A major challenge of molecular cell biology is to elucidate the biochemical mechanisms by which cells grow and divide. Given the complexities involved, there is no doubt that mathematical modeling will play an increasingly important role in meeting this challenge. The assembled team of investigators has recently developed a whole-cell modeling framework in which cellular biochemical dynamics and changes in cell morphology are inter-dependent, and have used it to construct a simple self-replicating cell model in which an established mechanism of eukaryotic cell cycle regulation is incorporated. The resulting system can produce stable self-replicating behavior, with cytoplasm volume and membrane surface area doubling in synchrony with the periodicity in component concentrations. Because the biochemical dynamics of processes that affect changes in cell morphology in vivo can be modeled, exploring in vivo mechanisms by which growing and dividing cells "pinch" during the division process (cytokinesis) is possible. Towards this end, the objective of the proposed project is to model cytokinesis as it occurs in animal cells and to compare such models to the known cytokinetic behavior of real cells. Cytokinesis occurs near the end of mitosis, and involves the rapid assembly and contraction of a membrane-associated actomyosin ring located at the cell equator. Cytokinesis is exquisitely choreographed with other mitotic events, such that the ring assembles as anaphase begins and daughter chromatids begin to separate. The ring contracts as the chromatids separate, and the ring completes contraction soon after separation is complete. The choreography of these terminal events of mitosis will be modeled. Biochemical reactions and kinetic parameters will be assumed based on what is known experimentally. Once developed, cytokinesis models will be installed into the existing whole-cell model in which cell cycle regulation is treated explicitly. In this way, cytokinesis can be modeled within an in vivo setting. Cell morphology changes involved in cell growth and division will also be estimated by minimizing membrane bending energies. In real cells, membrane composition is altered at the cleavage furrow during cytokinesis, and this aspect will be modeled and analyzed for its ability to promote cell division. These aspects will be integrated into a self-replicating whole-cell model to observe "pinching" behavior about its equator as the cell divides. All of this will be driven by an explicit biochemical mechanism and synchronized with other cell cycle events. The complexity of the models will be scaled in proportion to what is known experimentally, such that they will be closely connected to reality, possess predictive ability, and thus be useful to experimentalists. Models will be analyzed to assess the importance of a cytoskeletal contractile ring vs. local changes in membrane composition in effecting cytokinesis. This integrative approach results in a mathematical model which couples a system of ordinary and partial differential equations with a constrained minimization problem (associated with the determination of cell shape). The primary mathematical challenges stem from the need to determine system parameters within physically realistic ranges so that the solution to the mathematical model exhibits physically reasonable, stable self-replicating behavior. The project is significant because of the novelty of modeling animal-cell cytokinetics on the biochemical/mechanistic level and under both in vitro and in vivo settings. In the broadest sense, the project will assess the feasibility of building a comprehensive cell model piecemeal by designing individual cellular "modules" in vitro and installing them into a whole-cell frame once appropriate in vitro behavior is observed. Ultimately a comprehensive molecular-level cell model will be required to explore the pathogenesis of many human diseases, especially cancer, and to test the intended and unintended metabolic effects of new pharmaceuticals.
Living cells can be simplistically viewed as tiny sacs filled with water, salts and molecules such as DNA and proteins. One of the most fundamental aspects of such cells is their ability to self-replicate. To do this, a cell must grow to twice its original size, make a second copy of its DNA, move each copy of the DNA to different ends of the cell, and finally divide around its middle to form two cells. During the last part of this process (technically called "cytokinesis"), the cell constructs a little belt around its middle, but on the inside of itself such that the belt cannot be seen from the outside of the cell. This internal belt is constructed of many protein units of the same type, linked end-to-end like a stacked set of sticky blocks. Also, the belt is tied to the surface (called a membrane) of the cell. When the cell sends a signal to this belt, the belt starts to tighten around the belly of the cell (by removing blocks, one at a time) and it pulls the membrane in with it. This squeezing doesn't stop until the belt has constricted to a very small circumference, the membrane has pinched completely and two cells are made. The investigators have recently developed a new mathematical approach to modeling cell growth and division at the level of molecules reacting. In this project, this approach will be used to investigate the fine details (at the molecular level) of how this belt is assembled and how it squeezes. Another factor that appears to help this pinching process occur has to do with the types of molecules in the membrane right at the region where the belt is attached. Experiments have shown that the molecules in this region are different from those in the rest of the membrane, but no one understands why they are different. A second aspect of this project will be to investigate this question. Membranes are generally most stable when they are flat rather then bent. This pinching process during cell division requires that they bend a lot, which suggests that pinching might require a lot of energy. It is suspected that the different molecules found in this region help the membrane bend without requiring so much energy. Again, using a mathematical modeling approach, the researchers will investigate whether this might explain why different types of molecules are found in this region. These processes are not only significant from the perspective of basic cell biology, they are also involved in understanding diseases such as cancer. Cancerous cells grow and divide uncontrollably--something has gone awry with the cell division process described above. Modeling these processes using mathematics and computers is important because these processes are so complicated that it is literally impossible for any person to keep track of all the factors and understand how they interact as time changes. However, using mathematics and computers, these factors and interactions can be tracked, which permits the careful testing of what had previously been simply word-based explanations. By such careful testing, it might be possible to understand better how cells grow and divide, and how to reestablish control of uncontrolled cancerous cell growth.
