Hans G. Othmer

The topology of the regulatory interactions predicts the expression pattern of the segment polarity genes in Drosophila melanogaster.

Expression of the Drosophila segment polarity genes is initiated by a pre-pattern of pair-rule gene products and maintained by a network of regulatory interactions throughout several stages of embryonic development. Analysis of a model of gene interactions based on differential equations showed that wild-type expression patterns of these genes can be obtained for a wide range of kinetic parameters, which suggests that the steady states are determined by the topology of the network and the type of regulatory interactions between components, not the detailed form of the rate laws. To investigate this, we propose and analyse a Boolean model of this network which is based on a binary ON/OFF representation of mRNA and protein levels, and in which the interactions are formulated as logical functions. In this model the spatial and temporal patterns of gene expression are determined by the topology of the network and whether components are present or absent, rather than the absolute levels of the mRNAs and proteins and the functional details of their interactions. The model is able to reproduce the wild-type gene expression patterns, as well as the ectopic expression patterns observed in overexpression experiments and various mutants. Furthermore, we compute explicitly all steady states of the network and identify the basin of attraction of each steady state. The model gives important insights into the functioning of the segment polarity gene network, such as the crucial role of the wingless and sloppy paired genes, and the networks ability to correct errors in the pre-pattern.

Models of dispersal in biological systems

In order to provide a general framework within which the dispersal of cells or organisms can be studied, we introduce two stochastic processes that model the major modes of dispersal that are observed in nature. In the first type of movement, which we call the position jump or kangaroo process, the process comprises a sequence of alternating pauses and jumps. The duration of a pause is governed by a waiting time distribution, and the direction and distance traveled during a jump is fixed by the kernel of an integral operator that governs the spatial redistribution. Under certain assumptions concerning the existence of limits as the mean step size goes to zero and the frequency of stepping goes to infinity the process is governed by a diffusion equation, but other partial differential equations may result under different assumptions. The second major type of movement leads to what we call a velocity jump process. In this case the motion consists of a sequence of “runs” separated by reorientations, during which a new velocity is chosen. We show that under certain assumptions this process leads to a damped wave equation called the telegraphers equation. We derive explicit expressions for the mean squared displacement and other experimentally observable quantities. Several generalizations, including the incorporation of a resting time between movements, are also studied. The available data on the motion of cells and other organisms is reviewed, and it is shown how the analysis of such data within the framework provided here can be carried out.

Aggregation, blowup, and collapse: the abc's of taxis in reinforced random walks

In many biological systems, movement of an organism occurs in response to a diffusible or otherwise transported signal, and in its simplest form this can be modeled by diffusion equations with advection terms of the form first derived by Patlak [Bull. of Math. Biophys., 15 (1953), pp. 311--338]. However, other systems are more accurately modeled by random walkers that deposit a nondiffusible signal that modifies the local environment for succeeding passages. In these systems, one example of which is the myxobacteria, the question arises as to whether aggregation is possible under suitable hypotheses on the transition rules and the production of a control species that modulates the transition rates. Davis [Probab. Theory Related Fields, 84 (1990), pp. 203--229] has studied this question for a certain class of random walks, and here we extend this analysis to the continuum limit of such walks. We first derive several general classes of partial differential equations that depend on how the movement rules are...

The Diffusion Limit of Transport Equations Derived from Velocity-Jump Processes

In this paper we study the diffusion approximation to a transport equation that describes the motion of individuals whose velocity changes are governed by a Poisson process. We show that under an appropriate scaling of space and time the asymptotic behavior of solutions of such equations can be approximated by the solution of a diffusion equation obtained via a regular perturbation expansion. In general the resulting diffusion tensor is anisotropic, and we give necessary and sufficient conditions under which it is isotropic. We also give a method to construct approximations of arbitrary high order for large times. In a second paper (Part II) we use this approach to systematically derive the limiting equation under a variety of external biases imposed on the motion. Depending on the strength of the bias, it may lead to an anisotropic diffusion equation, to a drift term in the flux, or to both. Our analysis generalizes and simplifies previous derivations that lead to the classical Patlak--Keller--Segel--Alt...

Shaping BMP morphogen gradients in the Drosophila embryo and pupal wing

In the early Drosophila embryo, BMP-type ligands act as morphogens to suppress neural induction and to specify the formation of dorsal ectoderm and amnioserosa. Likewise, during pupal wing development, BMPs help to specify vein versus intervein cell fate. Here, we review recent data suggesting that these two processes use a related set of extracellular factors, positive feedback, and BMP heterodimer formation to achieve peak levels of signaling in spatially restricted patterns. Because these signaling pathway components are all conserved, these observations should shed light on how BMP signaling is modulated in vertebrate development.

