A Theory of Biological Pattern Formation Presented by Xia Fan

Introduction Formation of a spatial pattern of tissue structures is an essential process in embryology and regeneration. The formation of the pattern is generally assumed to result form a primary pattern of morphogen concentration. Several types of theories proposed for this primary pattern:  Auto- and cross- catalysis  Polar cells pump morphogens in one direction  Two periodic events of different wavelength

Introduction Two substance involved:  activator & inhibitor with mutual interaction ( they react auto- and cross catalytically on their sources. )  Different initial conditions lead to different stable states, which may represent states of differentiation.  Linear equations employing different diffusion terms produce unstable solutions. Stable states require non-linear reaction kinetics with restrictions. Three postulates (restrictions):  Short range activation  Long range inhibition, the inhibition being derived from, and extending into a wider area. (In most cases, long range inhibition has implications very similar to depletion of a substance, derived from a wider area, which is necessary for activation.)  Conceptual distinction between effective concentration and density of the source (density of sources chage at a slower rate than the production of effective activator or inhibitor.)

A Method for Generating Simple Theories of Pattern Formation Requirement: the equation should lead to patterns of morphogen concentration even with shallow gradients of source distributions and even initial distribution of activating and inhibiting substance. Notations: source density for activator source density for inhibitors activator concentration inhibitor concentration instead of inhibitor, the inhibiting effect of depletion of a substance ’ s concentration

A Method for Generating Simple Theories of Pattern Formation Assumption: synthesis of inhibitor depends on local activator concentration as a result of cross catalysis and inhibitor spreads and equilibrate fast within a wider area, thus inhibitor concentration can be approximated as a function of activator concentration averaged over the mean area from which the inhibitor is derived. It also depends on the mean inhibitor source density averaged over the area in which inhibitor is produced. Equation:  Activator concentration changes according to a rate given by the difference between production and destruction.  As a result of auto- and cross- catalysis both the production and destruction terms are assumed to be dependent on some powers of a and  Production rate is considered as proportional to the local activator source density.

A Method for Generating Simple Theories of Pattern Formation Criterium :  k>m to avoid negative values of a  n>m to meet the requirement of formation of primary pattern

A Method for Generating Simple Theories of Pattern Formation Proof :  Assuming a near-even distribution A psedudo-equilibrium is reached if n>m  if in a particular region, distribution is slightly above average We obtain  Resulting the firing of a gradient: after some time, a will be mainly confined to some fraction p of the total area, where a has an average value a*,whereas a in the remaining area will be small.

A Method for Generating Simple Theories of Pattern Formation Size regulation  Constant proportion of the parts of the pattern irrespective of total size  Generally, Without any limitation of either a* or p, the entire activation will concentrate on a infinitely small area. But with some limitations, a stationary stage can be reached. The relation between a* and p will be given approximately, This equation is essentially a relation between p and a*.

Special Models- Depletion Model Equation: Approximation: Simplest version

Special Models Activator-Inhibitor Models Equation: Approximation Simplest version  Common source  Different source

Special Models Activator-Inhibitor Models Chose this model to explain the size regualation

Examples of Basic Properties of Pattern Formation A Theory of Biological Pattern Formation.pdfA Theory of Biological Pattern Formation.pdf

Discussion Main aspects of biological pattern formation can be explained with this model. The model permits different versions and interpretation  Source gradients may be replaced by, or added to sink gradients.  Inhibition may be substituted by depletion.  Source may be either synthesizing systems, or special structures releasing activators and inhibitors.  Spreading in space can be diffusion or other mechanisms like convection.