Frederic Y. M. Wan

Do Morphogen Gradients Arise by Diffusion

Many patterns of cell and tissue organization are specified during development by gradients of morphogens, substances that assign different cell fates at different concentrations. Gradients form by morphogen transport from a localized site, but whether this occurs by simple diffusion or by more elaborate mechanisms is unclear. We attempt to resolve this controversy by analyzing recent data in ways that appropriately capture the complexity of systems in which transport, receptor interaction, endo- and exocytosis, and degradation occur together. We find that diffusive mechanisms of morphogen transport are much more plausible-and nondiffusive mechanisms much less plausible-than has generally been argued. Moreover, we show that a class of experiments, endocytic blockade, thought to effectively distinguish between diffusive and nondiffusive transport models actually fails to draw useful distinctions.

Extraction Costs in the Theory of Exhaustible Resources

This paper has two objectives. First, it reformulates the theory of optimal use of an exhaustible resource with more attention to the costs of extraction than has been customary in the literature. The output and shadow-price implications of optimal extraction are studied under these broader assumptions. Second, the paper provides some numerical solutions of a simple two-grade case, to give some feeling for the quantitative importance of changes in the supply of exhaustible resources. Our most striking result is, in fact, the suggestion that relatively large changes in resource availability generate very small changes in the sustainable level of final consumption.

Cell Lineages and the Logic of Proliferative Control

It is widely accepted that the growth and regeneration of tissues and organs is tightly controlled. Although experimental studies are beginning to reveal molecular mechanisms underlying such control, there is still very little known about the control strategies themselves. Here, we consider how secreted negative feedback factors (“chalones”) may be used to control the output of multistage cell lineages, as exemplified by the actions of GDF11 and activin in a self-renewing neural tissue, the mammalian olfactory epithelium (OE). We begin by specifying performance objectives—what, precisely, is being controlled, and to what degree—and go on to calculate how well different types of feedback configurations, feedback sensitivities, and tissue architectures achieve control. Ultimately, we show that many features of the OE—the number of feedback loops, the cellular processes targeted by feedback, even the location of progenitor cells within the tissue—fit with expectations for the best possible control. In so doing, we also show that certain distinctions that are commonly drawn among cells and molecules—such as whether a cell is a stem cell or transit-amplifying cell, or whether a molecule is a growth inhibitor or stimulator—may be the consequences of control, and not a reflection of intrinsic differences in cellular or molecular character.

On plate theories and Saint-Venant's principle

Abstract It is generally known (hat the classical Germain-Kirchhoff plate theory is the leading term of the outer (asymptotic expansion or interior) solution in a small thickness parameter for the linear elastostatics of thin, flat, isotropic bodies. This leading term (or the actual) outer solution alone cannot satisfy arbitrarily prescribed admissible edge-data. On the other hand, the complementary inner (asymptotic expansion or boundary layer) solution is determined by a sequence of boundary value problems which are nearly as difficult to solve as the original problem. For stress edge-data, Saint-Venants principle has been invoked to generate a set of stress boundary conditions for the classical plate theory as well as for higher order terms in the outer expansion (giving various thick plate theories) without any reference to the inner solution. Attempts to derive the corresponding boundary conditions for displacement and other types of edge-data in the literature for general shape plates have not been successful. The present study applies a general method developed by the authors to derive the correct set of boundary conditions for arbitrarily prescribed admissible edge-data (without an explicit solution of the inner (or boundary layer) solution) for a number of special cases of general interest, including cases with displacement edge-data. Our general results also show that, to be strictly correct, Saint-Venants principle should be applied only to the leading term outer solution, i.e. the classical plate theory.

The Measure of Success: Constraints, Objectives, and Tradeoffs in Morphogen-mediated Patterning

A large, diverse, and growing number of strategies have been proposed to explain how morphogen gradients achieve robustness and precision. We argue that, to be useful, the evaluation of such strategies must take into account the constraints imposed by competing objectives and performance tradeoffs. This point is illustrated through a mathematical and computational analysis of the strategy of self-enhanced morphogen clearance. The results suggest that the usefulness of this strategy comes less from its ability to increase robustness to morphogen source fluctuations per se, than from its ability to overcome specific kinds of noise, and to increase the fraction of a morphogen gradient within which robust threshold positions may be established. This work also provides new insights into the longstanding question of why morphogen gradients show a maximum range in vivo.

The Interior Solution for Linear Problems of Elastic Plates

1 Necessary conditions have been established recently for the prescribed data along the cylindrical edge(s) of an elastic flat plate to induce only an exponentially decaying elastostatic state. The present paper describes how these conditions may be used to determine the interior solution (or its various thin and thick plate theory approximations) of plate problems. The results in turn show that the necessary conditions for a decaying state are also sufficient conditions. Boundary conditions for the interior solution of circular plate problems with edgewise nonuniform boundary data are discussed in detail and then applied to two specific problems. One of them is concerned with a circular plate compressed by two equal and opposite point forces at the plate rim. The solution process for this problem illustrates for the first time how the stretching action in the plate interior induced by transverse loads can be I properly analyzed.