Facilitated Transport of a Dpp/Scw Heterodimer by Sog/Tsg Leads to Robust Patterning of the Drosophila Blastoderm Embryo

Patterning the dorsal surface of the Drosophila blastoderm embryo requires Decapentaplegic (Dpp) and Screw (Scw), two BMP family members. Signaling by these ligands is regulated at the extracellular level by the BMP binding proteins Sog and Tsg. We demonstrate that Tsg and Sog play essential roles in transporting Dpp to the dorsal-most cells. Furthermore, we provide biochemical and genetic evidence that a heterodimer of Dpp and Scw, but not the Dpp homodimer, is the primary transported ligand and that the heterodimer signals synergistically through the two type I BMP receptors Tkv and Sax. We propose that the use of broadly distributed Dpp homodimers and spatially restricted Dpp/Scw heterodimers produces the biphasic signal that is responsible for specifying the two dorsal tissue types. Finally, we demonstrate mathematically that heterodimer levels can be less sensitive to changes in gene dosage than homodimers, thereby providing further selective advantage for using heterodimers as morphogens.

THE DIFFUSION LIMIT OF TRANSPORT EQUATIONS II: CHEMOTAXIS EQUATIONS ∗

In this paper, we use the diffusion-limit expansion of transport equations developed earlier [T. Hillen and H. G. Othmer, SIAM J. Appl. Math., 61 (2000), pp. 751--775] to study the limiting equation under a variety of external biases imposed on the motion. When applied to chemotaxis or chemokinesis, these biases produce modification of the turning rate, the movement speed, or the preferred direction of movement. Depending on the strength of the bias, it leads to anisotropic diffusion, to a drift term in the flux, or to both, in the parabolic limit. We show that the classical chemotaxis equation---which we call the Patlak--Keller--Segel--Alt (PKSA) equation---arises only when the bias is sufficiently small. Using this general framework, we derive phenomenological models for chemotaxis of flagellated bacteria, of slime molds, and of myxobacteria. We also show that certain results derived earlier for one-dimensional motion can easily be generalized to two- or three-dimensional motion as well.

FROM INDIVIDUAL TO COLLECTIVE BEHAVIOR IN BACTERIAL CHEMOTAXIS

Bacterial chemotaxis is widely studied from both the microscopic (cell) and macroscopic (population) points of view, and here we connect these very different levels of description by deriving the classical macroscopic description for chemotaxis from a microscopic model of the behavior of individual cells. The analysis is based on the velocity jump process for describing the motion of individuals such as bacteria, wherein each individual carries an internal state that evolves according to a system of ordinary differential equations forced by a time- and/or space-dependent external signal. In the problem treated here the turning rate of individuals is a functional of the internal state, which in turn depends on the external signal. Using moment closure techniques in one space dimension, we derive and analyze a macroscopic system of hyperbolic differential equations describing this velocity jump process. Using a hyperbolic scaling of space and time, we obtain a single second-order hyperbolic equation for the...

Instability and dynamic pattern in cellular networks.

Abstract Dynamic instability of reaction and transport processes in groups of intercommunicating cells can lead to pattern formation and periodic oscillations. Turings 1952 analysis suggests a more general theory. A powerful new method for analyzing onset of instability in arbitrary networks of compartments or model cells is developed. Network structure is found to influence interaction of intracellular chemical reactions and intercellular transfers, and thereby the stability of uniform stationary states of the network. With the theory, effects of changes in network topology can be treated systematically. Several regular planar and polyhedral networks provide illustrations. Influences of boundary conditions and intercellular permeabilities on patterns of instability are illustrated in simple networks. Non-linear aspects of instability are not treated.

The BMP-Binding Protein Crossveinless 2 Is a Short-Range, Concentration-Dependent, Biphasic Modulator of BMP Signaling in Drosophila

In Drosophila, the secreted BMP-binding protein Short gastrulation (Sog) inhibits signaling by sequestering BMPs from receptors, but enhances signaling by transporting BMPs through tissues. We show that Crossveinless 2 (Cv-2) is also a secreted BMP-binding protein that enhances or inhibits BMP signaling. Unlike Sog, however, Cv-2 does not promote signaling by transporting BMPs. Rather, Cv-2 binds cell surfaces and heparan sulfate proteoglygans and acts over a short range. Cv-2 binds the type I BMP receptor Thickveins (Tkv), and we demonstrate how the exchange of BMPs between Cv-2 and receptor can produce the observed biphasic response to Cv-2 concentration, where low levels promote and high levels inhibit signaling. Importantly, we show also how the concentration or type of BMP present can determine whether Cv-2 promotes or inhibits signaling. We also find that Cv-2 expression is controlled by BMP signaling, and these combined properties enable Cv-2 to exquisitely tune BMP signaling.