Spectral properties of Eigen evolution matrices

Eigen [7] has employed deterministic kinetic-like equations to describe macromolecular replication and mutation leading to selection. The solutions to these equations and their physically interesting properties depend upon the spectrum of a type of matrix appearing in those equations. Below we explicitly solve for the spectrum of a fairly general class of such matrices. These solutions are obtained recursively for equations describing macromolecules of any length v.     

Analysis of Eigen’s equations for selection of biological molecules with fluctuating mutation rates

In this paper we consider Eigen’s equations for the selection and evolution of self-instructing, macromolecular systems. We construct exact, asymptotic solutions for the equations when the rate coefficients and error distributions are considered as functions of time. Implications for selection are discussed.     

A model for macromolecular selection in complementary instructing systems

Summary In this paper we consider a model for the selection and evolution of biological macromolecules when their reproduction is based on complementary instruction. The model is an extension of one of Eigen's models for selection and takes in account explicitly the formation of both single stranded and double stranded molecular complexes. We construct exact solutions to the rate equations for the case of constant rate parameters and error distributions. Criteria for selection are discussed.     

Energy and biological evolution—I. The equilibrium states of biochemical processes

The Thom gradient model of morphogenesis poses the followinga posteriori problem: “From the observed morphology of a given natural process (effect) determine the dynamics of the process (cause)”. In this paper we consider the classicala priori problem: “Given the cause (dynamics) determine the effect (resultant morphology)”. We find that in biochemical processes the mechanisms for energy activation, energy-matter interaction and energy dissipation determine the dynamics. Furthermore there exists basic energy mechanisms which drive the equilibrium states through the elementary catastrophes of Thom. A comparison with current theories shows that our models describe open ecological food chains and their dynamical systems generalize the equations of organisation posed by M. Eigen. Work supported by a Research Associateship of the International Centre for Theoretical Physics, P.O.B. 586, Miramare, 34100 Trieste, Italy.     

Asymptotic Behavior of Eigen’s Quasispecies Model

We study Eigen’s quasispecies model in the asymptotic regime where the length of the genotypes goes to \(\infty \) and the mutation probability goes to 0. A limiting infinite system of differential equations is obtained. We prove convergence of trajectories, as well as convergence of the equilibrium solutions. We give analogous results for a discrete-time version of Eigen’s model, which coincides with a model proposed by Moran.     

Stability analysis of a stochastic model for biomolecular selection

The technique of the probability generating function is used to derive the stochastic differential equations for a nonlinear model based on Eigen and Schuster's theory of biomolecular selection and evolution. The stabilities of various steady states are analyzed by using the linear stability approximation. The instability of a small starting population is investigated numerically. The minimum starting populations required for steady-state survival are then estimated for a wide range of parameters.     

Cumulant dynamics of a population under multiplicative selection, mutation, and drift

We revisit the classical population genetics model of a population evolving under multiplicative selection, mutation, and drift. The number of beneficial alleles in a multilocus system can be considered a trait under exponential selection. Equations of motion are derived for the cumulants of the trait distribution in the diffusion limit and under the assumption of linkage equilibrium. Because of the additive nature of cumulants, this reduces to the problem of determining equations of motion for the expected allele distribution cumulants at each locus. The cumulant equations form an infinite dimensional linear system and in an authored appendix Adam Prügel-Bennett provides a closed form expression for these equations. We derive approximate solutions which are shown to describe the dynamics well for a broad range of parameters. In particular, we introduce two approximate analytical solutions: (1) Perturbation theory is used to solve the dynamics for weak selection and arbitrary mutation rate. The resulting expansion for the system's eigenvalues reduces to the known diffusion theory results for the limiting cases with either mutation or selection absent. (2) For low mutation rates we observe a separation of time-scales between the slowest mode and the rest which allows us to develop an approximate analytical solution for the dominant slow mode. The solution is consistent with the perturbation theory result and provides a good approximation for much stronger selection intensities.     

