Cellular & Molecular Biology Letters

International Scientific Journal
Established 1996

Volume 4 (1999) No. 1

Foreword

International Workshop on Mathematics for Nonlinear Biophysics

Mathematics has always benefited from its involvement with developing sciences. In the middle thirties Werner Heisenberg said: ”The progress in physics will to a large extent depend on the progress of nonlinear mathematics, of methods to solve nonlinear equations .... and therefore we can learn by comparing different nonlinear problems”. Can we say the same about ”Nonlinear Biophysics”? We tried to answer this question, and some more, while in Zakopane on 9-16 May, 1998. The main message we would like to convey to the wide audience of active theoreticians is that Nonlinear Biophysics is a fast growing, well recognised, albeit not clearly defined, subject and is, to our mind, the most exciting application of Nonlinear Mathematics. By ”nonlinear” we mean describing such phenomena like bifurcation i.e. splitting of solution (when, at certain points, an actual phenomenon splits into two, qualitatively new ones), nonlinear stability i.e. when nonlinear term in analytical description can not be neglected for it would cause the system to become unstable, local existence and/or blow-up of solution i.e. with no asymptotic behaviour, chaos, fractals etc. namely, phenomena that can not be predicted by the classical, linear theories but are observed in nature and laboratories. This special issue of Cellular and Molecular Biology Letters contains mostly selected papers presented during the meeting. We have tried to show the spectrum of problems which should be recognised as nonlinear.

We would like to thank the sponsors: Committee for Scientific Research (KBN), and the Silesian Technical University in Gliwice. Special thanks are due to Prof. dr hab. inz. Remigiusz Sosnowski, Vice President of the STU, for his generous help in overcoming financial problems of printing this issue with no delay.

Gliwice - Kraków - Denver, December, 1998

Editorial

The Guest Editors for this issue of the Journal are Professors Zbigniew J. Grzywna (Gliwice, Poland), Andrzej Fulinski (Kraków, Poland) and T. Gregory Dewey (Denver, USA) who were co-organizers of the International Workshop on "Mathematics for Nonlinear Biophysics" held in Zakopane, Poland, May 9-16, 1998. They gathered the papers, had them subjected to peer review, requested revisions by the authors where appropriate, and assumed the entire function of Editors.

Jan Szopa
Arkadiusz Kozubek
Aleksander F. Sikorski

Volume 4 (1999) pp 7-18
Title	THE THERMODYNAMICS OF BIOMOLECULAR SEQUENCES
Authors	T. Gregory Dewey
Abstract	A statistical mechanical treatment of biopolymers is presented that includes the sequence information as an internal coordinate. This approach allows an assessment of the contribution of sequence information to the thermodynamic entropy. Even in cases where the sequence composition has no effect on the intersubunit interactions, the sequence composition contributes to the entropy of the system. Using a path integral representation, the canonical partition function can be represented as a product of a polymer configurational path integral and a sequence walk path integral. In most, biological instances the sequence composition influences the potential energy of intersubunit interaction. Consequently, the two path integrals are not separable, but rather "interact" via a sequence-dependent configurational potential. Biological constraints can also be built into the system and these effectively introduce an external potential. In proteins and RNA, the sequence walk occurs in dimensions greater than 3 and, therefore, will be an ideal "polymer". The Markovian nature of this walk can be exploited to show that all the structural information is contained in the sequence. This later effect is a result of the dimensionality of the sequence walk and is not necessarily a result of biological optimization of the system.
Adress and Contact Informations	Department of Chemistry and Biochemistry, University of Denver, Denver, Colorado 80208-2436 USA