a) We created a Model for a Cell-size Checkpoint: This cell-size sensing system is based on the spatial cellular distribution of two proteins, namely Pom1 and Cdr2. Cdr2 promotes mitosis and is localized at the midcell region. Pom1 inhibits Cdr2 and is primarily localized at the cell tips. Changes in these concentrations gradients are used to trigger the advancement from G2 to M phase of the cell cycle. In short, small fission yeast cells, Pom1 efficiently inhibits Cdr2. As the cell grows, the Pom1 concentration at midcell decreases. Once a particular cell length is achieved, Cdr2 becomes active and induces entry upon mitosis and the cell growth ceases. We modeled the Pom1:Cdr2 behavior using a deterministic reaction-diffusion convection system interacting with a deterministic model of microtubule dynamics. b) We developed and analyzed a mathematical whole cell model. In this work, we showed that a simplistic mathematical model could be created to account for cell growth with synchronized replication, taking into account cell shape during growth and division through the assumption that the bending energy of the cell membrane is minimized at each time step during the growth and division process. Synchronizing the cell processes involved the mathematically difficult problem of developing a model in which chemical species had periodic behavior that coincided with the total volume and surface area doubling of the cell during the growth and replication process. c) We compared the deterministic method and the stochastic method for a polymerization network when the number of available subunits is small. For the stochastic method, it is proved that there is a recursive method to compute the expected molecule numbers of various components in the reaction network, using the stationary probability distribution of molecule numbers which we illustrate to have a multivariate Poisson form. For the deterministic method, ordinary differential equations for the component concentrations are built following the mass action law. The steady state of the system is extracted to estimate the corresponding molecule numbers. Identities involving the propensity function parameters for the stochastic method and the reaction rate constants in the deterministic method are used to connect the two methods. d) We studied developed a mechanism for positioning the ring involving the MinC, MinD and MinE proteins, which oscillate between cell poles to inhibit ring assembly. Averaged over time, the concentration of the inhibitor MinC is lowest at midcell, restricting ring assembly to this region. A second positioning mechanism, called Nucleoid Occlusion, acts through protein SlmA to inhibit ring polymerization in the location of the nucleoid. A mathematical model was developed to explore the interactions between Min oscillations, nucleoid occlusion, Z ring assembly and positioning. SlmA was presumed to break long FtsZ filaments into shorter units. FtsZ subunits in membrane-bound filaments touch and align with other filaments. This alignment was critical in forming sharp stable rings. Simulations qualitatively reproduced experimental results showing the incorrect positioning of rings when Min proteins were not expressed, and the formation of multiple rings when FtsZ was overexpressed. e) We developed mathematical analysis of domain/boundary reaction diffusion systems. This work is an extension of long-standing work detailed in a recent monograph of Michel Pierre associated with global well-posedness for reaction diffusion systems with mass balance. The new aspect of this work is the inclusion of reaction diffusion systems on both the domain and boundary, with coupling through mass transport terms to incorporate chemical reactions involving components which reside in the domain and components which are restricted to the boundary. The work required extension of classical critical estimates of O. A. Ladyzhenskaja, as well as development of estimates mentioned by K. Brown associated with the relationship between the behavior of solutions to parabolic problems and Lp estimates of Neumann boundary data.
