equation defined on a multi-dimensional lattice, which can be considered as an extension to the coprimeness-preserving discrete KdV like equation in our previous paper. The equation is also interpreted as a higher-dimensional analog of the Hietarinta-Viallet equation, which is famous for its singularity confining property while having an exponential degree growth. As the main theorem, we prove the Laurent and the irreducibility properties of the equation in its "tau-function " form. From the theorem, the coprimeness of the equation follows. In Appendixes A-D, we review the coprimeness-preserving discrete KdV like equation, which is a base equation for our main system, and prove the properties such as the coprimeness.

The rule 184 fuzzy cellular automaton is regarded as a mathematical model of traffic flow because it contains the two fundamental traffic flow models, the rule 184 cellular automaton and the Burgers equation, as special cases. We show that the fundamental diagram (flux-density diagram) of this model consists of three parts: a free-flow part, a congestion part and a two-periodic part. The two-periodic part, which may correspond to the synchronized mode region, is a two-dimensional area in the diagram, the boundary of which consists of the free-flow and the congestion parts. We prove that any state in both the congestion and the two-periodic parts is stable, but is not asymptotically stable, while that in the free-flow part is unstable. Transient behaviour of the model and bottle-neck effects are also examined by numerical simulations. Furthermore, to investigate low or high density limit, we consider ultradiscrete limit of the model and show that any ultradiscrete state turns to a travelling wave state of velocity one in finite time steps for generic initial conditions.

Vascular endothelial cells (ECs) in angiogenesis exhibit inhomogeneous collective migration called "cell mixing", in which cells change their relative positions by overtaking each other. However, how such complex EC dynamics lead to the formation of highly ordered branching structures remains largely unknown. To uncover hidden laws of integration driving angiogenic morphogenesis, we analyzed EC behaviors in an in vitro angiogenic sprouting assay using mouse aortic explants in combination with mathematical modeling. Time-lapse imaging of sprouts extended from EC sheets around tissue explants showed directional cohesive EC movements with frequent U-turns, which often coupled with tip cell overtaking. Imaging of isolated branches deprived of basal cell sheets revealed a requirement of a constant supply of immigrating cells for ECs to branch forward. Anisotropic attractive forces between neighboring cells passing each other were likely to underlie these EC motility patterns, as evidenced by an experimentally validated mathematical model. These results suggest that cohesive movements with anisotropic cell-to-cell interactions characterize the EC motility, which may drive branch elongation depending on a constant cell supply. The present findings provide novel insights into a cell motility-based understanding of angiogenic morphogenesis.

In healthy cells, proteolysis is orderly executed to maintain basal homeostasis and normal physiology. Dyscontrol in proteolysis under severe stress condition induces cell death, but the dynamics of proteolytic regulation towards the critical phase remain unclear. Teleosts have been suggested an alternative model for the study of proteolysis under severe stress. In this study, horse mackerel (Trachurus japonicus) was used and exacerbated under severe stress conditions due to air exposure. Although the complete genome for T. japonicus is not available, a transcriptomic analysis was performed to construct a reference protein database, and the expression of 72 proteases were confirmed. Quantitative peptidomic analysis revealed that proteins related to glycolysis and muscle contraction systems were highly cleaved into peptides immediately under the severe stress. Novel analysis of the peptide terminome using a multiple linear regression model demonstrated profiles of proteolysis under severe stress. The results indicated a phase transition towards dyscontrol in proteolysis in T. japonicus skeletal muscle during air exposure. Our novel approach will aid in investigating the dynamics of proteolytic regulation in skeletal muscle of non-model vertebrates.

We introduce a class of recursions defined over the d-dimensional integer lattice. The discrete equations we study are interpreted as higher dimensional extensions to the discrete Toda lattice equation. We shall prove that the equations satisfy the coprimeness property, which is one of integrability detectors analogous to the singularity confinement test. While the degree of their iterates grows exponentially, their singularities exhibit a nature similar to that of integrable systems in terms of the coprimeness property. We also prove that the equations can be expressed as mutations of a seed in the sense of the Laurent phenomenon algebra.

Coprimeness property was introduced to study the singularity structure of discrete dynamical systems. In this paper we shall extend the coprimeness property and the Laurent property to further investigate discrete equations with complicated pattern of singularities. As examples we study extensions to the Somos-4 recurrence and the 2D discrete Toda equation. By considering their non-autonomous polynomial forms, we prove that their tau function analogues possess the extended Laurent property with respect to their initial variables and some extra factors related to the non-autonomous terms. Using this Laurent property, we prove that these equations satisfy the extended coprimeness property. This coprimeness property reflects the singularities that trivially arise from the equations.

