Flexing Protein muscles: How to Pull with a "Burning Rope" J. P. Keener 215 Joint International Conference on via Guangzhou University and Huaihua University Huaihua 8/15 p.1/28
Eukaryotic Chromosomal Segregation Before cells divide their chromosomes are duplicated and then moved to the center (midline) of the dividing cell. Once they are all centered, they are separated into two equal sets, enabling cell division. Question: How are chromosomes moved? Huaihua 8/15 p.2/28
Chromosome/microtubule interactions Chromosomes are connected to microtubules (MT) at the KINETO- CHORES where kinetochore structural complexes (such as Ndc8) bind the MT lattice. Huaihua 8/15 p.3/28
Chromosome/microtubule interactions Microtubules change their length by polymerizing and depolymerizing at their tip. DYNAMIC INSTABILITY describes the process by which microtubules switch between polymerizing and depolymerizing states. Chromosome velocities are related to the MT polymerization/depolymerization rates. Question: How can depolymerization/polymerization affect chromosomal movement? Huaihua 8/15 p.4/28
Ndc8 Complex Question: How can depolymerizing microtubules pull the chromosomes? Remark: It is known to not be due to walking molecular motors (i.e., dynein). Proposed Answer: Biased diffusion of Ndc8 proteins along the MT. Before now there has been no mechanistically satisfactory model of how biased diffusion actually works. Huaihua 8/15 p.5/28
Outline of this Talk The main question: how do kinetochores work, i.e., how to pull with a depolymerizing microtubule? Issues to discuss: Models for load-velocity curves Huxley model of actin-myosin crossbridges Binding and unbinding of flexible proteins Bond breaking - Bell s law Bond formation - OU process Sliding platelet in a flow Kinetochore motion Huaihua 8/15 p.6/28
Warmup Exercise: Skeletal Muscle Huaihua 8/15 p.7/28
Crossbridges Huaihua 8/15 p.8/28
The Power Stroke Kinesin Animation Huaihua 8/15 p.9/28
The Huxley Model Suppose a crossbridge can be bound (B) or unbound (U) U k on(x) B, k off (x) Thin filament Thick filament x Unstressed position v > is contraction Binding site Crossbridge Let n(x,t) be the probability density of bound crossbridges with extension x, n t = v n x +k on(x)(1 ndx) k off (x)n. k off k on h koff x The force generated is F = ρ kx n(x,t)dx. Huaihua 8/15 p.1/28
To find the load-velocity curve Pick a velocity v, Calculate steady state n(x;v) Calculate the force F, Plot v vs. F. Remark: Although not biophysically accurate, this is a "typical" load-velocity curve for molecular motors. Load-Velocity Curve 1..8 n.6.4.2 ñ1.5 ñ1. ñ.5.5 1. x/h 1..8 n.6 ñ1.5 ñ1. ñ.5.5 1. x/h 1..8 n.6 ñ1.5 ñ1. ñ.5.5 1. x/h 1..8 n.6.4.2 ñ1.5 ñ1. ñ.5.5 1. Speed of shortening 1..8.6.4.2 V = V =.1 V max V =.25 V max V = V max x/h.4.2.4.2.2.4.6.8 1. Tension Huaihua 8/15 p.11/28
Load Dependent Unbinding Bonds break in a load dependent fashion F U kon k off B, k off = βexp( F F ) (Bell s Law). k k G G δx Fx Huaihua 8/15 p.12/28
Consequences of Bell s Law Suppose the load is spread uniformly among all bonds S n αn βn S n+1, α n = (N n)k on, β n = nβexp( F nf ) leads to the deterministic equation dθ dt = (1 θ) κθexp(f θ ) with θ = n N, κ = β k on, f = F exhibits critical behavior. NF θ 1.8.6.4.2 steady state solutions κ =.5 κ =.1.5 1 1.5 f Remark: This is an interesting stochastic exit time problem. Huaihua 8/15 p.13/28
Position Dependent Binding Unbound flexible proteins undergo an Ornstein-Uhlenbeck process (diffusion with linear restoring force) ξdx = kxdt+dw x with spring constant k, have Gaussian equilibrium distribution, so that the binding rate is also Gaussian k on = κexp( kx2 2k B T ). Huaihua 8/15 p.14/28
Sliding in a Flow Vf V x Let n(x,t) be the density of bound binders with extension x, n t = V n x +α(x) ( N T α(x) = κexp( kx2 2k B T ), The force generated is F = k xn(x,t)dx. ) n(x, t)dx β(x)n, β(x) = k off exp( k x F ), Huaihua 8/15 p.15/28
