Interactions of a Signal Transduction Protein Investigated by Fluorescence Stopped-Flow Kinetics.

To understand cellular processes such as biochemical pathways and signaling networks, we need to understand binding and reaction rates of often competing reactions, their dependence on cellular concentrations of participating molecules, and the regulation of these rates through allostery, posttranslational modifications, or other mechanisms. To do so, we break these systems down into their elementary steps, which are almost invariably either unimolecular or bimolecular reactions that frequently occur on sub-second, often sub-millisecond, time scales. Rapid mixing techniques, which generally achieve mixing in less than 2 ms, are generally suitable for the study of such reactions. The application of these techniques to the study of enzyme mechanisms is described in several excellent texts (Cornish-Bowden, Fundamentals of enzyme kinetics, 1995; Gutfreund, Kinetics for the life sciences. Receptors, transmitters and catalysis, 1995); flow techniques are used to study individual steps by monitoring the approach to equilibrium (the pre-steady state) under single turnover conditions.The individual steps in complex biochemical reaction schemes determine how fast systems can respond to incoming signals and adapt to changed conditions [1, 2]. This chapter is concerned with in vitro techniques that have been developed to study fast reactions in solution, and we present the study of various interactions of calmodulin as an example. The kinetic information obtained with these techniques is indispensable for understanding the dynamics of biochemical processes and complements the static structural and thermodynamic information available from X-ray crystallography, NMR, and equilibrium binding studies.

Binding of Filamentous Actin to CaMKII as Potential Regulation Mechanism of Bidirectional Synaptic Plasticity by ß CaMKII in Cerebellar Purkinje Cells.

Calcium-calmodulin dependent protein kinase II (CaMKII) regulates many forms of synaptic plasticity, but little is known about its functional role during plasticity induction in the cerebellum. Experiments have indicated that the ß isoform of CaMKII controls the bidirectional inversion of plasticity at parallel fibre (PF)-Purkinje cell (PC) synapses in cerebellar cortex. Because the cellular events that underlie these experimental findings are still poorly understood, we developed a simple computational model to investigate how ß CaMKII regulates the direction of plasticity in cerebellar PCs. We present the first model of AMPA receptor phosphorylation that simulates the induction of long-term depression (LTD) and potentiation (LTP) at the PF-PC synapse. Our simulation results suggest that the balance of CaMKII-mediated phosphorylation and protein phosphatase 2B (PP2B)-mediated dephosphorylation of AMPA receptors can determine whether LTD or LTP occurs in cerebellar PCs. The model replicates experimental observations that indicate that ß CaMKII controls the direction of plasticity at PF-PC synapses, and demonstrates that the binding of filamentous actin to CaMKII can enable the ß isoform of the kinase to regulate bidirectional plasticity at these synapses.

Symmetry structure in discrete models of biochemical systems: natural subsystems and the weak control hierarchy in a new model of computation driven by interactions.

Interaction computing is inspired by the observation that cell metabolic/regulatory systems construct order dynamically, through constrained interactions between their components and based on a wide range of possible inputs and environmental conditions. The goals of this work are to (i) identify and understand mathematically the natural subsystems and hierarchical relations in natural systems enabling this and (ii) use the resulting insights to define a new model of computation based on interactions that is useful for both biology and computation. The dynamical characteristics of the cellular pathways studied in systems biology relate, mathematically, to the computational characteristics of automata derived from them, and their internal symmetry structures to computational power. Finite discrete automata models of biological systems such as the lac operon, the Krebs cycle and p53-mdm2 genetic regulation constructed from systems biology models have canonically associated algebraic structures (their transformation semigroups). These contain permutation groups (local substructures exhibiting symmetry) that correspond to 'pools of reversibility'. These natural subsystems are related to one another in a hierarchical manner by the notion of 'weak control'. We present natural subsystems arising from several biological examples and their weak control hierarchies in detail. Finite simple non-Abelian groups are found in biological examples and can be harnessed to realize finitary universal computation. This allows ensembles of cells to achieve any desired finitary computational transformation, depending on external inputs, via suitably constrained interactions. Based on this, interaction machines that grow and change their structure recursively are introduced and applied, providing a natural model of computation driven by interactions.

