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At least 19 recordsLinked to original sources
PubMed1979

Analysis of picosecond laser induced fluorescence phenomena in photosynthetic membranes utilizing a master equation approach.

A Pauli master equation is formulated and solved to describe the fluorescence quantum yield, phi, and the fluorescence temporal decay curves. F(t), obtained in picosecond laser excitation experiments of photosynthetic systems. It is assumed that the lowering of phi with increasing pulse intensity is due to bimolecular singlet exciton annihilation processes which compete with the monomolecular exciton decay processes; Poisson statistics are taken into account. Calculated curves of phi as a function of the number of photon hits per domain are compared with experimental data, and it is concluded that these domains contain at least two to four connected photosynthetic units (depending on the temperature), where each photosynthetic unit is assumed to contain approximately 300 pigment molecules. It is shown that under conditions of high excitation intensities, the fluorescence decays approximately according to the (time)1/2 law.

Kinetics↗
PubMed1976

Energy transfer in photosynthesis.

A theoretical model is presented to account for the physical mechanism of energy transfer from antenna molecules to the reaction centers in photosynthesis. The energy transfer is described by a generalized transport equation or "master equation". The solution of this equation for the proposed model gives a relationship between the antennae interaction energy and the transfer rate. The results are shown to be in agreement with inter-antenna transfer rates calculated from experimental fluorescence lifetimes. Previous theories were based either on the Förster mechanism, which is valid for very small interaction energies, or an exciton model valid for very large interactions, but experimental results seemed to indicate that the actual situation was intermediate between these two. The Förster theory and the exciton model are limiting cases of the master equation.

Electron Transport↗
PubMed1985

Non-equilibrium voltage noise generated by ion transport through pores.

In this paper, we describe a systematic approach to the theoretical analysis of non-equilibrium voltage noise that arises from ions moving through pores in membranes. We assume that an ion must cross one or two barriers in the pore in order to move from one side of the membrane to the other. In our analysis, we consider the following factors: a) surface charge as a variable in the kinetic equations, b) linearization of the kinetic equations, c) master equation approach to fluctuations. To analyze the voltage noise arising from ion movement through a two barrier (i.e., one binding site) pore, we included the effects of ions in the channel's interior on the voltage noise. The current clamp is considered as a white noise generating additional noise in the system. In contrast to what is found for current noise, at low frequencies the voltage noise intensity is reduced by increasing voltage across the membrane. With this approach, we demonstrate explicitly for the examples treated that, apart from additional noise generated by the current clamp, the non-equilibrium voltage fluctuations can be related to the current fluctuations by the complex admittance.

Electric Conductivity↗
PubMed1985

Complex membrane transport systems. A non-Markovian approach.

This paper suggests a method of how to deal with complex membrane transport systems such as ion channels or ion pumps formed by proteins. The complexity of these systems results from the fact that proteins may undergo an internal dynamics of conformational changes and may thereby affect the transmembrane transport. Usually, complex transport systems are mapped into multi-state graphs and couched in terms of Markovian master equations. It is shown in this paper how the dimensionality of such multi-state systems can be reduced. The resulting description may be expressed in the form of a generalized master equation with a memory function as integral kernel. The memory function reflects the protein's own dynamics and its overall effect on the transport. This formalism, non-Markovian in nature, is applied to describe the time-dependent action of ion pumps. A general model is constructed on the basis of the rate theory which contains all the essential parts of ion pumps such as a catalytic unit and a channel-like conduit for ion translocation and which is still analytically tractable. The short-time behaviour of the pumping process turns out to be of particular interest, since it reveals the dynamics of the catalytic unit itself. A strong correlation of the particle's motion over times less than a certain correlation time has been found. This result is compared with experimental findings on the proton pump of Halobacterium halobium. It is concluded that such a perfect short-time memory could be a generic property of active transport systems.

Adenosine Triphosphatases↗
PubMed1978

Current noise around steady states in discrete transport systems.

