Adding self-renewal in committed erythroid progenitors improves the biological relevance of a mathematical model of erythropoiesis

We propose a new mathematical model of erythropoiesis that takes a positive feedback of erythrocytes on progenitor apoptosis into account, and incorporates a negative feedback of erythrocytes on progenitor self-renewal. The resulting model is a system of age-structured equations that reduces to a system of delay differential equations where the delays account for progenitor compartment duration and cell cycle length. We compare this model with experimental data on an induced-anemia in mice that exhibit damped oscillations of the hematocrit before it returns to equilibrium. When we assume no self-renewal of progenitors, we obtain an inaccurate fitting of the model with experimental data. Adding self-renewal in the progenitor compartment gives better approximations, with the main features of experimental data correctly fitted. Our results indicate the importance of progenitor self-renewal in the modelling of erythropoiesis. Moreover, the model makes testable predictions on the lifespan of erythrocytes confronted to a severe anemia, and on the progenitors behavior.     

Analysis of a mathematical model for transport of I-albumin

The hypotheses and results given are motivated by the study of the distribution of albumin in man which represents a class of delay-differential systems. The approach used is to study the behavior of the solutions of nonlinear delay-differential systems with variable coefficients under the assumptions of continuity and boundedness of coefficients. The criterions are conditions on the roots of a certain “quasi-polynomial”, i.e., a polynomial in a variable and exponential of that variable. These criterions bear a resemblance to the ones in the case of constant coefficients and retardations and are applicable to this case also. The method is based on Lyapunov type functional with appropriate properties.     

Analysis of a mathematical model for the growth of tumors

 In this paper we study a recently proposed model for the growth of a nonnecrotic, vascularized tumor. The model is in the form of a free-boundary problem whereby the tumor grows (or shrinks) due to cell proliferation or death according to the level of a diffusing nutrient concentration. The tumor is assumed to be spherically symmetric, and its boundary is an unknown function r=s(t). We concentrate on the case where at the boundary of the tumor the birth rate of cells exceeds their death rate, a necessary condition for the existence of a unique stationary solution with radius r=R ₀ (which depends on the various parameters of the problem). Denoting by c the quotient of the diffusion time scale to the tumor doubling time scale, so that c is small, we rigorously prove that (i) lim inf _t→∞ s(t)>0, i.e. once engendered, tumors persist in time. Indeed, we further show that (ii) If c is sufficiently small then s(t)→R ₀ exponentially fast as t→∞, i.e. the steady state solution is globally asymptotically stable. Further, (iii) If c is not “sufficiently small” but is smaller than some constant γ determined explicitly by the parameters of the problem, then lim sup _t→∞ s(t)<∞; if however c is “somewhat” larger than γ then generally s(t) does not remain bounded and, in fact, s(t)→∞ exponentially fast as t→∞. Received: 25 February 1998 / Revised version: 30 April 1998     

A mathematical model for period-memorizing behavior in Physarum plasmodium

Mathematical models to describe period-memorizing behavior in Physarum plasmodium are reported. In constructing the model, we first examine the basic characteristics required for the class of models, then create a minimal linear model to fulfill these requirements. We also propose two modifications of the minimal model, nonlinearization and noise addition, which improve the reproducibility of experimental evidences. Differences in the mechanisms and in the reproducibility of experiments between our models and the previous models are discussed.     

