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Biomedical subjects

F W Cummings

Publications and source records attributed to F W Cummings.

8 recordsLinked to original sources
PubMed2001

The interaction of surface geometry with morphogens.

Expressions are given for the Gauss and Mean curvatures of a surface of thickness h. The two curvatures, (K and H), which are given at each point of the middle surface, are adequate to describe the surface. The sheet thickness varies with position in the middle surface bisecting the apical and basal surfaces. The definitions of K and H are in terms of radii of curvature, but such radii are not appropriate variables for determining how morphogens in the surface may couple to the geometry. More suitable expressions are developed here. Two important geometrical constraints must be satisfied, namely the famous Gauss-Bonnet theorem, and an inequality stemming from the definition of the two curvatures. It is argued that these constraints are of great usefulness in determining the form of the coupling of morphogens to the geometry. In particular, when two key morphogens suffice to determine surface geometry, explicit expressions are suggested to determine both Gauss (K) and Mean (H) curvatures as functions of invariant morphogen densities.

Animals↗
PubMed2000

A model of pattern formation based on signaling pathways.

A model of pattern formation in the early embryo is presented. It is motivated by the necessary interaction of kinases and phosphatases, and in particular by the family of Wnt kinase and RPTP phosphatase signaling pathways. It is seen as complementary to the more short-range patterning of the Delta/Notch (or juxtacrine) signaling pathway.

Animals↗
PubMed1999

Spatial patterning via PTP adhesive phosphatases.

Signaling pathways to the genome are a common way by which cells communicate with each other and their environment, and are often kineases or phosphatases. Patter formation is the differential spatial specification of gene activity necessary for multicellularity. It has been suspected that signaling pathways are crucial players in pattern formation in metazoans, but exactly how the pattern arise from the signaling systems has not been shown. The model discussed here is based on the protein tyrosine phosphatases, and it is shown how this important signaling system may straightforwardly produce patterns typical of early development. The protein tyrosine phosphatases have architectural characteristics basically different from those of the kinases, with the receptor tyrosine phosphatases displaying structural motifs of cell adhesion molecules. These membrane-spanning phosphatases then have a unique mission in cell growth, cell shape, and differentiation quite apart from that of the kineases. The complex intracellular biochemistry involved is modeled in the simplest way, with the intent that concepts be emphasized over biochemical detail.

Animals↗
PubMed1999

Waves of pattern formation and signal pathways.

The model addresses the question of the origin of traveling waves, mimicking waves of the Hh and Dpp proteins such as found in development of the fly eye. These two proteins, which are part of an important signaling pathway to the genome, are found in a diverse array of developmental situations, for example in the formation of the proximal-distal axes in limbs. The complex intracellular biochemistry involved is modeled in the simplest way in both cases, with the intent that concepts be emphasized over biochemical detail.

Animals↗
PubMed1996

A model of growth and form based on adhesion molecules.

A model is given for the generation of pattern and form in living systems, based on the assumption of two types of a single adhesion molecule that form homotypic cell-cell contacts. The time dependence of the model is different from that of the Turing models, instead viewing the change in time as being driven by the change in time of the total area A, along with the sequential gene activation which is called into play at discrete times as the organism grows and changes shape. Two equations are derived on the assumption of an energy minimum equilibrium being achieved at all times on a scale that is short compared with the rate of change over time of the total area. The two equations derived from the simple assumptions of the model are the (nonlinear) Helmholtz equation and the Laplace equation. It is argued that a suitable "morphogen" should attempt to satisfy certain rather specific conditions.

Animals↗
PubMed1990

A model of morphogenetic pattern formation.

A model for the morphogenetic movement of surfaces composed of cellular monolayers is proposed. The cells are presumed joined at their lateral surfaces. An otherwise unspecified substance called a "morphogen" is introduced which is the agent of change in the individual cell (or cell-like region). The distribution of these cellular deformations define a surface (the middle surface, through the middle of the cell heights) via equations given for the Gauss and Mean curvatures of the surface defined at each point. The Gauss curvature as a function of the morphogen level determines the metric of the surface "g(u, v)" in conformal co-ordinates u, v. A unique equation for the morphogen distribution over the survace is presented which has the property of size invariance, that is, the model "regulates" without need of further arguments. The two resulting coupled equations for the metric and the morphogen, eqns (4) and (2), both non-linear equations, are to be solved self-consistently, once the individual cell deformation as a function of morphogen is given. The surface geometry determines the morphogen distribution, and the morphogen distribution in turn affects the surface geometry. Extension of the model to two or more morphogens is straightforward, and the key property of "regulation" or size invariance of the model is retained. Numerical integration of the two coupled equations is carried out in the case of axial symmetry, and the results presented by the case that individual cells deform by changing the ratio of their apical to basal areas, as well as their heights. Gastrulation in small regulating holoblastic eggs (e.g. starfish, sea urchin and amphioxus) is discussed in light of the present model.

Animals↗
PubMed1989

On surface geometry coupled to morphogen.

An expression is derived for both the Gauss and the Mean curvature of a surface, in terms of three simple cell parameters. The surface is thought of as composed of a single-cell thick sheet of cells joined laterally. The three cellular parameters involved are the ratios of (linear) basal to apical dimension in two orthogonal directions, S1 and S2, and the cell thickness "h". These three parameters may be envisioned as functions of a morphogen or morphogens which vary from point to point over the (middle) surface. As an example, the "reaction-diffusion" equations which are often used to describe pattern-formation in early development can be seen as possible candidates for these morphogens, when the resultant surface deformations are given when the dependence of the three cellular parameters are specified as a function of morphogen concentration. The coupling back of the surface deformations to the set of reaction-diffusion equations is simply given, and is through the dependence on geometry of the Laplacian operator which enters these equations.

Cells↗