Subject of this paper is the transport noise in discrete systems. The transport systems are given by a number (n) of binding sites separated by energy barriers. These binding sites may be in contact outer reservoirs. The state of the systems is characterized by the occupation numbers of particles (current carriers) at these binding sites. The change in time of the occupation numbers is generated by individual "jumps" of particles over the energy barriers, building up the flux matter (for charged particles: the electric current). In the limit n leads to infinity continuum processes as e.g. usual diffusion are included in the transport model. The fluctuations in occupation numbers and other quantities linearly coupled to the occupation numbers may be treated with the usual master equation approach. The treatment of the fluctuation in fluxes (current) makes necessary a different theoretical approach which is presented in this paper under the assumption of vanishing interactions between the particles. This approach may be applied to a number of different transport systems in biology and physics (ion transport through porous channels in membranes, carriers mediated ion transport through membranes, jump diffusion e.g. in superionic conductors). As in the master equation approach the calculation of correlations and noise spectra may be reduced to the solution of the macroscopic equations for the occupation numbers. This result may be regarded as a generalization to non-equilibrium current fluctuations of the usual Nyquist theorem relating the current (voltage) noise spectrum in thermal equilibrium to the macroscopic frequency dependent admittance. The validity of the general approach is demonstrated by the calculation of the autocorrelation function and spectrum of current noise for a number of special examples (e.g. pores in membranes, carrier mediated ion transport).

Binding Sites↗
PubMed1986

Nonequilibrium voltage fluctuations in biological membranes. I. General framework of charge transport in discrete systems and related voltage noise.

This paper continues our work on the theory of nonequilibrium voltage noise generated by electric transport processes in membranes. Introducing the membrane voltage as a further variable, a system of kinetic equations linearized in voltage is derived by which generally the time-dependent behaviour of charge-transport processes under varying voltage can be discussed. Using these equations, the treatment of voltage noise can be based on the usual master equation approach to steady-state fluctuations of scalar quantities. Thus, a general theoretical approach to nonequilibrium voltage noise is presented, completing our approach to current fluctuations which had been developed some years ago. It is explicitly shown that at equilibrium the approach yields agreement with the Nyquist relation, while at nonequilibrium this relation is not valid. A further general property of voltage noise is the reduction of low-frequency noise with increasing number of transport units as a consequence of the interactions via the electric field. In a second paper, the approach will be applied for a number of special transport mechanisms, such as ionic channels, carriers or electrogenic pumps.

Kinetics↗
PubMed1978

Stochastic theory of ligand migration in biomolecules.

When ligand binding to proteins involves the presence of more than one ligand inside a given biomolecule, linear deterministic rate equations become useless. A stochastic approach, however, permits a treatment of the migration and binding of small molecules to proteins even at high ligand concentrations. An appropriate linear master equation and its analytic solution are given. As an example, the binding of carbon monoxide to myoglobin at partial pressures from 1 to 10(3) bars (0.1 to 100 MPa) is treated.

Binding Sites↗
PubMedDate not supplied

Dynamics of excitions created by a single picosecond pulse.

A theoretical analysis of bimolecular annihilation in finite domains is presented. A Pauli master equation is formulated for the case of varying incident delta function excitation sources. Expressions for the quantum fluorescence yield and its time dependence are derived. The relationship between the fluorescence yield and the number of hits per domain depends on two parameters: the rate constant of bimolecular exciton annihilation and the dimension of the domain in which this annihilation occurs. Recent experimental results imply that the exciton diffusion constant (D) is large (D approximately to greater than 10(-3) cm2 S-1) and that the photosystem II domains may contain as many as five photosynthetic units. An analysis of the time decay of the fluorescence indicates that, for a few hits per domain, the decay may be considered as exponential but for many hits it becomes non-exponential. Thus the fluorescence decay depends on the intensity of the excitation source and/or on the dimension of the domains. Conditions which change the effective size of the domain may change the shape of the fluorescence decay. Some biological consequences and experimental applications of this theory are presented.

Kinetics↗
PubMed1988

A reevaluation of the mathematical models for simulating single-channel and whole-cell ionic currents.

