Mechanics of Motor Proteins and the Cytoskeleton Jonathon Howard Chapter 10 Force generation 2 nd part. Andrea and Yinyun April 4 th,2012

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1 Mechanics of Motor Proteins and the Cytoskeleton Jonathon Howard Chapter 10 Force generation 2 nd part Andrea and Yinyun April 4 th,2012

2 I. Equilibrium Force Reminder: Dissociation constant Critical constant

3 I. Equilibrium Force Reminder: Mechanical Energy : Actin: Microtubule: Dissociation constant: K(F) (Fig 10.3)A compressive force opposing the polymerization of a polymer

4 I. Equilibrium Force Chemical-force equation Equilibrium force chemical potential ( concentration gradient ) is balanced by mechanic potential. interpretation: reversal force. Increasing the force through F_eq will change the direction of polymerization from net growth to net shrinkage.

(for Actin filaments) Answer: L= several microns (for microtubule, 500 time more rigid than actin

5 I. Equilibrium Force Prediction of equilibrium force: For actin ~7pN For microtubule~30pn Whether the cytoskeletal filaments are strong enough to exert forces while polymerizing? (for Actin filaments) Answer: L= several microns (for microtubule, 500 time more rigid than actin filaments) both actin filaments and microtubules are rigid enough to support the polymerization forces that are observed in cells.

II. Brownian Ratchet model Peskin et al,1993 Figure 10.4, The Brownian ratchet mechanism for force generation by polymerization. The particle is free to diffuse horizontally as shown.

6 II. Brownian Ratchet model Peskin et al,1993 Figure 10.4, The Brownian ratchet mechanism for force generation by polymerization. The particle is free to diffuse horizontally as shown. It is assumed that additional constraints (not shown) prevent vertical diffusion. Idea: a growing polymer pushing against an opposing force, the particle being pushed undergoes thermal motions sufficiently large to unblock the adjacent filament end and permit subunit

7 II. Brownian Ratchet model Reaction-diffusion equation(1) Comments: (a) steady state requires monomer concentration be constant; (b) assume there is a fixed (large) number of nucleation sites and that growth occurs equally well on all sites.

8 II. Brownian Ratchet model What do we care: elongation rate: Interpretation: (shrinkage rate) is not affected by the applied force, (growth rate) is weighted by the probability that the gap is larger than To find the, we need an expression of, how? Reaction-limited case & Diffusion limited case.

9 II. Brownian Ratchet model (1)Reaction Reaction-limited Polymerization Assume diffusion coefficient D is very large: Growth rate:

10 Example of reaction-limited polymerization Example 10.1 Actin polymerization is fast enough and powerful enough to drive the movement of listeria. Claim: Fastest velocity 1um/s, corresponds to 360 monomer/s. True? using rates from (table 11.1) 15% is free to polymerize (30uM), gives v=1 um/s

11 Example 10.3 Actin polymerization is fast enough in vitro to account for the speed of the acrosomal reaction in sea cucumber sperm. requires In reality, the actin concentration in the acrosomal cup is actually ~3mM Sufficiently high to account for the elongation rate.

12 II. Brownian Ratchet model diffusion-limited polymerization What do you mean by diffusion-limited? Diffusion is slow and the monomer addition rate is very fast => As soon as a gap opens up, a monomer will drop in. if Then Koff is negligible => Growth rate = flux (the rate at which a gap opens up

13 Diffusion-limited Polymerization The rate of gap creation is simply the time it takes the particle to diffuse a distance against the force: this is the mean passage problem of chapter 4(eqn. 4.17). Reminder: How long does it take a molecule to Diffuse through a given distance? We Call this time the first passage time. How long does it take for a molecule To diffuse over an energy barrier at x=x0 When the force is constant, U=-F X.

14 Diffusion-limited Polymerization The rate of gap creation is simply the time it takes the particle to diffuse a distance delta against the force: this is the mean passage problem of chapter 4(eqn. 4.17). (10.4) How long does it take a molecule to Diffuse through a given distance? We Call this time the first passage time. How long does it take for a molecule To diffuse over an energy barrier at x=x0 When the force is constant, U=-F X. Fig 10.5 Dependence of polymerization The diffusion-limited rate is the Reciprocal of the first-passage time.

15 Diffusion-limited Polymerization The rate of gap creation is simply the time it takes the particle to diffuse a distance delta against the force: this is the mean passage problem of chapter 4(eqn. 4.17). (10.4) Drag force for actin: 3pN (equilibrium F:7~8pN) Drag force for microtubule: 13pN(equilibrium F:30pN Fig 10.5 Dependence of polymerization Drag force is independent of viscosity! Because drag force corresponds to the maximum Force inherent to the polymerization Mechanism.

16 Examples of diffusion-limited polymerization Example 10.2 The diffusion limited speed of listeria Measured speed v in water: 0.1~1 um/s, which suggests that listeria s mot may not be diffusion limited. However, if v(cytoplasm) is 1/100 or 1/1000 than that in water, then Speed is reduced by a corresponding factor, 0.36~3.6 um/s, reasonable. Different speeds in different cell types. 3-fold difference could arise from differences in [A_1] if reaction-limited, or to difference in viscosity if diffusion-limited.

17 Other kinetic models Initial state (n-mer) Figure 10.6 Two possible transition rates for polymerization reaction. Kramer-like mechanism (Brownian ratchet model) Eyring-like mechanism (induced fit pathway) Attractive feature of Brownian ratchet model: explicit However: Bacterial motility: purely diffusive mechanism seems less likely.

18 summary Polymerization and depolymerization can generate forces when the monomer concentration differs from the critical concentration. Polymerization is fast enough to account for the polymerization-driven processes, such as the movement of listeria with the cytoplasm and the amoeboid movement of crawling cells.

19 Thanks!

20 II. Brownian Ratchet model Reaction-diffusion equation (2) Question: How to solve it and what do we want? Answer: Elongation rate. Gap flux is zero at the boundaries (There is no sinks and sources).

21 II. Brownian Ratchet model diffusion-limited polymerization Then Koff is negaligible => Growth rate = flux (the rate at which a gap opens up

22 Equations needed.

monomer polymer polymeric network cell

monomer polymer polymeric network cell 3.1 Motivation 3.2 Polymerization The structural stability of the cell is provided by the cytoskeleton. Assembling and disassembling dynamically, the cytoskeleton enables cell movement through a highly