A mathematical model of adaptive behavior in quadruped locomotion

Locomotion involves repetitive movements and is often executed unconsciously and automatically. In order to achieve smooth locomotion, the coordination of the rhythms of all physical parts is important. Neurophysiological studies have revealed that basic rhythms are produced in the spinal network called, the central pattern generator (CPG), where some neural oscillators interact to self-organize coordinated rhythms. We present a model of the adaptation of locomotion patterns to a variable environment, and attempt to elucidate how the dynamics of locomotion pattern generation are adjusted by the environmental changes. Recent experimental results indicate that decerebrate cats have the ability to learn new gait patterns in a changed environment. In those experiments, a decerebrate cat was set on a treadmill consisting of three moving belts. This treadmill provides a periodic perturbation to each limb through variation of the speed of each belt. When the belt for the left forelimb is quickened, the decerebrate cat initially loses interlimb coordination and stability, but gradually recovers them and finally walks with a new gait. Based on the above biological facts, we propose a CPG model whose rhythmic pattern adapts to periodic perturbation from the variable environment. First, we design the oscillator interactions to generate a desired rhythmic pattern. In our model, oscillator interactions are regarded as the forces that generate the desired motion pattern. If the desired pattern has already been realized, then the interactions are equal to zero. However, this rhythmic pattern is not reproducible when there is an environmental change. Also, if we do not adjust the rhythmic dynamics, the oscillator interactions will not be zero. Therefore, in our adaptation rule, we adjust the memorized rhythmic pattern so as to minimize the oscillator interactions. This rule can describe the adaptive behavior of decerebrate cats well. Finally, we propose a mathematical framework of an adaptation in rhythmic motion. Our framework consists of three types of dynamics: environmental, rhythmic motion, and adaptation dynamics. We conclude that the time scale of adaptation dynamics should be much larger than that of rhythmic motion dynamics, and the repetition of rhythmic motions in a stable environment is important for the convergence of adaptation. Received: 10 July 1997 / Accepted in revised form: 13 March 1998     

Use of a mathematical model of erythropoiesis to study the recovery process in mice exposed to acute x-ray irradiation]

The model described in the preceding paper was used (after introducing several further presumptions of a radiobiological nature) for the interpretation of the dynamics of erythropoietic recovery, experimentally studied by the authors in mice after acute X-irradiation. The solution of equations in the model was carried out by a digital computer, finding some (not yet experimentally determined) quantitative characteristics of the system by trial and error. A satisfactory degree of agreement in the behaviour of the model with the experimentally determined course of recovery seems to indicate the validity of the basic assumptions of the model, esp. the existence of the negative feedback between erythroid populations of the bone marrow and spleen.     

温室茄子（Solanum melongena L.）光合数学模型与光合生化模型模拟分析

针对植物光合与内外环境因子间的关系以及光合“午睡”现象中的气孔限制与非气孔限制问题,以温室茄子‘茄杂一号’为试材,对叶室温光组合方式下测定的净光合速率Pn对胞间CO2浓度Ci响应曲线,和人工增施CO2处理下测定的Pn日变化进程,进行了光合数学模型和Farquhar、von Caemmerer和Berry的光合生化动力学模型（简称为FvCB模型）模拟分析。采用美国思爱迪生态仪器有限公司的CI-301PS光合作用测定仪进行净光合速率（Pn）、光合有效辐射（PAR）、气温（Ta）、叶温（Tl）、环境二氧化碳浓度（Ca）、胞间二氧化碳浓度（Ci）和空气相对湿度（Hr）参数测定。其结果表明,无论是Pn对Ci的响应曲线还是光合日进程中,数学模型对Pn的拟合度明显优于为FvCB模型。因此,通过数学模型可以解析出光合日进程受单一环境因子（PAR、Ta、Ca、Hr）及其复合环境因子的综合影响。然而,FvCB模型模拟结果显示出,温光组合下受Rubisco（即RuBP羧化/加氧酶）数量与活性及动力学特性限制的羧化速率Ac、受RuBP（1,5-二磷酸核酮糖）再生限制的羧化速率Aj以及受TPU（磷酸丙糖）可利用量限制的羧化速率Ap对Ci响应的主控作用呈现交替变化趋势。其交替变化转折点胞间二氧化碳浓度Cicj在强光高温组合中较高,而在弱光低温组合中较低;同时还发现,Cicj和Cijp受叶温的影响强于光照。光合日进程中的FvCB模型模拟分析揭示出,早晨和傍晚弱光下为Aj限制时段;晴天上午和中午前后的充足日照下为Ac限制时段。多云和阴天下Aj的限制时段延长。增施CO2会延长Aj的限制时段,同时相应缩短Ac的限制时段;冬季2次增施CO2的出现了Ap限制时段。     

