Spin Glasses and Biology

The following sections are included:

• Preface

• Contents

Two seminal events for the subject of this book happened in the year 1969: the first paper entitled “spin glass” was published¹ in which I borrowed the term “magnetic glass” from “cet Mauvais Gaulois”² by Bryan Coles and shortened it; and StuartKauffman proposed a Boolean network computer model for evolution.³ The first mode was random and — in a primitive fashion — non-ergodic, but not of course biological; the second was not yet random nor non-ergodic but biological. Much the same period marks the beginnings of complex neutral network theory with ideas due to Cooper, Little, Rumelhart and others. These three strands of theory did not begin to interact until barely 10 years ago,yet an examination of the bibliographies of the various chapters to this book will reveal an already rich literature. I suppose, if nothing is done to prevent it, that some enterprising editor can be expected to start a journal on spin glasses, as in fact one already exists on neutral networks…

The following sections are included:

• Introduction
  • Historical Perspective
  • Why Statistical Physics?
  • Purpose and Outline of the Paper

• Basic Elements of Neural Network Models
  • The Biological Neuron
  • From the Biological to the Formal Neuron
  • The Formal Neuron
  • Network Architecture
  • Network Dynamics

• Basic Functions of Neural Network Models
  • Associative Memory
  • Learning
  • Categorization
  • Generalization
  • Optimization

• The Hopfield Model
  • Solution of the Model
  • The Merit of the Hopfield Model
  • Beyond the Standard Model

• The Gardner Approach
  • A Microcanonical Formulation
  • The Case of Biased Patterns
  • A Canonical Formulation
  • Constraints on the Synaptic Weights
  • Learning with Errors
  • Learning with Noise

• Hierarchically Correlated Data and Categorization
  • Hierarchical Data Structures
  • Storage of Hierarchical Data Structures
  • Categorization

• Generalization
  • Learning a Classification Task
  • The Reference Perceptron Problem
  • The Contiguity Problem

• Discussion – Issues of Relevance
  • The Notion of Attractors and Modes of Computation
  • The Nature of Attractors
  • Temporal versus Spatial Coding

• Acknowledgements

• References

The following sections are included:

• Introduction

• Fitness Landscapes in Sequence Space

• The NK Model of Rugged Fitness Landscapes
  • The NK Model of Random Epistatic Interactions

• The Rank Order Statistics on K = N − 1 Random Landscapes
  • The number of local optima is very large
  • The expected fraction of fitter 1-mutant neighbors dwindles by 1/2 on each improvement step
  • Walks to local optima are short and vary as a logarithmic function of N
  • The expected time to reach an optimum is proportional to the dimensionality of the space
  • The ratio of accepted to tried mutations scales as lnN/N
  • Any genotype can only climb to a small fraction of the local optima
  • A small fraction of the genotypes can climb to any one optimum

• Conflicting constraints cause a “complexity catastrophe”: as complexity increase accessible adaptive peaks fall toward the mean fitness
  • The “Tunable” NK Family of Correlated Landscapes
  • Other Combinatorial Optimization Problems and Their Landscapes

• Summary

• References

This paper addresses the problem of biological information by introducing ameasure of its value inspired by spin glasses. A model for the origin and selection of biological information is then discussed, first from the point of view of a general theory, and later with respect to a collection of computer simulations of a specific model. This model demonstrates some interesting properties of autocatalytic systems, establishing its plausibility as a model for prebiological systems.

The following sections are included:

• Introduction
  • The Genotype/Phenotype Problem
  • Population Genetics
  • Kauffman Model of the Genome
  • Robustness

• Robustness in Networks of Automata
  • Sensitivity to initial conditions^4,5
  • Patterns
  • Other Examples of Robustness

• Evolution of Species
  • The Paradox of Punctuated Equilibria
  • Equations for Population Dynamics
  • Competition
  • Evolution
  • Solution of the Differential System by Perturbation Methods

• References

Problems involving multi-peaked fitness or energy landscapes have attractedattention in evolutionary biology, structural biology (protein folding), condensed matter physics (glasses and spin glasses) and computer science (combinatorial optimization). Motivated by an important problem in the biology of the immune system, involving the somatic evolution of antibody molecules, we developand analyze a mathematical model for molecular evolution on rugged landscapes.First, we characterize evolution on random landscapes and use the mathematicalresults to interpret immunological experiments on affinity maturation by somatic mutation. This analysis provides an explanation for the previously unexplained phenomenon in which mutations initially lead to large improvements in antibody fitness, but later lead to no or little improvement. Second, we show how to extend our mathematical results to a family of correlated landscapes, whichwe believe will be more appropriate for the quantitative interpretation of immunological data.

The following sections are included:

• Foreword: Just What Problem Are We Trying to Solve?

• What is a Spin Glass?

• What a Protein Glass Might Be

• Properties of Spin Glasses
  • Magnetic Susceptibility
  • Specific Heat Measurements of Spin Glasses
  • Distribution of Relaxation Rates and Energy Levels in Spin Glasses
  • Hierarchical States
  • Ultrametricity

• Spin Glasses and Elastic Orientational Glasses
  • Dielectric Susceptibility
  • Distribution of Relaxation Times
  • Elastic Moduli

• Spin Glasses and Proteins
  • Low Temperature Recombination Kinetics and Distributed Kinetics
  • Rare-Earth Luminescence and Conformational Disorder
  • Specific Heat Measurements of Proteins
  • Dielectric Susceptibility
  • Elastic Moduli
  • Passage Through the Glass Transition: The Kinetic Signatures
  • Hierarchical States
  • Ultrametricity

• References

We do not yet understand how protein sequence is translated into three-dimensional protein structure. This article explains how spin glass ideas may be of relevance in solving this “protein folding problem”. Spin glass phase transitions may influence the rate of folding. We discuss this using astatistical protein model. Sophisticated spin glass theories of heteropolymer collapse are also discussed. Connections with Hopfield associative memory models may give predictive schemes for solving the problem.