We have developed a technique that allows for the simulation of both single-channel and whole-cell ionic currents given any arbitrary first-order kinetic scheme for the conformational states of an ion channel. The procedure is based on the solution of the master equation, which, in turn, is a general expression for a Markov process. The solution is expressed in terms of the eigenvalues and eigenvectors of the kinetic system and the system's deviation from equilibrium. Our derived expression provides a general recipe for the calculation of whole-cell currents. By further manipulation of this expression, we show how conditional probabilities are derived that can be used for the simulation of single-channel currents. We discuss computer implementation of the results so that complicated kinetic schemes can be solved numerically. Finally, we demonstrate the procedure by providing a worked example of a simple model of activation followed by inactivation.

Computer Simulation↗
PubMed1987

Exact results for the average dynamic behavior of some non-linear neural networks.

We have studied the global dynamic behavior of neural-like networks of synchronous threshold elements by writing a master equation as a function of parameter values using statistical methods. Exact results for highly connected networks and no correlation are obtained, showing that in this case (contrary to previous results) the average activity can only display simple stable behaviour, the sole exception being special cases of a slow passage through a tangent bifurcation, and a limit cycle of length two. By introducing an appropriate probabilistic hypothesis, we also study the average activity and correlation for highly connected networks with the topology of a (Cayley) tree. In this case the dynamic is ruled by a pair of coupled equations linking activity and correlation, and the tendency is for the correlation to disappear over time. However, under reasonable biological conditions, this tendency will be extremely slow, giving rise to a region of pseudo-stability.

Animals↗
PubMed1979

Theory of transport noise in membrane channels with open-closed kinetics.

A theoretical approach to transport noise in kinetic systems, which has recently been developed, is applied to electric fluctuations around steady-states in membrane channels with different conductance states. The channel kinetics may be simple two state (open-closed) kinetics or more complicated. The membrane channel is considered as a sequence of binding sites separated by energy barriers over which the ions have to jump according to the usual single-file diffusion model. For simplicity the channels are assumed to act independently. In the special case of ionic movement fast compared with the channel open-closed kinetics the results agree with those derived from the usual Master equation approach to electric fluctuations in nerve membrane channels. For the simple model of channels with one binding site and two energy barries the coupling between the fluctuations coming from the open-closed kinetics and from the jump diffusion is investigated.

Binding Sites↗
PubMed1990

Theory of exciton annihilation in complexes of a finite number of molecular sites.

A theory of the kinematics of singlet exciton annihilation in complexes of a finite number of molecular sites is developed. The theory is based on a specific scheme suggested earlier by Gülen, Wittmershaus, and Knox [Biophys J. 49:469-477 (1986)]. It is adequate to address the excitation kinetics and dynamics in such systems, especially under high excitation intensities. A Pauli master equation is formulated and is solved to give explicit expressions for observables such as quantum yield and fluorescence intensity. The excitation intensity dependence of the observables is taken into account by introducing Poisson statistics. Details relevant to its application to the annihilation of excitons in photosynthetic systems and its connection to earlier theories are presented.

Energy Transfer↗
PubMed1992

DNA damage caused by ionizing radiation.

A survey is given of continuous-time Markov chain models for ionizing radiation damage to the genome of mammalian cells. In such models, immediate damage induced by the radiation is regarded as a batch-Poisson arrival process of DNA double-strand breaks (DSBs). Enzymatic modification of the immediate damage is modeled as a Markov process similar to those described by the master equation of stochastic chemical kinetics. An illustrative example is the restitution/complete-exchange model. The model postulates that, after being induced by radiation, DSBs subsequently either undergo enzymatically mediated restitution (repair) or participate pairwise in chromosome exchanges. Some of the exchanges make irremediable lesions such as dicentric chromosome aberrations. One may have rapid irradiation followed by enzymatic DSB processing or have prolonged irradiation with both DSB arrival and enzymatic DSB processing continuing throughout the irradiation period. Methods for analyzing the Markov chains include using an approximate model for expected values, the discrete-time Markov chain embedded at transitions, partial differential equations for generating functions, normal perturbation theory, singular perturbation theory with scaling, numerical computations, and certain matrix methods that combine Perron-Frobenius theory with variational estimates. Applications to experimental results on expected values, variances, and statistical distributions of DNA lesions are briefly outlined. Continuous-time Markov chains are the most systematic of those radiation damage models that treat DSB-DSB interactions within the cell nucleus as homogeneous (e.g., ignore diffusion limitations). They contain virtually all other relevant homogeneous models and semiempirical summaries as special cases, limiting cases, or approximations. However, the Markov models do not seem to be well suited for studying spatial dependence of DSB interactions, which is known to be important in some situations.