Rapid mixing kinetic techniques.

Almost all of the elementary steps in a biochemical reaction scheme are either unimolecular or bimolecular processes that frequently occur on sub-second, often sub-millisecond, time scales. The traditional approach in kinetic studies is to mix two or more reagents and monitor the changes in concentrations with time. Conventional spectrophotometers cannot generally be used to study reactions that are complete within less than about 20 s, as it takes that amount of time to manually mix the reagents and activate the instrument. Rapid mixing techniques, which generally achieve mixing in less than 2 ms, overcome this limitation. This chapter is concerned with the use of these techniques in the study of reactions which reach equilibrium; the application of these methods to the study of enzyme kinetics is described in several excellent texts (Cornish-Bowden, Fundamentals of enzyme kinetics. Portland Press, 1995; Gutfreund, Kinetics for the life sciences. Receptors, transmitters and catalysis. Cambridge University Press, 1995).There are various ways to monitor changes in concentration of reactants, intermediates and products after mixing, but the most common way is to use changes in optical signals (absorbance or fluorescence) which often accompany reactions. Although absorbance can sometimes be used, fluorescence is often preferred because of its greater sensitivity, particularly in monitoring conformational changes. Such methods are continuous with good time resolution but they seldom permit the direct determination of the concentrations of individual species. Alternatively, samples may be taken from the reaction volume, mixed with a chemical quenching agent to stop the reaction, and their contents assessed by techniques such as HPLC. These methods can directly determine the concentrations of different species, but are discontinuous and have a limited time resolution.

Exploring the concept of interaction computing through the discrete algebraic analysis of the Belousov-Zhabotinsky reaction.

Interaction computing (IC) aims to map the properties of integrable low-dimensional non-linear dynamical systems to the discrete domain of finite-state automata in an attempt to reproduce in software the self-organizing and dynamically stable properties of sub-cellular biochemical systems. As the work reported in this paper is still at the early stages of theory development it focuses on the analysis of a particularly simple chemical oscillator, the Belousov-Zhabotinsky (BZ) reaction. After retracing the rationale for IC developed over the past several years from the physical, biological, mathematical, and computer science points of view, the paper presents an elementary discussion of the Krohn-Rhodes decomposition of finite-state automata, including the holonomy decomposition of a simple automaton, and of its interpretation as an abstract positional number system. The method is then applied to the analysis of the algebraic properties of discrete finite-state automata derived from a simplified Petri net model of the BZ reaction. In the simplest possible and symmetrical case the corresponding automaton is, not surprisingly, found to contain exclusively cyclic groups. In a second, asymmetrical case, the decomposition is much more complex and includes five different simple non-abelian groups whose potential relevance arises from their ability to encode functionally complete algebras. The possible computational relevance of these findings is discussed and possible conclusions are drawn.

Genetic algorithms and their application to in silico evolution of genetic regulatory networks.

A genetic algorithm (GA) is a procedure that mimics processes occurring in Darwinian evolution to solve computational problems. A GA introduces variation through "mutation" and "recombination" in a "population" of possible solutions to a problem, encoded as strings of characters in "genomes," and allows this population to evolve, using selection procedures that favor the gradual enrichment of the gene pool with the genomes of the "fitter" individuals. GAs are particularly suitable for optimization problems in which an effective system design or set of parameter values is sought.In nature, genetic regulatory networks (GRNs) form the basic control layer in the regulation of gene expression levels. GRNs are composed of regulatory interactions between genes and their gene products, and are, inter alia, at the basis of the development of single fertilized cells into fully grown organisms. This paper describes how GAs may be applied to find functional regulatory schemes and parameter values for models that capture the fundamental GRN characteristics. The central ideas behind evolutionary computation and GRN modeling, and the considerations in GA design and use are discussed, and illustrated with an extended example. In this example, a GRN-like controller is sought for a developmental system based on Lewis Wolpert's French flag model for positional specification, in which cells in a growing embryo secrete and detect morphogens to attain a specific spatial pattern of cellular differentiation.