We consider a two dimensional extension of the so-called linearizable mappings. In particular, we start from the Heideman-Hogan recurrence, which is known as one of the linearizable Somos-like recurrences, and introduce one of its two dimensional extensions. The two dimensional lattice equation we present is linearizable in both directions, and has the Laurent and the coprimeness properties. Moreover, its reduction produces a generalized family of the Heideman-Hogan recurrence. Higher order examples of two dimensional linearizable lattice equations related to the Dana Scott recurrence are also discussed.

We investigate an integrate and fire model for two cardiomyocytes interacting with each other. A feature of the model is to incorporate the refractory periods of the cardiomyocytes as well as the influence of firing of adjacent cells. The present model predicts that, if refractory periods of the two cells are nearly equal, the beating rhythms of the two cells always synchronize and their beating rate is tuned to the faster rate between the two cells. On the other hand, if their refractory periods significantly differ, they exhibit various kinds of harmonious beating rhythms. These results successfully explain the well known characteristics of synchronized beating of cultured cardiomyocytes. We also discuss effects of a delay time of cell-to-cell interaction, that gives further complicated phase diagrams for the beating rhythms. (C) 2017 Elsevier Ltd. All rights reserved.

In this article we investigate the coprimeness properties of one and two-dimensional discrete equations, in a situation where the equations are decomposable into several factors of polynomials. After experimenting on a simple equation, we shall focus on some higher power extensions of the Somos-4 equation and the (1-dimensional) discrete Toda equation. Our previous results are that all of the equations satisfy the irreducibility and the coprimeness properties if the r.h.s. is not factorizable. In this paper we shall prove that the coprimeness property still holds for all of these equations even if the r.h.s. is factorizable, although the irreducibility property is no longer satisfied.

The community effect of cardiomyocytes was investigated in silico by the change in number and features of cells, as well as configurations of networks. The theoretical model was based on experimental data and accurately reproduced recently published experimental results regarding coupled cultured cardiomyocytes. We showed that the synchronised beating of two coupled cells was tuned not to the cell with a faster beating rate, but to the cell with a more stable rhythm. In a network of cardiomyocytes, a cell with low fluctuation, but not a hight frequency, became a pacemaker and stabilised the beating rhythm. Fluctuation in beating rapidly decreased with an increase in the number of cells (N), almost irrespective of the configuration of the network, and a cell comes to have natural and stable beating rhythms, even for N of approximately 10. The universality of this community effect lies in the fluctuation-dissipation theorem in statistical mechanics.

We introduce a so-called coprimeness-preserving non-integrable extension to the two-dimensional Toda lattice equation. We believe that this equation is the first example of such discrete equations defined over a three-dimensional lattice. We prove that all the iterates of the equation are irreducible Laurent polynomials of the initial data and that every pair of two iterates is co-prime, which indicate confined singularities of the equation. By reducing the equation to two- or one-dimensional lattices, we obtain coprimeness-preserving non-integrable extensions to the one-dimensional Toda lattice equation and the Somos-4 recurrence. Published by AIP Publishing.

We present a quasi-integrable two-dimensional lattice equation: i.e., a partial difference equation which satisfies a test for integrability, singularity confinement, although it has a chaotic aspect in the sense that the degrees of its iterates exhibit exponential growth. By systematic reduction to one-dimensional systems, it gives a hierarchy of ordinary difference equations with confined singularities, but with positive algebraic entropy including a generalized form of the Hietarinta-Viallet mapping. We believe that this is the first example of such quasi-integrable equations defined over a two-dimensional lattice.

概要．血管新生を記述する単純な数理モデルを考察する．最近の血管新生における内皮細胞運動のタイムラプス撮影の実験を参考に，内皮細胞の新生血管先端での密度のみにより，新生血管の伸長と分岐が定まることを仮定すると，非線形連立常微分方程式によって記述されるモデルが導かれる．細胞分裂および VEGFなどの内皮細胞の運動を活性化する因子の影響も取り入れ，厳密解および数値シミュレーションの結果を示す．

The periodic discrete Toda equation defined over finite fields has been studied. We obtained the finite graph structures constructed by the network of states where edges denote possible time evolutions. We simplify the graphs by introducing a equivalence class of cyclic permutations to the initial values. We proved that the graphs are bi-directional and that they are composed of several arrays of complete graphs connected at one of their vertices. The condition for the graphs to be bi-directional is studied for general discrete equations.