Compact integration factor methods in high spatial dimensions

The dominant cost for integration factor (IF) or exponential time differencing (ETD) methods is the repeated vector-matrix multiplications involving exponentials of discretization matrices of differential operators. Although the discretization matrices usually are sparse, their exponentials are not, unless the discretization matrices are diagonal. For example, a two-dimensional system of N × N spatial points, the exponential matrix is of a size of N(2) × N(2) based on direct representations. The vector-matrix multiplication is of O(N(4)), and the storage of such matrix is usually prohibitive even for a moderate size N. In this paper, we introduce a compact representation of the discretized differential operators for the IF and ETD methods in both two- and three-dimensions. In this approach, the storage and CPU cost are significantly reduced for both IF and ETD methods such that the use of this type of methods becomes possible and attractive for two- or three-dimensional systems. For the case of two-dimensional systems, the required storage and CPU cost are reduced to O(N(2)) and O(N(3)), respectively. The improvement on three-dimensional systems is even more significant. We analyze and apply this technique to a class of semi-implicit integration factor method recently developed for stiff reaction-diffusion equations. Direct simulations on test equations along with applications to a morphogen system in two-dimensions and an intra-cellular signaling system in three-dimensions demonstrate an excellent efficiency of the new approach.

Wingless Directly Represses DPP Morphogen Expression via an Armadillo/TCF/Brinker Complex

Background Spatially restricted morphogen expression drives many patterning and regeneration processes, but how is the pattern of morphogen expression established and maintained? Patterning of Drosophila leg imaginal discs requires expression of the DPP morphogen dorsally and the wingless (WG) morphogen ventrally. We have shown that these mutually exclusive patterns of expression are controlled by a self-organizing system of feedback loops that involve WG and DPP, but whether the feedback is direct or indirect is not known. Methods/Findings By analyzing expression patterns of regulatory DNA driving reporter genes in different genetic backgrounds, we identify a key component of this system by showing that WG directly represses transcription of the dpp gene in the ventral leg disc. Repression of dpp requires a tri-partite complex of the WG mediators armadillo (ARM) and dTCF, and the co-repressor Brinker, (BRK), wherein ARM•dTCF and BRK bind to independent sites within the dpp locus. Conclusions/Significance Many examples of dTCF repression in the absence of WNT signaling have been described, but few examples of signal-driven repression requiring both ARM and dTCF binding have been reported. Thus, our findings represent a new mode of WG mediated repression and demonstrate that direct regulation between morphogen signaling pathways can contribute to a robust self-organizing system capable of dynamically maintaining territories of morphogen expression.

A spatial stochastic neuronal model with Ornstein-Uhlenbeck input current

Abstract. We consider a spatial neuron model in which the membrane potential satisfies a linear cable equation with an input current which is a dynamical random process of the Ornstein–Uhlenbeck (OU) type. This form of current may represent an approximation to that resulting from the random opening and closing of ion channels on a neurons surface or to randomly occurring synaptic input currents with exponential decay. We compare the results for the case of an OU input with those for a purely white-noise-driven cable model. The statistical properties, including mean, variance and covariance of the voltage response to an OU process input in the absence of a threshold are determined analytically. The mean and the variance are calculated as a function of time for various synaptic input locations and for values of the ratio of the time constant of decay of the input current to the time constant of decay of the membrane voltage in the physiological range for real neurons. The limiting case of a white-noise input current is obtained as the correlation time of the OU process approaches zero. The results obtained with an OU input current can be substantially different from those in the white-noise case. Using simulation of the terms in the series representation for the solution, we estimate the interspike interval distribution for various parameter values, and determine the effects of the introduction of correlation in the synaptic input stochastic process.

Spatial Dynamics of Multistage Cell Lineages in Tissue Stratification

In developing and self-renewing tissues, terminally differentiated (TD) cell types are typically specified through the actions of multistage cell lineages. Such lineages commonly include a stem cell and multiple progenitor (transit-amplifying) cell stages, which ultimately give rise to TD cells. As the tissue reaches a tightly controlled steady-state size, cells at different lineage stages assume distinct spatial locations within the tissue. Although tissue stratification appears to be genetically specified, the underlying mechanisms that direct tissue lamination are not yet completely understood. Herein, we use modeling and simulations to explore several potential mechanisms that can be utilized to create stratification during developmental or regenerative growth in general systems and in the model system, the olfactory epithelium of mouse. Our results show that tissue stratification can be generated and maintained through controlling spatial distribution of diffusive signaling molecules that regulate the proliferation of each cell type within the lineage. The ability of feedback molecules to stratify a tissue is dependent on a low TD death rate: high death rates decrease tissue lamination. Regulation of the cell cycle lengths of stem cells by feedback signals can lead to transient accumulation of stem cells near the base and apex of tissue.