Stochastic model of template-directed elongation processes in biology.

We present a novel modular, stochastic model for biological template-based linear chain elongation processes. In this model, elongation complexes (ECs; DNA polymerase, RNA polymerase, or ribosomes associated with nascent chains) that span a finite number of template units step along the template, one after another, with semaphore constructs preventing overtaking. The central elongation module is readily extended with modules that represent initiation and termination processes. The model was used to explore the effect of EC span on motor velocity and dispersion, and the effect of initiation activator and repressor binding kinetics on the overall elongation dynamics. The results demonstrate that (1) motors that move smoothly are able to travel at a greater velocity and closer together than motors that move more erratically, and (2) the rate at which completed chains are released is proportional to the occupancy or vacancy of activator or repressor binding sites only when initiation or activator/repressor dissociation is slow in comparison with elongation.

Simple stochastic simulation.

Stochastic simulations may be used to describe changes with time of a reaction system in a way that explicitly accounts for the fact that molecules show a significant degree of randomness in their dynamic behavior. The stochastic approach is almost invariably used when small numbers of molecules or molecular assemblies are involved because this randomness leads to significant deviations from the predictions of the conventional deterministic (or continuous) approach to the simulation of biochemical kinetics. Advances in computational methods over the three decades that have elapsed since the publication of Daniel Gillespie's seminal paper in 1977 (J. Phys. Chem. 81, 2340-2361) have allowed researchers to produce highly sophisticated models of complex biological systems. However, these models are frequently highly specific for the particular application and their description often involves mathematical treatments inaccessible to the nonspecialist. For anyone completely new to the field to apply such techniques in their own work might seem at first sight to be a rather intimidating prospect. However, the fundamental principles underlying the approach are in essence rather simple, and the aim of this article is to provide an entry point to the field for a newcomer. It focuses mainly on these general principles, both kinetic and computational, which tend to be not particularly well covered in specialist literature, and shows that interesting information may even be obtained using very simple operations in a conventional spreadsheet.

Do motifs reflect evolved function?--No convergent evolution of genetic regulatory network subgraph topologies.

Methods that analyse the topological structure of networks have recently become quite popular. Whether motifs (subgraph patterns that occur more often than in randomized networks) have specific functions as elementary computational circuits has been cause for debate. As the question is difficult to resolve with currently available biological data, we approach the issue using networks that abstractly model natural genetic regulatory networks (GRNs) which are evolved to show dynamical behaviors. Specifically one group of networks was evolved to be capable of exhibiting two different behaviors ("differentiation") in contrast to a group with a single target behavior. In both groups we find motif distribution differences within the groups to be larger than differences between them, indicating that evolutionary niches (target functions) do not necessarily mold network structure uniquely. These results show that variability operators can have a stronger influence on network topologies than selection pressures, especially when many topologies can create similar dynamics. Moreover, analysis of motif functional relevance by lesioning did not suggest that motifs were of greater importance to the functioning of the network than arbitrary subgraph patterns. Only when drastically restricting network size, so that one motif corresponds to a whole functionally evolved network, was preference for particular connection patterns found. This suggests that in non-restricted, bigger networks, entanglement with the rest of the network hinders topological subgraph analysis.

Automatic analysis of computation in biochemical reactions.

We propose a modeling and analysis method for biochemical reactions based on finite state automata. This is a completely different approach compared to traditional modeling of reactions by differential equations. Our method aims to explore the algebraic structure behind chemical reactions using automatically generated coordinate systems. In this paper we briefly summarize the underlying mathematical theory (the algebraic hierarchical decomposition theory of finite state automata) and describe how such automata can be derived from the description of chemical reaction networks. We also outline techniques for the flexible manipulation of existing models. As a real-world example we use the Krebs citric acid cycle.

Bio-logic: gene expression and the laws of combinatorial logic.

