Biological motors 18.S995 - L10

Transcription

1 Biological motors 18.S995 - L1

2 Reynolds numbers Re = UL µ = UL m the organism is mo

3 E.coli (non-tumbling HCB 437) Drescher, Dunkel, Ganguly, Cisneros, Goldstein (211) PNAS

4 Bacterial motors movie: V. Kantsler ~2 parts 2 nm source: wiki Berg (1999) Physics Today Chen et al (211) EMBO Journal

5 Chlamy 9+2 Merchant et al (27) Science

6 Eukaryotic motors Sketch: dynein molecule carrying cargo down a microtubule Yildiz lab, Berkeley

7 Microtubule filament tracks Dogic Lab, Brandeis Physical parameters (e.g. bending rigidity) from fluctuation Drosophila oocyte analysis Goldstein lab, PNAS 212

8 unlike dyneins (most) kinesins walk towards plus end of microtubule 25nm

9 Kinesin walks hand-over-hand Yildiz et al (25) Science

10 Kinesin walks hand-over-hand Yildiz et al (25) Science

11 Intracellular transport Chara corralina

12 wiki

13 Muscular contractions: Actin + Myosin G-Actin F-Actin helical filament (globular)

14 Actin-Myosin F-Actin helical filament Myosin

15 Actin-Myosin Myosin F-Actin helical filament myosin-ii myosin-v

16 R ESEARCH A RTICLES Myosin walks hand-over-hand Catalytic domain Hand over hand Cargo binding domain Light chain domain 74 nm Inchworm 37 nm 37 nm 37 nm 2 74 nm 37 nm 37 nm 37 nm 37 nm nm 37 nm 37 nm 37 nm 37 nm Fig. 3. Stepping traces of three different myosin V molecules displaying 74-nm steps and histogram (inset) of a total of 32 myosin V s taking 231 steps. Calculation of the standard deviation of step sizes can be found (14). Traces are for BR-labeled myosin V unless noted as Cy3 Myosin V. Lower right trace, see Movie S1. Yildiz et al (23) Science

17 Bacteria-driven motor Di Leonardo (21) PNAS

18 Feynman-Smoluchowski ratchet

19 generic model of a micro-motor

20 Basic ingredients for rectification some form of noise (not necessarily thermal) some form of nonlinear interaction potential spatial symmetry breaking non-equilibrium (broken detailed balance) due to presence of eternal bias, energy input, periodic forcing, memory, etc.

21 Eukaryotic motors Sketch: dynein molecule carrying cargo down a microtubule Yildiz lab, Berkeley

22 Most biological micro-motors operate in the low Reynolds number regime, where inertia is negligible. A minimal model can therefore be formulated in terms of an over-damped Ito-SDE dx(t) = U (X) dt + F (t)dt + p 2D(t) db(t). (1.116)

23 Most biological micro-motors operate in the low Reynolds number regime, where inertia is negligible. A minimal model can therefore be formulated in terms of an over-damped Ito-SDE dx(t) = U (X) dt + F (t)dt + p 2D(t) db(t). (1.116) Here, U is a periodic potential U() =U( + L) (1.117a) with broken reflection symmetry, i.e., there is no such that U( ) =U( + ). (1.117b)

24 Most biological micro-motors operate in the low Reynolds number regime, where inertia is negligible. A minimal model can therefore be formulated in terms of an over-damped Ito-SDE dx(t) = U (X) dt + F (t)dt + p 2D(t) db(t). (1.116) Here, U is a periodic potential U() =U( + L) (1.117a) with broken reflection symmetry, i.e., there is no such that U( ) =U( + ). (1.117b) A typical eample is U = U [sin(2 /L)+ 1 4 sin(4 /L)]. (1.117c) The function F (t) is a deterministic driving force, and the noise amplitude D(t) can be time-dependent as well.

25 66 P. Reimann / Physics Reports 361 (22) V()/V Fig Typical eample of a ratchet-potential V (), periodic in space with period L and with broken spatial symmetry. Plotted is the eample from (2.3) in dimensionless units. /L

26 time-dependent as well. The corresponding FPE for the associated PDF p(t, ) t p j, j(t, ) = {[U F (t)]p + D(t)@ p}, (1.118) and we assume that p is normalized to the total number of particles, i.e. N L (t) = d p(t, ) (1.119) gives the number of particles in [,L]. The quantity of interest is the mean particle velocity v L per period defined by v L (t) := 1 N L (t) d j(t, ). (1.12)

