SPATIAL DISTRIBUTION OF PROTEIN CONCENTRATION CAN CONTROL CELLULAR PATHWAY FLUXES BENG 221 Hooman Hefzi and Andrew Richards October 21, 2011 INTRODUCTION One of the goals of systems biology is to create models for intracellular pathway fluxes, which can then be used to design new strains or to predict cellular behavior for a strain. A pathway network can be mathematically reconstructed using singular perturbation theory. Data is gathered from in vitro experiments in which specific pathway components are sequentially altered in some way, such as a knockdown/knockout, or perhaps by an inhibitor molecule. Step by step, the observed effect of changing individual pathway components allows investigators to reverse engineer a metabolic map1. The steady state solutions and response characteristics of cells can then be found and modeled computationally2. These models often suggest strikingly coordinated behavior, such as robust bistability and stable limit cycles3. There are, however, limitations on the predictive power of such models due to certain assumptions, which makes it difficult to connect computational results to the real world biology. One such assumption concerns the spatial distribution of both enzymes within a given pathway, as well as small molecules such as pathway intermediates and signaling molecules. The concentrations of these species are often treated as spatially invariant, thereby treating the cell as a well-mixed system with homogeneous distribution of both enzymes and substrates. In reality, cells have a highly organized (as well as compartmentalized, in the case of eukaryotes), and the laws of mass transfer play a significant role in overall pathway fluxes4.Hooman Hefzi and Andrew Richards 2 Here we show that the spatial distribution of protein concentration can dramatically alter the flux through otherwise equivalent pathways using the same starting substrate, released in a finite burst and diffusing through the cell. One assumption that is physiologically accurate is that mass transfer in the cytoplasm is by diffusion only. The diffusion length of a small molecule is √ , where D is diffusivity (typically cm2/s) and t is time. The distance over which diffusion is effective for mass transport is therefore limited by the stability and half-life of the diffusing species. Typical cell diameters (~0.1 μm for prokaryotes, 1-10 μm for eukaryotes) are on the same order of magnitude as diffusion lengths for most small molecules. ANALYTICAL SOLUTION We began by modeling the cell as a rectangular slab with width L. A small molecule u is released from a plane representing an organelle in the center of the cell at x = L/2 and diffuses outwards. At the edges of the cell, the small molecule rapidly leaks into extracellular environment, fixing the concentration at zero for x = 0 and x = L. We modeled the initial concentration U(x,0) as a fourth-power sine function to approximate a finite pulse of the molecule (Note: We use U to refer to both the small molecule itself as well as its concentration). We then considered two protein pathways which use our small molecule as a starting substrate, Pathways A and Pathway B. As we were concerned only with the consumption of small molecule U and which pathway consumes it, only the starting proteins in each pathway were relevant to our model; we refer to these as Protein A and Protein B, respectively. We first considered the case in which A and B are both distributed homogenously (constant concentration). Furthermore, we set the total amount of A and B equal, and assumed that their binding affinities and rate constants were the same. The consumption of U was therefore modeled as a driving source term, ( ) in the differential equation. Finally, we assumed no convective terms, such that all mass transport within the cell was due to random-walk diffusion only, as described by Fick's second law of diffusion. The partial differential describing the dynamic behavior of our system was thereforeHooman Hefzi and Andrew Richards 3 ( ) ( ) { ( ) ( ) ( ) ( ) We used separation of variables to arrive at general form for our solution: ( ) ( ) ( ) Plugging in for U: ( ) ( ) Rearrange: ( ) Assuming an exponential function, we solved for G: ( ) Solution for F where = 0: ( ) ( ) ( ) ( ) = 0 therefore results in a trivial solution. Assuming > 0: (√ ) (√ ) ( ) ( ) ( ) ( ) ( ) ( ) (√ ) (√ )Hooman Hefzi and Andrew Richards 4 Because A = B = 0 is a trivial solution, we assumed that (√ ) : (√ ) ( ) F can therefore be represented by an infinite series on the integer : ( ) ∑ ( ) Our solution for U combines F and G by principle of superposition: ( ) ∑ ( ) ( ) ( ) where Bn is an constant coefficient for every n. We solve for Bn by noting that at time t = 0, our solution is represented by a Fourier series. By multiplying each side by ( ) and taking the integral from 0 to L, we can arrive at a general expression for Bn: ∫ ( ) ( ) ∑∫ ( ) ( ) Noting the orthogonality of sines, ∫ ( ) ( ) { we can see that only those terms for which m=n contribute to our sum. Our expression therefore simplifies to give us an expression for Bn: ∫ ( ) ( ) ∫ ( ) ∫ ( ) ( ) ∫ ( ) ∫ ( ) ( ) ∫ ( ) ( )Hooman Hefzi and Andrew Richards 5 Our analytical solution can now be expressed in terms of x, t, L, D, R, A, and B: ( ) ∑ ∫ ( ) ( ) ( ) [ ( ) ( ) ] RESULTS Figure 1 shows the surface plot for the analytical solution using 20 terms for the Fourier series. Four different protein distributions were then numerically simulated using MATLABs pdepe solver. Each case is discussed in turn. Homogenous Distribution of Protein A/Homogenous Distribution of Protein B Without biological cues, the distribution of proteins within a cell will approach a homogenous mixture due to simple Brownian motion. Setting the (linear) concentration of Protein A and Protein B to 0.375 for all x