Volume 4 (1999) pp 19-36
Title	DISORDER EFFECTS IN BRIDGED MOLECULAR SYSTEMS. RANDOM MATRIX THEORY APPROACH
Authors	Ewa Gudowska-Nowak1, JÜrgen Brickmann2 and Gabor Papp3
Abstract	Advances in nanotechnologies and molecular assembly techniques have brought much attention to the problem of molecular wires studied with respect to disorder and to increased electronic connectivity. In this communication we aim to use the techniques of Random Matrix Theory (RMT) in the formalism of Free Random Variables (FRV) to analyze and predict electronic properties of one-dimensional disordered bridged molecular systems. We discuss possible application of the method in biological and chemical context. As an example, based on recent achievements in the theory of nonhermitian ensembles of random matrices, we outline here an efficient procedure to calculate electron transfer matrix in electron conducting disordered materials. The approach can be applied to the variety of problems like analysis of final state-selected spectra in unimolecular chemical reactions or population dynamics of biological species.
Adress and Contact Informations	1-Institute of Physics, Jagiellonian University, 30-059 Kraków, Poland 2-Institute for Physical Chemistry Technische Univesität Darmstadt, Petersenstr.20, D-64287 Germany 3-ITP, Universitat Heidelberg, Philosophenweg 19, D-6912 Heidelberg, Germany & Institute for Theoretical Physics, Eötvös University, H-1088 Budapest, Hungary

Volume 4 (1999) pp 37-54
Title	CHAOS IN THE POTASSIUM CURRENT THROUGH CHANNELS OF LOCUST MUSCLE MEMBRANE
Authors	Zbigniew J. Grzywna1* , Zuzanna Siwy1, Andrzej Fuliński2, Ian Mellor3 and Peter N.R. Usherwood3
Abstract	The nonlinear, pseudo-periodic current of potassium ions through a high conductance locust K+ channel (BK channel) has been modelled by a two-parameter logistic map (“crowd model”). Data obtained by the patch clamp for different values of potential difference has been correlated with a mechanism of transport incorporating dynamical structure and morphology of pores in a membrane. The ordering influence of applied voltage upon ionic current behaviour has been found and explained within the “crowd model”.
Adress and Contact Informations	1-Department of Physical Chemistry and Technology of Polymers Section of Physics, Mathematics and Computer Science, Silesian Technical University, 44-100 Gliwice, Poland 2-M. Smoluchowski Institute of Physics, Jagellonian University, Reymonta 4, 30-059 Kraków, Poland 3-University of Nottingham, Department of Zoology, University Park, Nottingham NG7 2 RD, U.K. *corresponding author

Volume 4 (1999) pp 55-86
Title	A GENERAL PROBABILISTIC APPROACH TO THE UNIVERSAL RELAXATION RESPONSE OF COMPLEX SYSTEMS
Authors	Agnieszka Jurlewicz1 and Karina Weron2
Abstract	A new probabilistic representation of the multichannel relaxation mechanism, which generates the universal two power-law relaxation response with the stretched exponential and the classical exponential decays as special cases, is presented. The consideration of irreversible stochastic transitions of complex systems is based on a general probabilistic formalism applied to the analysis of the first passage of a system. By means of limit theorems the origins of the universality of relaxation responses are indicated. This approach, without referring to the conventional stochastic transition description, allows us to derive explicitly the intensity of transition from an initial state for a complex system in the most general case of parallel channel relaxation with a random number of transition channels, each characterized by an individual relaxation rate. The nonexponential relaxation is shown to result from general properties of transition channels only, namely, from the asymptotical self-similar behavior of their relaxation rate distributions. For the reader’s convenience a survey of limit theorems of probability theory is included in the Appendix.
Adress and Contact Informations	1-Institute of Mathematics, Wroclaw University of Technology, Wyspiańskiego 27, 50-370 Wroclaw, Poland 2-Institute of Physics, Wroclaw University of Technology, Wyspiańskiego 27, 50-370 Wrocław, Poland