Angiogenesis is the morphogenetic phenomenon in which new blood vessels emerge from an existing vascular network and configure a new network. In consideration of recent experiments with time-lapse fluorescent imaging in which vascular endothelial cells exhibit cell-mixing behavior even at a tip of newly generated vascular networks, we propose a discrete mathematical model for the dynamics of vascular endothelial cells in angiogenic morphogenesis. The model incorporates two-body interaction between endothelial cells which induces cell-mixing behavior and length of the generating blood vessel shows temporal power-law scaling behavior. Numerical simulation of the model successfully reproduces elongation and bifurcation of blood vessels in the early stage of angiogenesis.

We introduce a series of discrete mappings, which is considered to be an extension of the Hietarinta-Viallet mapping with one parameter. We obtain the algebraic entropy for this mapping by obtaining the recurrence relation for the degrees of the iterated mapping. For some parameter values the mapping has a confined singularity, in which case the mapping is equivalent to a recurrence relation between six irreducible polynomials. For other parameter values, the mapping does not pass the singularity confinement test. The properties of irreducibility and co-primeness of the terms play crucial roles in the discussion.

The pH-induced conformational changes of proteins are systematically studied in the framework of a hydrophobic-polar (HP) model, in which proteins are dramatically simplified as chains of hydrophobic (H) and polar (P) beads on a lattice. We express the electrostatic interaction, the principal driving force of pH-induced unfolding that is not included in the conventional HP model, as the repulsive energy term between P monomers. As a result of the exact enumeration of all of the 14- to 18-mers, it is found that lowest-energy states in many sequences change from single "native" conformations to multiple sets of "denatured" conformations with an increase in the electrostatic repulsion. The switching of the lowest-energy states occurs in quite a similar way to real proteins: it is almost always between two states, while in a small fraction of >= 16-mers it is between three states. We also calculate the structural fluctuations for all of the denatured states and find that the denatured states contain a broad range of incompletely unfolded conformations, similar to "molten globule" states referred to in acid or alkaline denatured real proteins. These results show that the proposed model provides a simple physical picture of pH-induced protein denaturation.

We propose the analysis of a general coupled system of semidiscrete (differential-difference form) Korteweg-de Vries equations. Its integrability is proved using bilinear Backlund transformations and Lax pairs. Starting from the Hirota bilinear form, we construct integrable discretizations (in difference-difference form) having two different nonlinear forms. Motivation is also given by showing that a modular genetic network with activation and repression coupling between genes can be modeled by such a system.

We reformulate the singularity confinement, which is one of the most famous integrability criteria for discrete equations, in terms of the algebraic properties of the general terms of the discrete Toda equation. We show that the coprime property, which has been introduced in our previous paper as one of the integrability criteria, is appropriately formulated and proved for the discrete Toda equation. We study three types of boundary conditions (semi-infinite, molecule, periodic) for the discrete Toda equation, and prove that the same coprime property holds for all the types of boundaries. (C) 2015 AIP Publishing LLC.

We study the Laurent property, the irreducibility and co-primeness of discrete integrable and non-integrable equations. First we study a discrete integrable equation related to the Somos-4 sequence, and also a non-integrable equation as a comparison. We prove that the conditions of irreducibility and co-primeness hold only in the integrable case. Next, we generalize our previous results on the singularities of the discrete Korteweg-de Vries (dKdV) equation. In our previous paper (Kanki et al 2014 J. Phys. A: Math. Theor. 47 065201) we described the singularity confinement test (one of the integrability criteria) using the Laurent property, and the irreducibility, and co-primeness of the terms in the bilinear dKdV equation, in which we only considered simplified boundary conditions. This restriction was needed to obtain simple (monomial) relations between the bilinear form and the nonlinear form of the dKdV equation. In this paper, we prove the co-primeness of the terms in the nonlinear dKdV equation for general initial conditions and boundary conditions, by using the localization of Laurent rings and the interchange of the axes. We assert that co-primeness of the terms can be used as a new integrability criterion, which is a mathematical re-interpretation of the confinement of singularities in the case of discrete equations.

We study the distribution of singularities for partial difference equations, in particular, the bilinear and nonlinear form of the discrete version of the Korteweg-de Vries (dKdV) equation. Using the Laurent property, and the irreducibility, and co-primeness of the terms of the bilinear dKdV equation, we clarify the relationship of these properties with the appearance of zeros in the time evolution. The results are applied to the nonlinear dKdV equation and we formulate the famous integrability criterion (singularity confinement test) for nonlinear partial difference equations with respect to the co-primeness of the terms.

