Biomedical subjects

A S Fokas

Publications and source records attributed to A S Fokas.

7 recordsLinked to original sources

PubMed2006

Integrable nonlinear evolution partial differential equations in 4 + 2 and 3 + 1 dimensions.

The derivation and solution of integrable nonlinear evolution partial differential equations in three spatial dimensions has been the holy grail in the field of integrability since the late 1970s. The celebrated Korteweg-de Vries and nonlinear Schrödinger equations, as well as the Kadomtsev-Petviashvili (KP) and Davey-Stewartson (DS) equations, are prototypical examples of integrable evolution equations in one and two spatial dimensions, respectively. Do there exist integrable analogs of these equations in three spatial dimensions? In what follows, I present a positive answer to this question. In particular, I first present integrable generalizations of the KP and DS equations, which are formulated in four spatial dimensions and which have the novelty that they involve complex time. I then impose the requirement of real time, which implies a reduction to three spatial dimensions. I also present a method of solution.

Journal Article↗

PubMed2006

Strict rules determine arrangements of strands in sandwich proteins.

From a computer analysis of the spatial organization of the secondary structures of beta-sandwich proteins, we find certain sets of consecutive strands that are connected by hydrogen bonds, which we call "strandons." The analysis of the arrangements of strandons in 491 protein structures that come from 69 different superfamilies reveals strict regularities in the arrangements of strandons and the formation of what we call "canonical supermotifs." Six such supermotifs account for approximately 90% of all observed structures. Simple geometric rules are described that dictate the formation of these supermotifs.

Amino Acid Motifs↗

PubMed2006

Reconstruction algorithm for single photon emission computed tomography and its numerical implementation.

The modern imaging techniques of positron emission tomography and of single photon emission computed tomography are not only two of the most important tools for studying the functional characteristics of the brain, but they now also play a vital role in several areas of clinical medicine, including neurology, oncology and cardiology. The basic mathematical problems associated with these techniques are the construction of the inverse of the Radon transform and of the inverse of the so-called attenuated Radon transform, respectively. An exact formula for the inverse Radon transform is well known, whereas that for the inverse attenuated Radon transform was obtained only recently by R. Novikov. The latter formula was constructed by using a method introduced earlier by R. Novikov and the first author in connection with a novel derivation of the inverse Radon transform. Here, we first show that the appropriate use of that earlier result yields immediately an analytic formula for the inverse attenuated Radon transform. We then present an algorithm for the numerical implementation of this analytic formula, based on approximating the given data in terms of cubic splines. Several numerical tests are presented which suggest that our algorithm is capable of producing accurate reconstruction for realistic phantoms such as the well-known Shepp-Logan phantom.

Algorithms↗

PubMed2005

A geometric construction determines all permissible strand arrangements of sandwich proteins.

For a large class of proteins called sandwich-like proteins (SPs), the secondary structures consist of two beta-sheets packed face-to-face, with each beta-sheet consisting typically of three to five beta-strands. An important step in the prediction of the three-dimensional structure of a SP is the prediction of its supersecondary structure, namely the prediction of the arrangement of the beta-strands in the two beta-sheets. Recently, significant progress in this direction was made, where it was shown that 91% of observed SPs form what we here call "canonical motifs." Here, we show that all canonical motifs can be constructed in a simple manner that is based on thermodynamic considerations and uses certain geometric structures. The number of these structures is much smaller than the number of possible strand arrangements. For instance, whereas for SPs consisting of six strands there exist a priori 900 possible strand arrangements, there exist only five geometric structures. Furthermore, the few motifs that are noncanonial can be constructed from canonical motifs by a simple procedure.

Amino Acid Motifs↗

PubMed2004

Prediction of the structural motifs of sandwich proteins.

We investigate the supersecondary structure of a large group of proteins, the so-called sandwich proteins. The analysis of a large number of such proteins has led us to propose a set of rules that can be used to predict the possible arrangements of strands in the two beta-sheets forming a given sandwich structure. These rules imply the existence of certain invariant supersecondary substructures common to all sandwich proteins. Furthermore, they dramatically restrict the number of permissible arrangements. For example, whereas for proteins consisting of three strands in each beta-sheet 180 possible strand arrangements exist a priori, our rules imply that only 15 of them are permissible. Five of these predicted arrangements describe all currently known sandwich proteins with six strands.

Amino Acid Sequence↗

PubMed2001

Solution of the modified Helmholtz equation in a triangular domain and an application to diffusion-limited coalescence.

A new transform method for solving boundary value problems for linear and integrable nonlinear partial differential equations recently introduced in the literature is used here to obtain the solution of the modified Helmholtz equation q(xx)(x,y)+q(yy)(x,y)-4 beta(2)q(x,y)=0 in the triangular domain 0< or =x< or =L-y< or =L, with mixed boundary conditions. This solution is applied to the problem of diffusion-limited coalescence, A+A<==>A, in the segment (-L/2,L/2), with traps at the edges.

Journal Article↗

PubMed1991

Mathematical model of granulocytopoiesis and chronic myelogenous leukemia.

We present a mathematical model of granulocytopoiesis that depends on certain physiologically meaningful parameters. By choosing different values of these parameters, the model describes both the normal process and that in chronic myelogenous leukemia (CML). The model fits all the available experimental data tested. Furthermore, it shows how the CML cells can ultimately outnumber the normal cells and how this process can be very slow. The model provides a quantitative approach to the relationship between proliferation and maturation and resolves the apparent contradiction between decreased proliferation and increased production, by assuming that a greater fraction of CML cells is produced by division rather than by maturation. The model should be helpful in designing experiments to better define the abnormalities of proliferation and maturation in CML and in seeking to define the specific alterations in the cell regulatory networks resulting from the production of the chimeric p210bcr-abl protein characteristic of CML.

Granulocytes↗