And the Answer is... 1.5 n(x,v) Contours.18 1.16.14.5.12 x Force f.1.8 -.5.6.4-1.2-1.5.5 1 1.5 2 2.5 3 3.5 4 V 2 4 6 8 1 12 14 16 18 2 Velocity v with the motion determined by the force balance equation biphasic Force vs. Velocity curve, ξ(v f V) F =. Velocity vs. V f has hysteretic behavior (explaining the statickinetic friction transition). Velocity v 2 18 16 14 12 1 8 6 4 2 2 4 6 8 1 12 14 16 18 2 Velocity v f Huaihua 8/15 p.16/28
Kinetochore Motion How can depolymerizing microtubules pull? kmt n kt binders Let n(z,y,t) be the density of bound kinetochore binders with rest position y and binding position z, n t = ( z (vn)+k nt on(z,y) L k on = κexp ( k(y z)2 2k B T ) H(l z), l T l ) ndz z L y k off (z,y)n, k off = κ off exp ( k z y F ), Huaihua 8/15 p.17/28
Protofilament Shape What is the shape of the depolymerizing protofilament? i) l < ii) < l < L iii) l > L f > f > f < u θ f z Total energy for a protofilament is E = l ( α 2 ( θ φ) 2 + k ) lat 2 u2 +zf ds, du ds = sinθ, dz ds = cosθ. To find the equilibrium shape, set δe =. Huaihua 8/15 p.18/28
Shape Equations Shape Equations (linearized): d 2 w dz 2 = k latθ, α d2 θ dz 2 = w+ρθ, dρ dz = f, f(z) = k L (y z)n(z,y)dy n p with boundary conditions θ = φ, w =, ρ =, at z = l θ =, u =, at z = Remark: With f = these reduce to the classical beam equation d 4 θ k dz 4 lat α θ =. Huaihua 8/15 p.19/28
Depolymerization Rate Curled or loaded filaments break more easily: k break (z) = βexp(κ b θ ρ ρ ) because of curvature and load - (Bell s law). Consequently, the depolymerization velocity is v d = l k break (z)dz, and total force generated by the bonds is F = k L l (y z)n(z,y,t)dzdy. Huaihua 8/15 p.2/28
Depolymerization Rate For an end-loaded protofilament: 3 1 Displacement u (nm) 25 2 15 1 5-5 Relative Depolymerization Rate.9.8.7.6.5.4.3-1 -15-1 -5 z (nm).2 2 4 6 8 1 End-Load (nondimensional) Typical of "catch-bonds" Huaihua 8/15 p.21/28
Finding the solution Pick a value of v and l; Find n(z,y;v): = z (vn)+k on(z,y)( ntl l ndz ) k off (z,y)n Calculate f(z): Find protofilament shape, Find v d : f(z) = k n p L v d = l k break(z)dz (y z)n(z,y)dy Adjust l until v d = v (a fixed point problem) Calculate the total force: F = k L Make lots of plots. l (y z)n(z,y,t)dzdy. Huaihua 8/15 p.22/28
And the Answer is... 8 1 1 7 6 1 Load (pn) 5 4 3 2 a b c Velocity (µ m s -1 ) 1-1 1-2 c b 1 a 1-3 1-2 1-1 1 1 1 Velocity (µ m s -1 ) 1-3 -1 -.8 -.6 -.4 -.2.2.4.6.8 1 δ = l/l Remark: There is no stall force. There is "disconnect" at high loads. Huaihua 8/15 p.23/28
How Does that Work? 1 8 protofilament load force density (pn µ m -1 ) u 5-5 a b c -1.2.4.6.8 1 y/l i) l < ii) < l < L iii) l > L f > f > f < θ f load(pn µ m -1 ) 6 4 2-2 -4-1 -5 5 z(nm) Microtubule protofilament shape 25 u, Displacement (nm) 2 15 1 5 a a b b c c z -5-2 -15-1 -5 5 z(nm) Huaihua 8/15 p.24/28
Effects of Parameters? γ = L2 k α (protein stiffness/protofilament bending stiffness): 8 1 1 7 γ=7 6 1 Load (pn) 5 4 3 γ=14 Velocity (µ m s -1 ) 1-1 γ=14 2 1 1-2 γ=7 1-3 1-2 1-1 1 1 1 Velocity (µ m s -1 ) 1-3 -1 -.8 -.6 -.4 -.2.2.4.6.8 1 k (Ndc8 spring constant): There is an "optimal" k. δ = l/l 8 1 1 7 κ=6 6 1 Load (pn) 5 4 3 κ=1 Velocity (µ m s -1 ) 1-1 κ=1 κ=6 κ=15 2 1-2 1 κ=15 1-3 1-2 1-1 1 1 1 Velocity (µ m s -1 ) 1-3 -3.5-3 -2.5-2 -1.5-1 -.5.5 1 δ = l/l Huaihua 8/15 p.25/28
Effects of Parameters? ρ (Bell s law coefficient for protofilaments): 8 1 1 7 ρ= 6 ρ=1 1 Load (pn) 5 4 3 ρ=5 Velocity (µ m s -1 ) 1-1 ρ=5 2 1-2 ρ=1 1 ρ= 1-3 1-2 1-1 1 1 1 Velocity (µ m s -1 ) 1-3 -1 -.8 -.6 -.4 -.2.2.4.6.8 1 Remark: Notice the "flat" load-velocity profile. δ = l/l Huaihua 8/15 p.26/28
Conclusion Protein flexibility plays an important role in their binding and unbinding, etc., and may lead to critical behaviors (i.e., thresholds, bifurcations, etc.). Biased diffusion of Ndc8 proteins enables depolymerizing microtubules to pull chromosomes, answering the question of how to pull with a "burning rope." Huaihua 8/15 p.27/28
Thanks! Thanks to Blerta Shtylla, Pomona College NSF (funding) Huaihua 8/15 p.28/28