We investigate the discrete Painleve equations (dP(II) and qP(II)) over finite fields. We first show that they are well defined by extending the domain according to the theory of the space of initial conditions. Then we treat them over local fields and observe that they have a property that is similar to the good reduction of dynamical systems over finite fields. We can use this property, which can be interpreted as an arithmetic analogue of singularity confinement, to avoid the indeterminacy of the equations over finite fields and to obtain special solutions from those defined originally over fields of characteristic zero.

We investigate the discrete Painleve II equation over finite fields. We treat it over local fields and observe that it has a property that is similar to the good reduction over finite fields. We can use this property, which seems to be an arithmetic analogue of singularity confinement, to avoid the indeterminacy of the equations over finite fields and to obtain special solutions from those defined originally over fields of characteristic zero.

We investigate the multi-soliton solutions to the generalized discrete KdV equation. In some cases a soliton with smaller amplitude moves faster than that with larger amplitude unlike the soliton solutions of the KdV equation. This phenomenon is intuitively understood from its ultradiscrete limit, where the system turns to the box ball system with a carrier.

We all use path routing everyday as we take shortcuts to avoid traffic jams, or by using faster traffic means. Previous models of traffic flow of RNA polymerase II (RNAPII) during transcription, however, were restricted to one dimension along the DNA template. Here we report the modeling and application of traffic flow in transcription that allows preferential paths of different dimensions only restricted to visit some transit points, as previously introduced between the 5 ' and 3 ' end of the gene. According to its position, an RNAPII protein molecule prefers paths obeying two types of time-evolution rules. One is an asymmetric simple exclusion process (ASEP) along DNA, and the other is a three-dimensional jump between transit points in DNA where RNAPIIs are staying. Simulations based on our model, and comparison experimental results, reveal how RNAPII molecules are distributed at the DNA-loop-formation-related protein binding sites as well as CTCF insulator proteins (or exons). As time passes after the stimulation, the RNAPII density at these sites becomes higher. Apparent far-distance jumps in one dimension are realized by short-range three-dimensional jumps between DNA loops. We confirm the above conjecture by applying our model calculation to the SAMD4A gene by comparing the experimental results. Our probabilistic model provides possible scenarios for assembling RNAPII molecules into transcription factories, where RNAPII and related proteins cooperatively transcribe DNA.

Ultradiscrete Ai and Bi functions are directly derived through the ultradiscrete limit from q-difference analogues of the Ai and Bi functions, respectively. An infinite number of identities among the number of restricted partitions are obtained as by-products. A direct relationship between a class of special solutions for the ultradiscrete Painleve II equation and those of the q-Painleve II equation which have a determinantal structure is also established.

Discrete integrable equations over finite fields are investigated. The indeterminacy of the equation is resolved by treating it over a field of rational functions instead of the finite field itself. The main discussion concerns a generalized discrete KdV equation related to a Yang-Baxter map. Explicit forms of soliton solutions and their periods over finite fields are obtained. Relation to the singularity confinement method is also discussed.

Existence of global solutions to initial value problems for a discrete analogue of a d-dimensional semilinear heat equation is investigated. We prove that a parameter a in the partial difference equation plays exactly the same role as the parameter of nonlinearity does in the semilinear heat equation. That is, we prove non-existence of a non-trivial global solution for 0 < alpha <= 2/d, and, for alpha > 2/d, existence of non-trivial global solutions for sufficiently small initial data.

We propose a new method to construct an isotropic cellular automaton corresponding to a reaction-diffusion equation. The method consists of replacing the diffusion term and the reaction term of the reaction-diffusion equation with random walk of microscopic particles and a discrete vector field which defines time evolution of the particles. The obtained cellular automaton can retain isotropy and therefore reproduces the patterns found in the numerical solutions of the reaction-diffusion equation. As a specific example, we apply the method to the Belousov-Zhabotinsky reaction in excitable media.

We construct generalized solutions to the ultradiscrete KdV equation, including the so-called negative solition solutions. The method is based on the ultradiscretization of soliton solutions to the discrete KdV equation with gauge transformation. The conserved quantities of the ultradiscrete KdV equation are shown to be constructed in a similar way to those for the box-ball system.

We investigate correlation functions in a periodic box-ball system. For the second and the third nearest neighbor correlation functions, we give explicit formulae obtained by combinatorial methods. A recursion formula for a specific N-point functions is also presented.

We present a model which aims at describing the morphology of colonies of Proteus mirabilis and Bacillus subtilis. Our model is based on a cellular automaton which is obtained by the adequate discretisation of a diffusion-like equation, describing the migration of the bacteria, to which we have added rules simulating the consolidation process. Our basic assumption, following the findings of the group of Chuo University, is that the migration and consolidation processes are controlled by the local density of the bacteria. We show that it is possible within our model to reproduce the morphological diagrams of both bacteria species. Moreover, we model some detailed experiments done by the Chuo University group, obtaining a fine agreement. (C) 2010 Elsevier B.V. All rights reserved.