Animals↗
PubMed1992

Structured Biological Modelling: a method for the analysis and simulation of biological systems applied to oscillatory intracellular calcium waves.

In biology signal and information processing networks are widely known. Due to their inherent complexity and non-linear dynamics the time evolution of these systems can not be predicted by simple plausibility arguments. Fortunately, the power of modern computers allows the simulation of complex biological models. Therefore the problem becomes reduced to the question of how to develop a consistent mathematical model which comprises the essentials of the real biological system. As an interface between the phenomenological description and a computer simulation of the system the proposed method of Structured Biological Modelling (SBM) uses top-down levelled dataflow diagrams. They serve as a powerful tool for the analysis and the mathematical description of the system in terms of a stochastic formulation. The stochastic treatment, regarding the time evolution of the system as a stochastic process governed by a master equation, circumvents most difficulties arising from high dimensional and non-linear systems. As an application of SBM we develop a stochastic computer model of intracellular oscillatory Ca2+ waves in non-excitable cells. As demonstrated on this example, SBM can be used for the design of computer experiments which under certain conditions can be used as cheap and harmless counterparts to the usual time-consuming biological experiments.

Calcium↗
PubMed1987

Group selection of early replicators and the origin of life.

A major problem of the origin of life has been that of information integration. As Eigen (1971) has shown, a mutant distribution of RNAs replicating without the aid of a replicase cannot integrate sufficient information for the functioning of a higher-level unit utilizing several types of encoded enzymes. He proposed the hypercycle model to bridge this gap in prebiology. It can be shown by a nonlinear game model, incorporating mutation of a hypercycle, that the selection properties of hypercycles make them inefficient information integrators as they cannot compete favourably with all kinds of less efficient information carriers or mutationally coupled hypercycles. The stochastic corrector model is presented as an alternative resolution of Eigen's paradox. It assumes that replicative templates are competing within replicative compartments, whose selective values depend on the internal template composition via a catalytic acid in replication and "metabolism". The dynamics of template replication are analyzed by numerical simulation of master equations. Due to the stochasticity in replication and compartment fission the best compartment types recur. An Eigen equation at the compartment level is set up and calculated. Even selfish template mutants cannot destroy the system though they make it less efficient. The genetic information of templates is evaluated at both levels, and the higher (compartment) level successfully constrains the lower (template) one. Compartmentation together with stochastic effects is sufficient to integrate information dispersed in competitive replicators. Compartment selection is considered to be group selection of replicators. Implications for the origin of life are discussed.

Biological Evolution↗
PubMed1988

Influence of the hypercycle on the error threshold: a stochastic approach.

The role of fluctuations on the error threshold of the hypercycle has been studied by a stochastic approach on a very simplified model. For this model, the master equation was derived and its unique steady state calculated. This state implies the extinction of the system. But the actual time necessary to reach the steady state may be astronomically long whereas for times of experimental interest the system could be near some quasi-stationary states. In order to explore this possibility a Gillespie simulation of the stochastic process has been carried out. These quasi-stationary states correspond to the deterministic steady states of the system. The error threshold shifts towards higher values of the quality factor Q. Moreover, information about the fluctuations around the quasi-stationary states is obtained. The results are discussed in relation to the deterministic states.

Animals↗
PubMed1972

Dynamics of fluorescence polarization in macromolecules.

Reexamination of the theory of fluorescence time dependence owing to rotational diffusion of rigid macromolecules reveals deficiencies or hidden restrictions in each of the previous treatments. The correct master equation has five exponential decay terms, with preexponential factors that depend upon the diffusion constants and, in a completely symmetrical fashion, upon the orientations of absorbing and emitting dipoles.

Diffusion↗