At the heart of the development of fertilized eggs into fully formed organisms and the adaptation of cells to changed conditions are genetic regulatory networks (GRNs). In higher multicellular organisms, signal selection and multiplexing are performed at the cis-regulatory domains of genes, where combinations of transcription factors (TFs) regulate the rates at which the genes are transcribed into mRNA. To be able to act as activators or repressors of gene transcription, TFs must first bind to target sequences on the regulatory domains. Two TFs that act in concert may bind entirely independently of each other, but more often binding of the first one will alter the affinity of the other for its binding site. This article presents a systematic investigation into the effect of TF binding dependences on the predicted regulatory function of this bio-logic. Four extreme scenarios, commonly used to classify enzyme activation and inhibition patterns, for the binding of two TFs were explored: independent (the TFs bind without affecting each other's affinities), competitive (the TFs compete for the same binding site), ordered (the TFs bind in a compulsory order), and joint binding (the TFs either bind as a preformed complex, or binding of one is virtually impossible in the absence of the other). The conclusions are: (1) the laws of combinatorial logic hold only for systems with independently binding TFs; (2) systems formed according to the other scenarios can mimic the functions of their Boolean logical counterparts, but cannot be combined or decomposed in the same way; and (3) the continuously scaled output of systems consisting of competitively binding activators and repressors can be controlled more robustly than that of single TF or (quasi-)logical multi-TF systems.

Genetic regulatory network models of biological clocks: evolutionary history matters.

We study the evolvability and dynamics of artificial genetic regulatory networks (GRNs), as active control systems, realizing simple models of biological clocks that have evolved to respond to periodic environmental stimuli of various kinds with appropriate periodic behaviors. GRN models may differ in the evolvability of expressive regulatory dynamics. A new class of artificial GRNs with an evolvable number of complex cis-regulatory control sites--each involving a finite number of inhibitory and activatory binding factors--is introduced, allowing realization of complex regulatory logic. Previous work on biological clocks in nature has noted the capacity of clocks to oscillate in the absence of environmental stimuli, putting forth several candidate explanations for their observed behavior, related to anticipation of environmental conditions, compartmentation of activities in time, and robustness to perturbations of various kinds or to unselected accidents of neutral selection. Several of these hypotheses are explored by evolving GRNs with and without (Gaussian) noise and blackout periods for environmental stimulation. Robustness to certain types of perturbation appears to account for some, but not all, dynamical properties of the evolved networks. Unselected abilities, also observed for biological clocks, include the capacity to adapt to change in wavelength of environmental stimulus and to clock resetting.

Circular dichroism and its application to the study of biomolecules.

Circular dichroism (CD) is an excellent method for the study of the conformations adopted by proteins and nucleic acids in solution. Although not able to provide the beautifully detailed residue-specific information available from nuclearmagnetic resonance (NMR) and X-ray crystallography, CD measurements have two major advantages: they can be made on small amounts of material in physiological buffers and they provide one of the best methods for monitoring any structural alterations that might result from changes in environmental conditions, such as pH, temperature, and ionic strength. This chapter describes the important basic steps involved in obtaining reliable CD spectra: careful instrument and sample preparation, the selection of appropriate parameters for data collection, and methods for subsequent data processing. The principal features of protein and nucleic acid CD spectra are then described, and the main applications of CD are discussed. These include: methods for analyzing CD data to estimate the secondary structure composition of proteins, methods for following the unfolding of proteins as a function of temperature or added chemical denaturants, the study of the effects of mutations on protein structure and stability, and methods for studying macromolecule-ligand and macromolecule-macromolecule interactions.

Rapid kinetic techniques.

The elementary steps in complex biochemical reaction schemes (isomerization, dissociation, and association reactions) ultimately determine how fast any system can react in responding to incoming signals and in adapting to new conditions. Many of these steps have associated rate constants that result in subsecond responses to incoming signals or externally applied changes. This chapter is concerned with the techniques that have been developed to study such rapidly reacting systems in vitro and to determine the values of the rate constants for the individual steps. We focus principally on two classes of techniques: (1) flow techniques, in which two solutions are mixed within a few milliseconds and the ensuing reaction monitored over milliseconds to seconds, and (2) relaxation techniques, in which a small perturbation to an existing equilibrium is applied within a few microseconds and the response of the system is followed over microseconds to hundreds of milliseconds. These reactions are most conveniently monitored by recording the change in some optical signal, such as absorbance or fluorescence. We discuss the instrumentation that is (commercially) available to study fast reactions and describe a number of optical probes (chromophores) that can be used to monitor the changes. We discuss the experimental design appropriate for the different experimental techniques and reaction mechanisms, as well as the fundamental theoretical concepts behind the analysis of the data obtained.