Hopf bifurcations in multiple-parameter space of the Hodgkin-Huxley equations I. Global organization of bistable periodic solutions

 The Hodgkin-Huxley equations (HH) are parameterized by a number of parameters and shows a variety of qualitatively different behaviors depending on the parameter values. We explored the dynamics of the HH for a wide range of parameter values in the multiple-parameter space, that is, we examined the global structure of bifurcations of the HH. Results are summarized in various two-parameter bifurcation diagrams with I _ext (externally applied DC current) as the abscissa and one of the other parameters as the ordinate. In each diagram, the parameter plane was divided into several regions according to the qualitative behavior of the equations. In particular, we focused on periodic solutions emerging via Hopf bifurcations and identified parameter regions in which either two stable periodic solutions with different amplitudes and periods and a stable equilibrium point or two stable periodic solutions coexist. Global analysis of the bifurcation structure suggested that generation of these regions is associated with degenerate Hopf bifurcations. Received: 23 April 1999 / Accepted in revised form: 24 September 1999     

A note on a more general solution of Eigen's rate equation for selection

In this note we examine Eigen's nonlinear rate equations for the relation of coupled biomacromolecules. We obtain an exact solution to the equations for constant overall population densities and constant rate parameters. We conclude that there is only one stationary solution where all molecular species coexist when they are coupled by mutation.     

Timing regulation in a network reduced from voltage-gated equations to a one-dimensional map

 We discuss a method by which the dynamics of a network of neurons, coupled by mutual inhibition, can be reduced to a one-dimensional map. This network consists of a pair of neurons, one of which is an endogenous burster, and the other excitable but not bursting in the absence of phasic input. The latter cell has more than one slow process. The reduction uses the standard separation of slow/fast processes; it also uses information about how the dynamics on the slow manifold evolve after a finite amount of slow time. From this reduction we obtain a one-dimensional map dependent on the parameters of the original biophysical equations. In some parameter regimes, one can deduce that the original equations have solutions in which the active phase of the originally excitable cell is constant from burst to burst, while in other parameter regimes it is not. The existence or absence of this kind of regulation corresponds to qualitatively different dynamics in the one-dimensional map. The computations associated with the reduction and the analysis of the dynamics includes the use of coordinates that parameterize by time along trajectories, and “singular Poincaré maps” that combine information about flows along a slow manifold with information about jumps between branches of the slow manifold. Received: 19 May 1997 / Revised version: 6 April 1998     

Improved neuronal models for studying neural networks

Previous neuronal models used for the study of neural networks are considered. Equations are developed for a model which includes: 1) a normalized range of firing rates with decreased sensitivity at large excitatory or large inhibitory input levels, 2) a single rate constant for the increase in firing rate following step changes in the input, 3) one or more rate constants, as required to fit experimental data for the adaptation of firing rates to maintained inputs. Computed responses compare well with the types of neuronal responses observed experimentally. Depending on the parameters, overdamped increases and decreases, damped oscillatory or maintained oscillatory changes in firing rate are observed to step changes in the input. The integrodifferential equations describing the neuronal models can be represented by a set of first-order differential equations. Steady-state solutions for these equations can be obtained for constant inputs, as well as the stability of the solutions to small perturbations. The linear frequency response function is derived for sufficiently small time-varying inputs. The linear responses are also compared with the computed solutions for larger non-linear responses.     

The thyroid-pituitary homeostatic mechanism

The paper develops a mathematical theory of thyroid-pituitary interaction. It is assumed that the pituitary gland produces thyrotropin, which activates an enzyme of the thyroid gland. The rate of production of thyroid hormone is considered to be proportional to the concentration of that enzyme. It is further assumed that in the absence of the thyroid hormone the rate of production of thyrotropin is constant, but, in general, it is a linear function of the concentration of the thyroid hormone. This picture leads to a system of non-linear differential equations, which present great difficulties. This system, however, may be conveniently “linearized”, by considering that the relations between different variables are linear, but that within different ranges of the variables the coefficients are different. Using this approximation, it is possible to show that the system admits periodic solutions of the nature of relaxation oscillations. Such oscillations are actually observed in some mental disorders, such as periodic catatonia. The study of the effects of different parameters of the system suggests different possible approaches to clinical treatment. In the light of this theory, the experimental determination of the parameters of the system becomes desirable and important.     