27 time-dependent as well. The corresponding FPE for the associated PDF p(t, ) t p j, j(t, ) = {[U F (t)]p + D(t)@ p}, (1.118) and we assume that p is normalized to the total number of particles, i.e. N L (t) = d p(t, ) (1.119) gives the number of particles in [,L]. The quantity of interest is the mean particle velocity v L per period defined by v L (t) := 1 N L (t) d j(t, ). (1.12) Inserting the epression forj, we find for spatially periodic solutions with p(t, ) = p(t, + L) that v L = 1 N L (t) d [F (t) U ()] p(t, ). (1.121)

28 = 1 +L e [U() (+L)F ]/D [U(z) (z+l)f ]/D dz e Tilted Smoluchowski-Feynman ratchet As a first eample, assume that F = const. and D = const. This case can be considered as a (very) simple model for kinesin or dynein walking along a polar microtubule, with the constant force F accountingforthepolarity. Wewouldliketodeterminethemean transport velocity v L for this model. To evaluate Eq. (1.121), we focus on the long-time limit, noting that a stationary solution p 1 () of the corresponding FPE (1.118) must yield a constant current-density j 1, i.e., where j 1 = [(@ )p 1 + D@ p 1 ] (1.122) () =U() F (1.123) is the full e ective potential acting on the walker. By comparing 2 with (1.85), one finds that the desired constant-current solution is given by p 1 () = 1 e ()/D +L This solution is spatially periodic, as can be seen from p 1 ( + L) = 1 +2L e [U(+L) (+L)F ]/D dy e [U(y) V eff () 1 dy e (y)/d. (1.124) +L = 1 +L e [U() (+L)F ]/D dz e [U(z+L) -1 yf]/d (z+l)f ]/D Fig Typical eample of an eective potential from P. Reimann / Physics Reports 361 (22)

29 1.6.1 Tilted Smoluchowski-Feynman ratchet As a first eample, assume that F = const. and D = const. This case can be considered as a (very) simple model for kinesin or dynein walking along a polar microtubule, with the constant force F accountingforthepolarity. Wewouldliketodeterminethemean transport velocity v L for this model. To evaluate Eq. (1.121), we focus on the long-time limit, noting that a stationary solution p 1 () of the corresponding FPE (1.118) must yield a constant current-density j 1, i.e., j 1 = [(@ )p 1 + D@ p 1 ] (1.122) where () =U() F (1.123) is the full e ective potential acting on the walker. By comparing with (1.85), one finds that the desired constant-current solution is given by p 1 () = 1 e ()/D +L dy e (y)/d. (1.124)

30 Constant current solution v L (t) := 1 N L (t) d j(t, ) = 1 N L (t) d [F (t) U ()] p(t, ) j 1 = [(@ )p 1 + D@ p 1 ] p 1 () = 1 e ()/D +L dy e (y)/d. (1.124) This solution is spatially periodic, as can be seen from p 1 ( + L) = 1 +2L e [U(+L) (+L)F ]/D dy e [U(y) +L = 1 e +L [U() (+L)F ]/D dz e [U(z+L) = 1 e +L [U() (+L)F ]/D dz e [U(z) yf]/d (z+l)f ]/D (z+l)f ]/D = p 1 (), (1.125) 1 where we have used the coordinate transformation z = y line. Inserting p 1 () intoeq.(1.121)gives L 2 [, + L] after the first

31 v L (t) := 1 N L (t) d j(t, ) = 1 N L (t) d [F (t) U ()] p(t, ) j 1 = [(@ )p 1 + D@ p 1 ] where we have used the coordinate transformation z = y line. Inserting p 1 () intoeq.(1.121)gives L 2 [, + L] after the first v L = = = 1 N L 1 N L D N L d (@ ) p 1 d (@ ) e e +L ()/D ()/D +L dy e (y)/d dy e (y)/d. (1.126)

32 v L (t) := 1 N L (t) d j(t, ) = 1 N L (t) d [F (t) U ()] p(t, ) j 1 = [(@ )p 1 + D@ p 1 ] where we have used the coordinate transformation z = y line. Inserting p 1 () intoeq.(1.121)gives L 2 [, + L] after the first v L = = = 1 N L 1 N L D N L d (@ ) p 1 d (@ ) e e ()/D +L dy e (y)/d +L ()/D dy e (y)/d. (1.126) Integrating by parts, this can be simplified to D L +L v L = d e dy e (y)/d N L D L = d e ()/D e (+L)/D e ()/D N L D L [ (+L) ()]/D = d 1 e N L D L F [(+L) ]/D = d 1 e N L = DL 1 e FL/D, (1.127) N L