Methods for simulating the dynamics of complex biological processes.

In this chapter, we provide the basic information required to understand the central concepts in the modeling and simulation of complex biochemical processes. We underline the fact that most biochemical processes involve sequences of interactions between distinct entities (molecules, molecular assemblies), and also stress that models must adhere to the laws of thermodynamics. Therefore, we discuss the principles of mass-action reaction kinetics, the dynamics of equilibrium and steady state, and enzyme kinetics, and explain how to assess transition probabilities and reactant lifetime distributions for first-order reactions. Stochastic simulation of reaction systems in well-stirred containers is introduced using a relatively simple, phenomenological model of microtubule dynamic instability in vitro. We demonstrate that deterministic simulation [by numerical integration of coupled ordinary differential equations (ODE)] produces trajectories that would be observed if the results of many rounds of stochastic simulation of the same system were averaged. In Section V, we highlight several practical issues with regard to the assessment of parameter values. We draw some attention to the development of a standard format for model storage and exchange, and provide a list of selected software tools that may facilitate the model building process, and can be used to simulate the modeled systems.

An elastically tethered viscous load imposes a regular gait on the motion of myosin-V. Simulation of the effect of transient force relaxation on a stochastic process.

Myosin-V is a processive molecular motor that moves membrane vesicles along actin tracks. In the simple model for motor and cargo motion investigated here, an elastic connection between motor and cargo transiently absorbs the abrupt mechanical transitions of the motor, and allows smooth relaxation of the cargo to a new position. We use a stochastic description to model motor stepping, with kinetics that depends on the instantaneous force exerted on the motor through the elastic connection. Tether relaxation is modelled as a continuous process, in which the rate is determined by the viscous drag of the cargo and the stiffness profile of the connection. Quantitative combined stochastic-continuous simulation of the dynamics of this system shows that bulky loads can impose a highly regular gait on the motor. If the characteristics of the elastic connection are similar to those of the myosin-II coiled-coil domain, the myosin-V motor, tether and cargo form a true escapement, in which the motor only escapes from its current position after one or more force thresholds have been crossed. Multiple thresholds limit the variation in tether length to values below that of the total step size.

CellML2SBML: conversion of CellML into SBML.

UNLABELLED: CellML and SBML are XML-based languages for storage and exchange of molecular biological and physiological reaction models. They use very similar subsets of MathML to specify the mathematical aspects of the models. CellML2SBML is implemented as a suite of XSLT stylesheets that, when applied consecutively, convert models expressed in CellML into SBML without significant loss of information. The converter is based on the most recent stable versions of the languages (CellML version 1.1; SBML Level 2 Version 1), and the XSLT used in the stylesheets adheres to the XSLT version 1.0 specification. Of all 306 models in the CellML repository in April 2005, CellML2SBML converted 91% automatically into SBML. Minor manual changes to the unit definitions in the originals raised the percentage of successful conversions to 96%. AVAILABILITY: http://sbml.org/software/cellml2sbml/. SUPPLEMENTARY INFORMATION: Instructions for use and further documentation available on http://sbml.org/software/cellml2sbml/

New computational approaches for analysis of cis-regulatory networks.

The investigation and modeling of gene regulatory networks requires computational tools specific to the task. We present several locally developed software tools that have been used in support of our ongoing research into the embryogenesis of the sea urchin. These tools are especially well suited to iterative refinement of models through experimental and computational investigation. They include: BioArray, a macroarray spot processing program; SUGAR, a system to display and correlate large-BAC sequence analyses; SeqComp and FamilyRelations, programs for comparative sequence analysis; and NetBuilder, an environment for creating and analyzing models of gene networks. We also present an overview of the process used to build our model of the Strongylocentrotus purpuratus endomesoderm gene network. Several of the tools discussed in this paper are still in active development and some are available as open source.

