Physics of Cellular materials: Filaments

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1 Physics of Cellular materials: Filaments Tom Chou Dept. of Biomathematics, UCLA, Los Angeles, CA (Dated: December 6, ) The basic filamentary structures in a cell are reviewed. Their basic structures and chemistry are discussed. An application of exclusion processes is also presented. I. FILAMENTARY STRUCTURES Besides polymeric macromolecules and memranes, another major component in cells are stiff filaments, typically denoted as part of the cytoskeleton. These are classified into three types: Actin: These are the simplest filaments. They are 5nm in diameter and require hydrolysis of ATP ADP to depolymerize. Figure : Fluorescence image of actin filaments (red) and tubulin (green). Actin is a major filament in muscle. Actin is a left-handed helix of monomers. Intermediate filaments: These are made up of a few protofilaments and are about 8-nm in diameter. They are common in areas where structural strength is needed, such as in the interior of nuclear membranes and in keratin. Figure : Hierarchy of structures comprising intermediate filaments. Unlike actin an tubulin, intermediate filaments are made up of long, intertwinned proteins, rather than globular monomer subunits. Microtubules: Microtubules such as Tubulin is a hollow cylinder very much like nanotubes. It is made up of dimers of an and a subunit. They are typically 5-5nm in diameter.

2 Figure 3: Electron micrograph of tubulin. Schematic of of tubuin in nerve cell axons. A drawing showing the subunits in tubulin. II. THE ASEP AND TRANSPORT ALONG FILAMENTS The simultaneous, overdamped transport of many particles in one dimension can be modeled by simple exclusion processes. The asymmetric exclusion process was first solved by Derrida and Schütz and Domany in 993. The model with open boundaries (finite sized lattice) is defined as follows. A lattice has N sites which can be either empty or filled with at most one particle. At any given time step, a particle is chosen and an attempt to move it is made. It cvan move to the right with probability p only if the site to teh right is unoccupied. It can move to the left with probability q p only if the site to the left is unoccupied. Particle enter the very first site from a reservoir with rate and are extracted from teh first site to the left reservoir with rate γ. Similarly, particles enter site N from the right reservoir with rate δ and exit from site N into the right reservoir with rate. ε q γ i σ i Figure 4: Schematic of the asymmetric exclusion process p {,} in δ Normally, one would have to solve a N -vector master equation by enumerating the N N transition matrix. However, Derrida used a matrix product ansatz, Schütz and Domany used recursion relation to find exact solutions. One important quantity is the nonequilibrium steady-state (NESS) current through the chain. For the totally asymmetric case, (q γ δ ), no particle can move backwards. The NESS current in this case is given by J N p S N N (p/) S N (p/), where S N (x) S N (p/) S N (p/) k (N k)(n + k )! x N k+. () N!k! In the N limit, current separates into three different regimes and a nonequilibrium phase diagram develops: When when > and /p < /, the system is in the low density regime and the current is a function only of and p: J ( /p). Conversely, if < and /p > /, the lattice is nearly full, and the current is a function only of the rate-limiting step: J ( /p). Only when both /p, /p > / do we have a current that is independent of the boundary rates,. This is the maximal current phase with J p/4. The standard ASEP assumes fixed transition rates for particles whose step sizes are their diameters. However, in numerous applications, the driven particles and the underlying length scales of the microscopic transitions (steps) are not identical, i.e., the particles are often larger than the lattice sizes of the track on which they move. For example, the particle may large and occlude d lattice sites. Specific biophysical examples include ribosomes moving along mrna and large molecules or vesicles that are shuttled by motor proteins along microtubules. Ribosomes are nm in diamater, but move by nm steps, codon by codon. Motor proteins that carry vesicles ( 5nm) typically move with 5nm steps along microtubules. For these applications d. The matrix product ansatz used by Derrida

3 3 (B) (A) desorption /p.5 J( /p) Jp/4 J( /p) (C).5 injection /p Figure 5: Phase diagram for the TASEP d3 N i i+ i+... i+n j N i i+ i+... i+n Figure 6: (a) Constraints and a configuration associated with the calculation of Z(n, L). (b) Typical configuration for the calculation of Z(n, L d). cannot be used when d >. Therefore we use a mean field approach. For the maximal current regime, we assume translational invariance and apply an equilibrium Tonks gas. The partition function for n particles of length d confined to a length of L nd lattice sites is which leads to L (d )n Z(n, L), () n P (x i+ > j + d x i j) Z(n, L ) Z(n, L) where ρ n/l. The steady-state current in a uniform phase is thus L (d )n n L (d )n ρd ρ(d ), (3) ρd J P (x i j)p (x i+ > j + d x i j) ρ ρ(d ). (4) The maximal current J max and its associated density ρ max is found by setting ρ J, which yields