33 =solution d corresponding ewith p(t, ) = p(t, e FPE e yield a constan ()/D p1 ()solutions of the (1.118) must ber of particles in [, L]. Thethe quantity of interest is the particleperiodic velocity Inserting epression for j, we findmean for spatially + L) defined by i.e., NL that L D 1 [ j (+L) 1 [(@()]/D )p1 + D@ p1 ] vl = d [F (t) U(1.12) ()] p(t,d ). = 1 e 1 = (1.121) vl (t) := d j(t, ). NL (t) NL (t) NL + LL) pression for j, we1.6.1 find for spatially periodic solutions with p(t, ) = p(t, 26 Tilted Smoluchowski-Feynman D ratchet F [(+L) ]/D = d 1 e As a Lfirst eample, assume that F = const. N and D = const. This case can be considered L 1as a (very) L +L simple model for kinesin or dynein walking along a polar microtubule, with the vl = d [F (t) U ()] p(t, ). (1.121) NL (t) DL We would likef to constant force F accounting for the polarity. determine the mean ()/D L/D 1 e, transport velocity vl for this model. L = N L ted Smoluchowski-Feynman To evaluate Eq. ratchet (1.121), we focus on the long-time limit, L noting that a stationary solution p1 () of the corresponding FPE (1.118) must yield a constant current-density j1, mple, assume that F = const. and D = const. This case can be considered L i.e.,n can be epressed as where mple model for kinesin orldynein walking along a polar microtubule, with the ()/D (+L)/D 27 F accounting for the polarity. We wouldj1 like= to [(@ determine the mean D@ p1 ] (1.122) )p1++l city vl for this model. 1 () (y)]/d. d26a stationary dy e [ L (1.128) te Eq. (1.121), we focus on the long-time N limit, L = noting that current-density ) of the corresponding FPE (1.118) must yield a constant j1, L = e N dy Integrating by parts, this can be simplified to D v = d dy N D = d e e N D [ (+L) ()]/D We thus obtain the final result = (1.122) d 1 e j = [(@ )p + D@ p ] N 1 Le v = DL R, (1.129) R 26 d D dy e L F [(+L) ]/D = d 1 e which holds for arbitrary periodic potentials U (). Note that there is no net-current at N L equilibrium F =. DL F L/D = 1 e, Temperature ratchet NL F L/D L L +L [ () (y)]/d As we have seen in the preceding sections, the combination of noise and nonlinear dynamics can yield surprising transport e ects. Another eample is the so-called temperatureratchet, which can be captured by the minimal SDE model p 27

34 Tilted Feynman-Smoluchowski ratchet P. Reimann / Physics Reports 361 (22) V eff () 1-1. <> F Fig Typical eample of an eective potential from (2.35) tilted to the left, i.e. F. Plotted is the eample from (2.3) in dimensionless units (see Section A.4 in Appendi A) with L = V = 1 and F = 1, i.e. V e () = sin(2)+:25 sin(4)+. Fig Steady state current ẋ from (2.37) versus force F for the tilted Smoluchowski Feynman ratchet dynamics (2.5), (2.34) with the potential (2.3) in dimensionless units (see Section A.4 in Appendi A) with = L = V = k B = 1 and T = :5. Note the broken point-symmetry. st

35 1.6.2 Temperature ratchet As we have seen in the preceding sections, the combination of noise and nonlinear dynamics can yield surprising transport e ects. Another eample is the so-called temperatureratchet, which can be captured by the minimal SDE model dx(t) =[F U (X)] dt + p 2D(t) db(t), (1.13a) where D(t) =D(t + T ) is now a time-dependent noise amplitude, such as for instance D(t) = D {1+A sign[sin(2 t/t )]}, (1.13b) where A < 1. Such a temporally varying noise strength can be realized by heating and cooling the ratchet system periodically. Transport can be quantified in terms of the combined spatio-temporal average hẋi := 1 T = 1 T t+t t t+t t ds ds d j(t, ) d [F U ()] p(t, ). (1.131) can be solved numerically

36 Time-dependent temperature P. Reimann / Physics Reports 361 (22) <> F Fig Average particle current ẋ versus force F for the temperature ratchet dynamics (2.3), (2.34), (2.47), (2.5) in dimensionless units (see Section A.4 in Appendi A). Parameter values are = L = T = k B = 1, V =1=2, T =:5, A = :8. The time- and ensemble-averaged current (2.53) has been obtained by numerically evolving the Fokker Planck equation (2.52) until transients have died out. Fig The basic working mechanism of the temperature ratchet (2.34), (2.47), (2.5). The gure illustrates how Brownian particles, initially concentrated at (lower panel), spread out when the temperature is switched to a very high value (upper panel). When the temperature jumps back to its initial low value, most particles get captured again in the basin of attraction of, but also substantially in that of + L (hatched area). A net current of particles to the right, i.e. ẋ results. Note that practically the same mechanism is at work when the temperature is kept ed and instead the potential is turned on and o (on o ratchet, see Section 4.2).