Mathematical Models of Actin Formation

Thomas Pollard, Sterling Professor and former Chair of Molecular, Cellular & Developmental Biology, recently published a review on the use of mathematical modeling and simulations in actin filament dependant processes. The Pollard laboratory, which studies cellular movement, focuses on understanding the roles of actin and myosin motors on a molecular level.

Actin is a protein involved in many cellular functions, most notably maintaining the cellular cytoskeleton and mediating cellular movement. Muscular contraction, cell division, cytokinesis (cytoplasm division), and cell signaling are all dependant on actin filaments.

The globular actin monomer (G-actin) is a protein with four subdomains. The head of the globular protein is referred to as the pointed end, while the opposite side is called the barbed end. A deep nucleotide binding cleft, which interacts with either adenosine triphosphate or adenosine diphosphate, is present near the interface of the head subdomains.

Actin monomers spontaneously polymerize into helical actin filaments (F-actin) at physiological conditions. Polymerization is a reversible process in which the monomers add on to (or fall off) either end. However, under most conditions, there is a net loss of monomers at the pointed end and a net gain at the barbed end.

Many processes involving multiple mechanisms govern actin filament polymerization. Models are particularly useful in examining such a multi-component process that has competing and coupled reactions, as it is difficult to predict the results of these complicated dynamic systems. Models and simulations help by combining empirically determined parameters (for example, rate and equilibrium constants) for various reactions with knowledge about the protein structures.

As Pollard writes, mathematical models were crucial in explaining why ATP hydrolysis by F-actin makes elongation much more favorable at barbed ends than at pointed ends. Monte Carlo simulations (repeated simulations with randomness in input parameters) have shown that the difference in the rate of elongation is primarily due to lower affinity of phosphate for terminal subunits at the pointed ends. In this case, experimentation provided the observed results (differences in elongation rates), but it did not explain why. The computer simulation allowed a rigorous approach to ascertain the answer.

Additionally, Pollard and colleagues tackled the task of determining the predominant pathway of actin filament branch formation with the aid of mathematical models. To do this, they used a model with established rate constants for the interactions among various proteins and the actin monomer and filaments. Many of these constants were taken from previous studies but adjusted to fit the conditions of the simulation, and the simulations were run with estimated cellular reactant concentrations. They found that the predominant pathway involves the association of the Arp2/3complex with actin filaments, a fast reaction because of the high local concentration of actin filaments. Ultimately, the model illuminated the dominant pathway, distinguishing it from many other proposed alternate pathways.

Models allow prediction and determination of new rate constants for certain mechanisms by matching simulation results against experimental data. They also shed light on explaining observed experimental results, providing a deeper understanding of complicated mechanisms. Even more, models can be used to supplement intuition, quantitatively evaluating proposed pathways. Due to the prevalence of complicated systems in molecular biology, mathematical models will certainly become an important tool for other areas in molecular biology as well.