4 4 i 3 4 i N 4 N 3 N N N p p (a) (b) Figure 7: Mean field scheme for particles of size d 4 near boundaries. (a) d + distinct states at the entrance region. (b) d + distinct states of the exit region. All the states that are within d micorscopic steps of the first state are included. ρ max and J max d( d + ) ( d + ). (5) As expected, (5) reduces to the standard TASEP result when d and that J max, ρ max /d as d. The steady-state current will be the maximal current J max when the ASEP is forced under periodic boundary conditions (a ring) or as long as the injection and extraction rates of an open boundary ASEP are large enough such that internal moves are the overall rate limiting steps. First consider the entrance site at the left of the chain as shown in Fig. 7(a). Upon balancing the probability currents into the first state in Fig. 7(a), J L P (x > d) P (x d, x > d), (6) where the most probable state in the entrance rate limited regime is the first d sites unoccupied. For example, the probability that the first d sites are unoccupied is P (x > d) ( θ) d dθ( θ) d + ( θ) d, (7) where θ is the relative fraction that a site is occupied by a particle and the normalisation dθ( θ) d + ( θ) d is the weight (counting only the first d lattice sites) of all possible states depicted in Fig. 7(a). The probability that the first particle is at site d and that the second particle is far enough away (x d + ), is P (x d, x > d) θ( θ) d dθ( θ) d + ( θ) d. (8) We have assumed a slowly varying density so that θ represents a mean over the particle diameter and is thus a constant over the first d sites. The mean occupation at the first d sites is thus Upon using Eq. (6), θ, and ρ θ( θ) d dθ( θ) d + ( θ) d θ + (d )θ. (9) ρ + (d ), J ( ) L + (d ). ()

5 5..5 MC: d MC: d4 MC: d J Figure 8: Currents as functions of and for fixed d, 4, 8. (a) Fixed. (b) Fixed. The points correspond to MC simulations, the thick solid curves that fit the simulations represent the optimal mean field predictions (), for d, 4, 8. To find the mean field exit rate-limited currents, we must consider two type of sites. One set of sites are spaced d sites apart starting from site N, and the intermediate sites comprising the second class of sites. Our final mean field result can be summarized (I) < ( ), J L d + + (d ) (II) <, J R ρ R (III) (, ) (, ) J max ( ) + (d ) ρ L ρ + R ( ρ d + ) N + (d ) + (d ) d( d + ). () The mean field currents are exact: and the NESS current phase diagram retains its qualitative behavior: (a) Simple(thin) and refined(thick) mean field phase diagrams for d5 (I) (II) (III) , critical entrance/exit rates (b), Tonks MFT, simple MFT, simple MFT 5 5 d Figure 9: (a) The phase diagram for the TASEP with particles of size d 5 derived from the Tonks distribution and the refined mean field (). The circles represent the phase boundaries estimated from Monte Carlo simulations. Boundaries across which first order phase transitions occur are shown with solid curves, while those across which second order transitions occur are drawn with dashed segments. (b) The critical values and at which the (I) - (III) and (II) - (III) phase boundaries occur. For the Tonks gas and refined mean field approaches, /( d + ).

6 (a) particle density profiles for d3., ρ i d (b).9.8, site i (c) Figure : Normalized density (ρ i d) profiles in three representative current regimes. (a) Entrance rate limited regime (., ). (b) Maximal current regime ( ). (c) Exit rate limited regime (,.). The boundary regions are shown in the insets. The density profiles, however, have some additional features: REFERENCES [] J. Howard, Mechanics of Motor Proteins and the Cytoskeleton, (Sinauer, ). A good introductory textbook with all the important references. [] Derrida, B., M. R. Evans, V. Hakim, and V. Pasquier Exact solution of a D asymmetric exclusion model using a matrix formulation. J. Phys. A6: [3] Derrida, B., and M. R Exact correlation functions in an asymmetric exclusion model with open boundaries. Journal de Physique I, 3(), 3-3. [4] Derrida, B. and M. R. Evans The asymmetric exclusion model: Exact results through a matrix approach in Nonequilibrium Statistical Mechanics in One Dimension. Cambridge University Press, Cambridge, UK. [5] Essler, F. H. L., and V. Rittenberg Representations of the quadratic algebra and partially asymmetric diffusion with open boundaries. J. Phys. A, 9, [6] Kolomeisky, A.B Asymmetric simple exclusion model with local inhomogeneity. J. Phys. A 3: ASEP with a defect site. [7] Schutz, G. and E. Domany Phase Transitions in an Exactly Soluble One-Dimensional Exclusion Process. J. Stat. Phys. 7:77-96.