Mathematical modeling of axonal formation. Part I: Geometry.

A stochastic model is proposed for the position of the tip of an axon. Parameters in the model are determined from laboratory data. The first step is the reduction of inherent error in the laboratory data, followed by estimating parameters and fitting a mathematical model to this data. Several axonogenesis aspects have been investigated, particularly how positive axon elongation and growth cone kinematics are coupled processes but require very different theoretical descriptions. Preliminary results have been obtained through a series of experiments aimed at isolating the response of axons to controlled gradient exposures to guidance cues and the effects of ethanol and similar substances. We show results based on the following tasks; (A) development of a novel filtering strategy to obtain data sets truly representative of the axon trail formation; (B) creation of a coarse graining method which establishes (C) an optimal parameter estimation technique, and (D) derivation of a mathematical model which is stochastic in nature, parameterized by arc length. The framework and the resulting model allow for the comparison of experimental and theoretical mean square displacement (MSD) of the developing axon. Current results are focused on uncovering the geometric characteristics of the axons and MSD through analytical solutions and numerical simulations parameterized by arc length, thus ignoring the temporal growth processes. Future developments will capture the dynamic growth cone and how it behaves as a function of time. Qualitative and quantitative predictions of the model at specific length scales capture the experimental behavior well.

A mathematical model for timing the release from sequestration and the resultant Brownian migration of SeqA clusters in E. coli.

DNA replication in Escherichia coli is initiated by DnaA binding to oriC, the replication origin. During the process of assembly of the replication factory, the DnaA is released back into the cytoplasm, where it is competent to reinitiate replication. Premature reinitiation is prevented by binding SeqA to newly formed GATC sites near the replication origin. Resolution of the resulting SeqA cluster is one aspect of timing for reinitiation. A Markov model accounting for the competition between SeqA binding and methylation for one or several GATC sites relates the timing to reaction rates, and consequently to the concentrations of SeqA and methylase. A model is proposed for segregation, the motion of the two daughter DNAs into opposite poles of the cell before septation. This model assumes that the binding of SeqA and its subsequent clustering results in loops from both daughter nucleoids attached to the SeqA cluster at the GATC sites. As desequestration occurs, the cluster is divided in two, one associated with each daughter. As the loops of DNA uncoil, the two subclusters migrate apart due to the Brownian ratchet effect of the DNA loop.

Analyses of mechanisms for force generation during cell septation in Escherichia coli.

Escherichia coli is a rod-shaped bacterium that divides at its midplane, partitioning its cellular material into two roughly equal parts. At the appropriate time, a septum forms, creating the two daughter cells. Septum formation starts with the appearance of a ring of FtsZ proteins on the cell membrane at midplane. This Z-ring causes an invagination in the membrane, which is followed by growth of two new endcaps for the daughter cells. Invagination occurs against a cell overpressure of several atmospheres. A model is presented for the shape of the cell as determined by the tension in the Z-ring. This model allows the calculation of the force required for invagination. Then three possible models to generate the force necessary to achieve invagination are presented and analyzed. These models are based on converting GTP-bound FtsZ polymeric structures to GDP-bound FtsZ structures, which then leave the polymer. Each model is able to generate the force by relating the hydrolyzation to an irreversible molecular binding event, resulting in a net motion of putative anchors for the structures. All three models show that cross-linking the FtsZ protofilaments into a polymer structure allows the removal of GDP-FtsZ without interrupting the structure during force generation, as would happen for a simple polymeric chain.

A polymerization-depolymerization model that accurately generates the self-sustained oscillatory system involved in bacterial division site placement.

Determination of the proper site for division in Escherichia coli and other bacteria involves a unique spatial oscillatory system in which membrane-associated structures composed of the MinC, MinD and MinE proteins oscillate rapidly between the two cell poles. In vitro evidence indicates that this involves ordered cycles of assembly and disassembly of MinD polymers. We propose a mathematical model to explain this behavior. Unlike previous attempts, the present approach is based on the expected behavior of polymerization-depolymerization systems and incorporates current knowledge of the biochemical properties of MinD and MinE. Simulations based on the model reproduce all of the known topological and temporal characteristics of the in vivo oscillatory system.

A mathematical model for elongation of a peptide chain.

A mathematical model is presented for the steps in the elongation process, and the steady-state elongation rate as a function of the amino acid concentrations is found. In addition, the reset sub-process of the elongation process is modeled. The rate of elongation of peptide chains is found to be a function of the concentration of the amino acid to be bound and the concentration of all other amino acids. In addition, the overall elongation rate depends on the concentrations of elongation factors.