A mathematical model of heterogeneous behavior of single muscle fibres

A mathematical model of contracting muscle fibre is studied. The model is composed of an array of segments placed in series; any segment has an elastic element (PE _i ) and a contractile element (CE _i ) that describes the cross bridge kinetics.The corresponding system of nonlinear partial differential equations of the model is analyzed. Existence, uniqueness and continuous dependence of the solution are proven.This work has been supported by Ministero Pubblica Istruzione (fondi per la ricerca scientifica) and by Istituto di Analisi Numerica del C.N.R., Pavia     

Turing instabilities in a mathematical model for signaling networks

GTPase molecules are important regulators in cells that continuously run through an activation/deactivation and membrane-attachment/membrane-detachment cycle. Activated GTPase is able to localize in parts of the membranes and to induce cell polarity. As feedback loops contribute to the GTPase cycle and as the coupling between membrane-bound and cytoplasmic processes introduces different diffusion coefficients a Turing mechanism is a natural candidate for this symmetry breaking. We formulate a mathematical model that couples a reaction–diffusion system in the inner volume to a reaction–diffusion system on the membrane via a flux condition and an attachment/detachment law at the membrane. We present a reduction to a simpler non-local reaction–diffusion model and perform a stability analysis and numerical simulations for this reduction. Our model in principle does support Turing instabilities but only if the lateral diffusion of inactivated GTPase is much faster than the diffusion of activated GTPase.     

Analysis of pathological defects in methionine metabolism using a simple mathematical model

Derangements in methionine metabolism are a hallmark of cancers and homocystinuria, an inborn error of metabolism. In this study, the metabolic consequences of the pathological changes associated with the key pathway enzymes, methionine adenosyl transferase (MAT), glycine N-methyl transferase (GNMT) and cystathionine beta-synthase (CBS) as well as an activation of polyamine metabolism, were analyzed using a simple mathematical model describing methionine metabolism in liver. The model predicts that the mere loss of allosteric regulation of CBS by adenosylmethionine (AdoMet) leads to an increase in homocysteine concentration. This is consistent with the experimental data on the corresponding genetic defects, which specifically impair allosteric activation but not basal enzyme activity. Application of the characteristics of transformed hepatocytes to our model, i.e., substitution of the MAT I/III isozyme by MAT II, loss of GNMT activity and activation of polyamine biosynthesis, leads to the prediction of a significantly different dependence of methionine metabolism on methionine concentrations. The theoretical predictions were found to be in good agreement with experimental data obtained with the human hepatoma cell line, HepG2.     

A multi-scale model of erythropoiesis

In this paper, a multi-scale mathematical model of erythropoiesis is proposed in which erythroid progenitors are supposed to be able to self-renew. Three cellular processes control erythropoiesis: self-renewal, differentiation and apoptosis. We describe these processes and regulatory networks that govern them. Two proteins (ERK and Fas) are considered as the basic proteins participating in this regulation. All erythroid progenitors are divided into several sub-populations depending on their maturity level. Feedback regulations by erythropoietin, glucocorticoids and Fas ligand (FasL) are introduced in the model. The model consists of a system of ordinary differential equations describing intracellular protein concentration evolution and cell population dynamics. We study steady states and their stability. We carry out computer simulations of an anaemia situation and analyse the results.     

On a mathematical model for body tissue inflammation

In the present paper I will try to prove the mathematical validity of a model on the localized bacterial infection for tissue inflammation. This model was proposed by Lauffenburger and Kennedy [3], and it describes the inflammatory response to bacterial invasion of body tissue. I prove the mathematical validity of the model by means of a positivity theorem, an existence theorem and a uniqueness theorem. In spite of the apparent simplicity of the problem, the solution requires a delicate set of techniques. It seems very difficult to extend these techniques to a model in more than one dimension.