Neural network firing-rate models on integral form

Firing-rate models describing neural-network activity can be formulated in terms of differential equations for the synaptic drive from neurons. Such models are typically derived from more general models based on Volterra integral equations assuming exponentially decaying temporal coupling kernels describing the coupling of pre- and postsynaptic activities. Here we study models with other choices of temporal coupling kernels. In particular, we investigate the stability properties of constant solutions of two-population Volterra models by studying the equilibrium solutions of the corresponding autonomous dynamical systems, derived using the linear chain trick, by means of the Routh–Hurwitz criterion. In the four investigated synaptic-drive models with identical equilibrium points we find that the choice of temporal coupling kernels significantly affects the equilibrium-point stability properties. A model with an α-function replacing the standard exponentially decaying function in the inhibitory coupling kernel is in most of our examples found to be most prone to instability, while the opposite situation with an α-function describing the excitatory kernel is found to be least prone to instability. The standard model with exponentially decaying coupling kernels is typically found to be an intermediate case. We further find that stability is promoted by increasing the weight of self-inhibition or shortening the time constant of the inhibition.     

Inversion of Markov processes to determine rate constants from single-channel data.

The determination of rate constants from single-channel data can be very difficult, in part because the single-channel lifetime distributions commonly analyzed by experimenters often have a complicated mathematical relation to the channel gating mechanism. The standard treatment of channel gating as a Markov process leads to the prediction that lifetime distributions are exponential functions. As the number of states of a channel gating scheme increases, the number of exponential terms in the lifetime distribution increases, and the weights and decay constants of the lifetime distributions become progressively more complicated functions of the underlying rate constants. In the present study a mathematical strategy for inverting these functions is introduced in order to determine rate constants from single-channel lifetime distributions. This inversion is easy for channel gating schemes with two or fewer states of a given conductance, so the present study focuses on schemes with more states. The procedure is to derive explicit equations relating the parameters of the lifetime distribution to the rate constants of the scheme. Such equations can be derived using the equality between symmetric functions of eigenvalues of a matrix and sums over principle minors, as well as expressions for the moments, derivatives, and weights of a lifetime distribution. The rate constants are then obtained as roots to this system of equations. For a gating scheme with three sequential closed states and a single gateway state, exact analytical expressions were found for each rate constant in terms of the parameters of the three-exponential closed-time distribution. For several other gating schemes, systems of equations were found that could be solved numerically to obtain the rate constants. Lifetime distributions were shown to specify a unique set of real rate constants in sequential gating schemes with up to five closed or five open states. For kinetic schemes with multiple gating pathways, the analysis of simulated data revealed multiple solutions. These multiple solutions could be distinguished by examining two-dimensional probability density functions. The utility of the methods introduced here are demonstrated by analyzing published data on nicotinic acetylcholine receptors, GABA(A) receptors, and NMDA receptors.     

Dynamic modelling of aerobic bioprocesses in gel particles with immobilised cells

A dynamic model for aerobic growing cells immobilised into gel beads is developed and its operation is illustrated for the case of gluconic acid production by a strictly aerophilic strain of Gluconobacter oxydans. The model consists of both kinetic and mass transfer equations predicting the time course of bulk and intraparticle concentrations of substrates, products, and biomass. The model includes a product inhibition term. The parameter values are taken from own studies and from the literature. A sensitivity analysis of the model shows that the most significant parameters for the process are the biotransformation rate constant, the specific cell growth rate in the bulk, and the Thiele modulus for glucose. The computer simulation reveals that depending on the parameter values the gel particles might perform as a source or a sink of the product, thus enhancing or retarding the net process. For a specific parameter selection, the biotransformation in the pellets can prevail compared with the bulk in the beginning of the process as long as the direction of the product diffusion flux is from the beads toward the bulk. Since the process in the free culture dominates, the system is more sensitive to parameters associated with the bulk phase (aeration rate, specific microbial growth rate, oxygen uptake rate). The model can be applied for prediction and fast evaluation of the performance of aerobic processes accomplished by immobilised growing cells.     