Volume 4 (1999) pp 87-104
Title	DETERMINICTIC AND NOISE-INDUCED SIGNAL AMPLIFICATION AND SIGNAL TRANSFER IN COUPLED NONLINEAR SYSTEMS
Authors	Friedmann Kaiser
Abstract	Nonlinear oscillatory processes are discussed under the influence of external signals to improve the understanding of signal interaction with and within biological systems. The biological endogenous thythms are modelled by self-sustained oscillations (limit cycles). Main emphasis is on the combined influence of very slow and very fast stimuli compared to the relevant internal frequencies and on additional effects caused by external and internal noise sources. The models represent arrays of coupled passive and active nonlinear oscillators, an external harmonic signal stimulates the input oscillator (initial stage of signal chain). Signal transfer through the pathway is studied under the influence of noise. Different noise contributions are considered, including spatially-coherent and spatially-incoherent sources. Results reveal a stochastic resonance kind of behaviour at different stages of the signal transfer, the harmonic signal is transduced through the whole system of copled oscillators. The combined action of different noise exhibits constructive as well as destructive influences on signal amplification. In addition, the influence of noise on the synchronous behaviour of coupled active systems is investigated. Noise-induced synchronization as well as desynchronization of the output signal to the external drive result. Besides signal amplification the systems exhibit the property to decode the frequency encoded information.
Adress and Contact Informations	Institute of Applied Physics – Nonlinear dynamics, Darmstadt University of Technology, 64289 Darmstadt , Germany

Volume 4 (1999) pp 105-116
Title	A COMPARISON BETWEEN DETERMINISTIC AND PROBABILISTIC APPROACHES TO THE PHENOMENON OF ANOMALOUS DIFFUSION IN TISSUE
Authors	Malgorzata Kotulska
Abstract	Two differently rooted descriptions of anomalous diffusion that can be observed in tissue dielectric response, in the range of dispersion ?, are presented. The probabilistic approach assumes a random nature of ion jumps and waiting times in the hopping conduction that is responsible for this dispersion. The other description stems from the assumption of deterministic character of ion transport equation. Results are supported by computer simulations.
Adress and Contact Informations	Wroclaw University of Technology, Faculty of Basic Problems of Technology, Division of Measuring & Medical Electronic Instruments, Wybrzeze Wyspianskiego 27, 50-370 Wroclaw, e-mail: kotulska@pwr.wroc.pl

Volume 4 (1999) pp 117-130
Title	IMPORTANCE OF INTRAMOLECULAR PROTEIN DYNAMICS TO KINETICS OF BIOCHEMICAL PROCESSES
Authors	Michal Kurzynski
Abstract	Several points seem essential for construction of the future statistical theory of biochemical processes. (a) the native proteins involved in these processes reveal a purely stochastic intramolecular dynamics of conformational transitions much slower than the usual vibrational dynamics. At least in the range from 10 to 10 s the relaxation time spectrum of conformational transition dynamics is practically quasi-continuous. (b) the majority of reactions involving proteins are controlled and, presumably, also gated by this stochastic dynamics. (c) of special importance is the short initial-condition dependent stage of biochemical reactions, neglected in the description of the reaction in terms of the standard kinetics. This stage is directly observed in experiments in which especially prepared initial conformational substates of the protein are confined to the reaction transition state. (d) the initial-condition dependent stage, and not that described by the standard kinetics, is responsible for the coupling of component reactions in the complete catalytic cycles proceeding in the steady-state and more complex processes of biological free energy transduction.
Adress and Contact Informations	Institute of Physics, A. Mickiewicz University, Umultowska 85, 61-614 Poznan, Poland, e-mail: kurzphys@phys.amu.edu.pl

Volume 4 (1999) pp 131-146
Title	KINETIC MODEL FOR OSCILLATIONS IN A CYCLE OF ENZYMATIC REACTIONS RELATED TO METHOXYPHENOLS TRANSFORMATION IN RHODOCOCCUS ERYTHROPOLIS CULTURE
Authors	Jan Sielewiesiuk1, Albin Czubla1, Elzbieta Malarczyk2 and Marzanna Pazdzioch2
Abstract	The four-membered cycle of enzymatic reactions with repression of enzyme synthesis in the presence of cyclic symmetry is presented. Experimental premises, formulation of the model, analytical analysis, bifurcations diagrams and numerical solutions are shown. By Hopf bifurcation theory, the conditions of oscillations were found for two kinds of the model: fully symmetric and extended one. The latter of them reconstructs better the experimental results than the former one. The models are more general and can be related to rings of coupled biological oscillators.
Adress and Contact Informations	1-Department of Biophysics, Institute of Physics 2-Department of Biochemistry, M. Curie-Sklodowska University, 20-031 Lublin, Poland