Modeling electrophoresis of peptides, proteins, DNA, blood cells and colloids is based on classical electrokinetic theory. The coupled field equations-Poisson, Navier-Stokes or Brinkman, and ion transport equations are solved numerically to calculate the electrophoretic mobilities. First, free solution electrophoretic mobility expressions are derived for weakly charged rigid bead arrays. Variables include the number of beads (N), their size (radius), charge, distribution (configuration), salt type, and salt concentration. We apply these mobility expressions to rings, rigid rods, and wormlike chain models and then apply the approach to the electrophoretic mobilities and translational diffusion constants of weakly charged peptides. It is shown that our bead model can predict the electrophoretic mobilities accurately. In order to make the method applicable at higher salt concentrations and/or to models consisting of larger sized subunits, account is taken of the finite size of the beads making up the model structure. For highly charged particles, it is also necessary to account for ion relaxation. This ion relaxation effect is accounted for by correcting "unrelaxed" mobilities on the basis of model size and average electrostatic surface, or "zeta" potential. With these corrections our model can be applied to the system with absolute electrophoretic mobilities exceeding approximately 0.20 cm2/kV sec and also models involving larger subunits. This includes bead models of duplex DNA. Along somewhat different lines, we have investigated the electrophoresis of colloidal particles with an inner hard core and an outer diffusive layer ("hairy" particles). An electrokinetic gel layer model of a spherical, highly charged colloid particle developed previously, is extended in several ways. The charge of the particle is assumed to arise from the deprotonation of acidic groups that are uniformly distributed over a portion (or all) of the gel layer. Free energy considerations coupled with Poisson-Boltzmann theory is used to calculate the change of the local pKa of the acidic groups depending on the local electrostatic environment. Based on the modeling of electrophoresis and viscosity, we predict that the thickness of the gel layer decreases as the salt concentration increases. And only the outermost portion of the gel layer is charged.

On the bead modeling methodology, or BMM, a macromolecule is modeled as a rigid, non-overlapping bead array with arbitrary radii. The BMM approach was pioneered by Kirkwood and coworkers (Kirkwood, J.G., Macromolecules, E.P. Auer (Ed.), Gordon and Breach, New York, 1967; Kirkwood, J.G., Riseman, J., J. Chem. Phys., 1948, 16, 565) and applied to such transport properties as diffusion, sedimentation, and viscosity. With the availability of computers, a number of investigators extended the work to account for the detailed shape of biomolecules in the 1970s. A principle objective of my research has been to apply the BMM approach to more complex transport phenomena such as transport in a gel, electrophoresis (free solution and in a gel), and also transport in more complex media (such as the viscosity of alkanes and benzene). Variables considered by the BMM include the number of beads (N), the radii of the beads, net charge and charge distribution, conformations, salt type, and salt concentration. The BMM has been extended to: (1) account for the existence of a gel; (2) characterize the charge and secondary structure of macromolecules; (3) account more accurately for hydrodynamic interaction (remove the orientationnal preaveraging approximation of hydrodynamic interaction); (4) study the effect of ion relaxation for particles in arbitrary size, shape, and charge; (5) consider the salt dependence of electrokinetic properties; (6) account for the formation of possible complex between guest ions and BGE ions. We also did diffusion constant measurement by NMR for amino acids and short peptides in 10%D2O-90% H2O at room temperature and applied to our modeling study by BMM.

Le modèle de dimères, également connu sous le nom de modèle de couplage parfait, est un modèle probabiliste introduit à l'origine dans la mécanique statistique. Une configuration de dimères d'un graphe est un sous-ensemble des arêtes tel que chaque sommet est incident à exactement une arête. Un poids est attribué à chaque arête et la probabilité d'une configuration est proportionnelle au produit des poids des arêtes présentes. Dans cette thèse, nous étudions principalement deux modèles qui sont liés au modèle de dimères, et plus particulièrement leur comportements limites. Le premier est le modèle des forêts couvrantes enracinées sur des cycles (CRSF) sur le tore, qui sont en bijection avec les configurations de dimères via la bijection de Temperley. Dans la limite quand la taille du tore tend vers l'infini, la mesure sur les CRSF converge vers une mesure de Gibbs ergodique sur le plan tout entier. Nous étudions la connectivité de l'objet limite, prouvons qu'elle est déterminée par le changement de hauteur moyen de la mesure de Gibbs ergodique et donnons un diagramme de phase. Le second est le modèle de perles, un processus ponctuel sur $\mathbb{Z}\times\mathbb{R}$ qui peut être considéré comme une limite à l'échelle du modèle de dimères sur un réseau hexagonal. Nous formulons et prouvons un principe variationnel similaire à celui du modèle dimère \cite{CKP01}, qui indique qu'à la limite de l'échelle, la fonction de hauteur normalisée d'une configuration de perles converge en probabilité vers une surface $h_0$ qui maximise une certaine fonctionnelle qui s'appelle "entropie". Nous prouvons également que la forme limite $h_0$ est une limite de l'échelle des formes limites de modèles de dimères. Il existe une correspondance entre configurations de perles et (skew) tableaux de Young standard, qui préserve la mesure uniforme sur les deux ensembles. Le principe variationnel du modèle de perles implique une forme limite d'un tableau de Young standard aléatoire. Ce résultat généralise celui de \cite{PR}. Nous dérivons également l'existence d'une courbe arctique d'un processus ponctuel discret qui encode les tableaux standard, defini dans \cite{Rom}. / The dimer model, also known as the perfect matching model, is a probabilistic model originally introduced in statistical mechanics. A dimer configuration of a graph is a subset of the edges such that every vertex is incident to exactly one edge of the subset. A weight is assigned to every edge, and the probability of a configuration is proportional to the product of the weights of the edges present. In this thesis we mainly study two related models and in particular their limiting behavior. The first one is the model of cycle-rooted-spanning-forests (CRSF) on tori, which is in bijection with toroidal dimer configurations via Temperley's bijection. This gives rise to a measure on CRSF. In the limit that the size of torus tends to infinity, the CRSF measure tends to an ergodic Gibbs measure on the whole plane. We study the connectivity property of the limiting object, prove that it is determined by the average height change of the limiting ergodic Gibbs measure and give a phase diagram. The second one is the bead model, a random point field on $\mathbb{Z}\times\mathbb{R}$ which can be viewed as a scaling limit of dimer model on a hexagon lattice. We formulate and prove a variational principle similar to that of the dimer model \cite{CKP01}, which states that in the scaling limit, the normalized height function of a uniformly chosen random bead configuration lies in an arbitrarily small neighborhood of a surface $h_0$ that maximizes some functional which we call as entropy. We also prove that the limit shape $h_0$ is a scaling limit of the limit shapes of a properly chosen sequence of dimer models. There is a map form bead configurations to standard tableaux of a (skew) Young diagram, and the map is measure preserving if both sides take uniform measures. The variational principle of the bead model yields the existence of the limit shape of a random standard Young tableau, which generalizes the result of \cite{PR}. We derive also the existence of an arctic curve of a discrete point process that encodes the standard tableaux, raised in \cite{Rom}.