Systems of Colour in Biology Björn Birnir Director of Center for Nonlinear Science Department of Mathematics University of California, Santa Barbara

Collaborators •Jean DeRousseaux •Dept. of Anthropology, UCSB •Expert on growth and evolution of rhesus monkeys •Christina Levyssohn-Sylva •Undergraduate researcher at UCSB •Currently in UC education abroad program in Ghana

•Systems of colour (SOC) are systems that magnify the white noise in the environment and are then driven by large coloured noise •The mathematical models that have been developed to describe such systems consist of stochastic partial differential equations driven by coloured noise •Such equations are notoriously hard to solve numerically. There are computational pitfalls and the standard methods fail SOC a new paradigm in biology

? Self-organized-critical systems ? Ill-posedness and Stochasticity ? Stochastic PDEs and the width function ? The phenotype of rhesus monkeys ? Scaling exponents of males and females ? Free-ranging versus caged populations ? Conclusion Outline

What is a SOC system? Heuristically, a system (governed by a nonlinear PDE or a map, or cellular automata) that has some statistical quantity, for example the ensamble averaged width, that possesses a scaling

Interpretations • There is an equivalence between time and distance in space t ~ |x|^z given by the dynamic exponent z • The system (width function) roughens initially as a power of t given by the temporal roughness coefficient beta • Eventually the system gets into a statistically stationary state where it does not roughen any more, but spatial fluctuations scale with a power of the lag variable (correlation length), given by the spatial roughness coefficient chi • Only two of these coefficients are independent

The map from the genotype to the phenotype is a random variable • Genes are pleiotropic, a single gene simultaneously influence several aspects of the phenotype • Components of the phenotype are polygenically determined, or affected by many genes • The processes determining the phenotype are very complex. Many components are involved in a complex interaction that take place in a noisy environment

The processes may be ill-posed • Small noise that is always present in the system gets magnified and drives the system • This phenomenon may be essential for the process, without it, it may not to take place at all • The system by which a phenotype component is made by a gene complex may be a System of Colour (SOC) • It is not a deterministic but a random process constrained by the domain of genes • How do we characterize this random variable?

How do nonlinear PDEs generate noise? •The initial value problem is ill-posed • Small perturbations grow exponentially •The nonlinearities saturate the exponential growth

Physical Characteristics of Rhesus Monkeys •Free Ranging •Naturally formed social groups •Veterinary policy of non-intervention •Housed indoors in single gang cages •Breeding program •Routine veterinary attention Cayo Santiago, Puerto Rico Primate Res. Ctr. Wisconsin Two different Populations

The measured phenotypes, Weight, Breadth of the head, Sitting height, Arm length, Leg length, Abdominal circumference, Calf circumference, Globe (eye) diameter These characteristics were measured and tabulated as a function of age in months

Computations of width function • We compute the width function as a function of the time lag variable t between two instances (ages) in time •This measures how the growth rates at different times are indicated •The variables are plotted on a log-log plot log(V) versus log(t) •The scaling exponents are the slopes of the straight lines

Observations •The width functions possess scalings •The scaling exponents are different for different phonetype •One phenotype (abdominal circumference) has the scaling of Brownian motion •Another phenotype (globe diameter) does not scale •The scalings of the sexes are different for the same phenotype

The Mathematical SOC Theory • A heat equation driven by white noise W(dy,ds) Where the A^k_t are mean zero Gaussian Markov processes, independent for different j They are called Ornstein-Uhlenbeck proceses

Edward-Wilkinson Theory Theorem The width function has the form There exists an initial transient state where V grows as towards a stationary state where In the stationary state V scales with distance x (between any two points) as

SOC Processes • The process possesses a scaling, so the width function V = scales with a temporal roughness exponent beta during the initial transients and the spatial roughness exponent chi in the statistically stationary state • The process projects the dynamics to a subspace H' of the original phase space H as t tends to infinity • There exists a unique invariant measure mu on this subspace • The process is multifractal so that the higher moments rho_n, n > 2 scale with exponents beta_n, which are independent of beta, during the initial transients A stochastic process is a SOC process if it possesses both a transient growth state and a statistically stationary state satisfying the following four conditions:

The Invariant Measure Let H' denote the subspace There exists an invariant measure on H'

Conclusions •Some bio-systems are deterministic but not predictable •Statistical quantities (variogram, moments) are predictable •There exist transient and stationary states characterized by scalings •The phenotypes of the rhesus monkeys are dermined by random variables, some SOC, some white •The sexes co-evolve, the phenotypes of caged animals being less complex or white