We investigate correlation functions in a periodic box-ball system. For the two-point functions of short distance, we give explicit formulae obtained by combinatorial methods. We give expressions for general N-point functions in terms of ultradiscrete theta functions.

We present an expression for the solution to the initial value problem for the ultradiscrete periodic Toda equation. The expression provides explicit forms of all dependent variables of the equation, while the previously known solutions give only half of the dependent variables while the others have to be determined implicitly using the conserved quantities.

In this paper, we propose the ultradiscrete optimal velocity model, a cellular-automaton model for traffic flow, by applying the ultradiscrete method for the optimal velocity model. The optimal velocity model, defined by a differential equation, is one of the most important models; in particular, it successfully reproduces the instability of high-flux traffic. It is often pointed out that there is a close relation between the optimal velocity model and the modified Korteweg-de Vries (mkdV) equation, a soliton equation. Meanwhile, the ultradiscrete method enables one to reduce soliton equations to cellular automata which inherit the solitonic nature, such as an infinite number of conservation laws, and soliton solutions. We find that the theory of soliton equations is available for generic differential equations and the simulation results reveal that the model obtained reproduces both absolutely unstable and convectively unstable flows as well as the optimal velocity model.

A differential equation exhibiting replicative time-evolution patterns is derived by inverse ultra-discretizatrion of Fredkin's game, which is one of the simplest replicative cellular automaton (CA) in two dimensions. This is achieved by employing a certain filter aid a clock function in the equation. These techniques are applicable to the inverse ultra-discretization (IUD) of other CA and stabilize the time-evolution of the obtained differential equation. Application to the game of life, another CA in two dimensions, is also presented.

Any state of the box-ball system (BBS) together with its time evolution is described by the N-soliton solution (with appropriate choice of N) of the ultradiscrete KdV equation. It is shown that simultaneous elimination of all '10'-walls in a state of the BBS corresponds exactly to reducing the parameters that determine 'the size of a soliton' by one. This observation leads to an expression for the solution to the initial-value problem (IVP) for the BBS. Expressions for the solution to the IVP for the ultradiscrete Toda molecule equation and the periodic BBS are also presented.

We propose a new cellular automaton (CA) model, which reproduces isotropic time-evolution patterns observed in the Belousov-Zhabotinsky reaction. Although several CA models have been proposed exhibiting isotropic patterns of the reaction, most of them need complicated rules and a large number of neighboring cells. Our model can produce isotropic patterns from a simple probabilistic rule among a few (4 or 8) neighboring cells. (c) 2008 Elsevier B.V. All rights reserved.

A periodic box-ball system (pBBS) is obtained by ultradiscretizing the periodic discrete Toda equation (pd Toda equation). We show the relation between a Young diagram of the pBBS and a spectral curve of the pd Toda equation. The formula for the fundamental cycle of the pBBS is obtained as a corollary.

In a recent paper (Phys. Rev. E 72, 035102 (2005)) we have proposed a stochastic optimal velocity model which includes two exactly solvable stochastic models. It can be regarded as a stochastic version of the optimal velocity model. We find that the model shows striking metastability (i.e. long-lived metastable states, dynamical phase transition, and sharp spontaneous metastability breaking) as well as solvability. In this work, we present additional explanations of the solvability and metastability, which are helpful in understanding the traffic dynamics of the model.

We show that the initial value problem of a periodic box-ball system can be solved in an elementary way using simple combinatorial methods.

In this paper, we study an exact solution of the asymmetric simple exclusion process on a periodic lattice of finite sites with two typical updates, i.e., random and parallel. Then, we find that the explicit formulae for the partition function and the average velocity are expressed by the Gauss hypergeometric function. In order to obtain these results, we effectively exploit the recursion formula for the partition function for the zero-range process. The zero-range process corresponds to the asymmetric simple exclusion process if one chooses the relevant hop rates of particles, and the recursion gives the partition function, in principle, for any finite system size. Moreover, we reveal the asymptotic behaviour of the average velocity in the thermodynamic limit, expanding the formula as a series in system size.

We investigate the fundamental cycle of a periodic box-ball system (PBBS) from a relation between the PBBS and a solvable lattice model. We show that the fundamental cycle of the PBBS is obtained from eigenvalues of the transfer matrix of the solvable lattice model.