A provisional regulatory gene network for specification of endomesoderm in the sea urchin embryo.

We present the current form of a provisional DNA sequence-based regulatory gene network that explains in outline how endomesodermal specification in the sea urchin embryo is controlled. The model of the network is in a continuous process of revision and growth as new genes are added and new experimental results become available; see http://www.its.caltech.edu/~mirsky/endomeso.htm (End-mes Gene Network Update) for the latest version. The network contains over 40 genes at present, many newly uncovered in the course of this work, and most encoding DNA-binding transcriptional regulatory factors. The architecture of the network was approached initially by construction of a logic model that integrated the extensive experimental evidence now available on endomesoderm specification. The internal linkages between genes in the network have been determined functionally, by measurement of the effects of regulatory perturbations on the expression of all relevant genes in the network. Five kinds of perturbation have been applied: (1) use of morpholino antisense oligonucleotides targeted to many of the key regulatory genes in the network; (2) transformation of other regulatory factors into dominant repressors by construction of Engrailed repressor domain fusions; (3) ectopic expression of given regulatory factors, from genetic expression constructs and from injected mRNAs; (4) blockade of the beta-catenin/Tcf pathway by introduction of mRNA encoding the intracellular domain of cadherin; and (5) blockade of the Notch signaling pathway by introduction of mRNA encoding the extracellular domain of the Notch receptor. The network model predicts the cis-regulatory inputs that link each gene into the network. Therefore, its architecture is testable by cis-regulatory analysis. Strongylocentrotus purpuratus and Lytechinus variegatus genomic BAC recombinants that include a large number of the genes in the network have been sequenced and annotated. Tests of the cis-regulatory predictions of the model are greatly facilitated by interspecific computational sequence comparison, which affords a rapid identification of likely cis-regulatory elements in advance of experimental analysis. The network specifies genomically encoded regulatory processes between early cleavage and gastrula stages. These control the specification of the micromere lineage and of the initial veg(2) endomesodermal domain; the blastula-stage separation of the central veg(2) mesodermal domain (i.e., the secondary mesenchyme progenitor field) from the peripheral veg(2) endodermal domain; the stabilization of specification state within these domains; and activation of some downstream differentiation genes. Each of the temporal-spatial phases of specification is represented in a subelement of the network model, that treats regulatory events within the relevant embryonic nuclei at particular stages.

The temperature dependence of P680(+) reduction in oxygen-evolving photosystem II.

The temperature dependence for the reduction of the oxidized primary electron donor P680(+) by the redox active tyrosine Y(Z) has been studied in oxygen-evolving photosystem II preparations from spinach. The observed temperature dependence is found to vary markedly with the S-state of the manganese cluster. In the higher oxidation states, S(2) and S(3), sub-microsecond P680(+) reduction exhibits activation energies of about 260 meV. In contrast, there is only a small temperature dependence for the sub-microsecond reaction in the S(0) and S(1) states (an activation energy of approximately 50 meV). Slower microsecond components of P680(+) reduction show an activation energy of about 250 meV which, within experimental error, is independent of the oxidation state of the Mn cluster. By combining these values with measurements of DeltaG for electron transfer, the reorganization energies for each component of P680(+) reduction have been calculated. High activation and reorganization energies are found for sub-microsecond P680(+) reduction in S(2) and S(3), demonstrating that these electron transfers are coupled to significant reorganization events which do not occur in the presence of the lower S-states. One interpretation of these results is that there is an increase in the net charge on the manganese cluster on the S(1) to S(2) transition which acts as a barrier to electron transfer in the higher S-states. This argues against the electroneutrality requirement for some models of the function of the manganese cluster and hence against a role for Y(Z) as a hydrogen abstractor on all S-state transitions. An alternative or additional possibility is that there are proton (or other ion) motions in the sub-microsecond phases in S(2) and S(3) which contribute to the large reorganization energies observed, these motions being absent in the S(0) and S(1) states. Indeed charge accumulation may directly cause the increased reorganization energy.