A program for least squares analysis of reassociation and hybridization data.

A computer program is described for the rapid calculation of least squares solutions for data fitted to different functions normally used in reassociation and hybridization kinetic measurements. The equations for the fraction not reacted as a function of Cot follow: First order, exp(-kCot); second order, (1+kCot)-1; variable order, (1+kCot)-n; approximate fraction of DNA sequence remaining single stranded, (1+kCot)-.44; and a function describing the pairing of tracer when the rate constant for the tracer (k) is distinct from the driver rate constant (kd): (formula: see text). Several components may be used for most of these functional forms. The standard deviations of the individual parameters at the solutions are calculated.     

Optimal pH control of batch processes for production of curdlan by Agrobacterium species

We sought an optimal pH profile to maximize curdlan production in a batch fermentation of Agrobacterium species. The optimal pH profile was calculated using a gradient iteration algorithm based on the minimum principle of Pontryagin. The model equations describing cell growth and curdlan production were developed as functions of pH, sucrose concentration, and ammonium concentration, since the specific rates of cell growth and curdlan production were highly influenced by those parameters. The pH profile provided the strategy to shift the culture pH from the optimal growth condition (pH 7.0) to the optimal production one (pH 5.5) at the time of ammonium exhaustion. By applying the optimal pH profile in the batch process, we obtained significant improvement in curdlan production (64 g L⁻¹) compared to that of constant pH operation (36 g L⁻¹). Received 24 November 1998/ Accepted in revised form 17 June 1999     

Replicator Equations and the Principle of Minimal Production of Information

Many complex systems in mathematical biology and other areas can be described by the replicator equation. We show that solutions of a wide class of replicator equations minimize the KL-divergence of the initial and current distributions under time-dependent constraints, which in their turn, can be computed explicitly at every instant due to the system dynamics. Therefore, the Kullback principle of minimum discrimination information, as well as the maximum entropy principle, for systems governed by the replicator equations can be derived from the system dynamics rather than postulated. Applications to the Malthusian inhomogeneous models, global demography, and the Eigen quasispecies equation are given.     

On a population pathogen model incorporating species dispersal with temporal variation in dispersal rate

In the present paper, we consider a mathematical model of ecosystem population interaction where the population suffers from a susceptible–infectious–susceptible disease. Dispersal of both the susceptible and the infective is incorporated using reaction–diffusion equations. We first study the stability criteria of the basic (non-spatial) model around the disease-free and the infected steady states. We find that the loss rate of the infective species controls disease prevalence. Also without predation pressure, the disease will continue to exist among the population. Then we analyze the spatial model with species dispersal in constant as well as in time-varying form. It is observed that though constant dispersal is unable to generate diffusion-driven instability, dispersal with sinusoidal variation in dispersion rate can generate diffusive instability when the wave number of the perturbation lies within a given range. Numerical simulations are performed to illustrate analytical studies.     

Stochastic analysis of a nonlinear model for selection of biological macromolecules

A stochastic analysis of a nonlinear selection model is presented. The model, based on Eigen and Schuster's theory of selection and evolution of biological macromolecules, considers the effects of fluctuations on the individual concentrations of macromolecules as well as the total population numbers in constrained systems. Our analysis shows that one of the models most often treated deterministically (referred to as constant organization in the literature) becomes unstable when fluctuations in the total population number are considered. An alternative model which apparently has built in self-regulating properties is analyzed and proves to be stable except for some special cases of degeneracy.