Biological motors 18.S995 - L10 Reynolds numbers Re = UL = UL - PowerPoint PPT Presentation

Biological motors 18.S995 - L10

Reynolds numbers Re = ρ UL = UL µ ν dunkel@math.mit.edu

E.coli (non-tumbling HCB 437) Drescher, Dunkel, Ganguly, Cisneros, Goldstein (2011) PNAS dunkel@math.mit.edu

Bacterial motors movie: V. Kantsler ~20 parts 20 nm source: wiki Berg (1999) Physics Today Chen et al (2011) EMBO Journal dunkel@math.mit.edu

Chlamy 9+2 dunkel@math.mit.edu Merchant et al (2007) Science

Eukaryotic motors Sketch: dynein molecule carrying cargo down a microtubule http://www.plantphysiol.org/content/127/4/1500/F4.expansion.html Yildiz lab, Berkeley dunkel@math.mit.edu

Microtubule filament “tracks” Dogic Lab, Brandeis Physical parameters (e.g. bending rigidity) from fluctuation analysis Drosophila oocyte Goldstein lab, PNAS 2012 dunkel@math.mit.edu

unlike dyneins (most) kinesins walk towards plus end of microtubule 25nm dunkel@math.mit.edu

Kinesin walks hand-over-hand e total- The dif- nm, to with the a alternat- displacements, experi- a head 1B) e Yildiz et al (2005) Science dunkel@math.mit.edu

Kinesin walks hand-over-hand e individual total- The dif- nm, to mole- with molecules the a S43C- alternat- displacements, experi- is a his- head 1B) e Yildiz et al (2005) Science dunkel@math.mit.edu

Intracellular transport Chara corralina http://damtp.cam.ac.uk/user/gold/movies.html dunkel@math.mit.edu

wiki dunkel@math.mit.edu

Muscular contractions: Actin + Myosin F-Actin G-Actin helical filament (globular) dunkel@math.mit.edu

Actin-Myosin F-Actin helical filament Myosin dunkel@math.mit.edu

Actin-Myosin F-Actin Myosin helical filament myosin-II myosin-V dunkel@math.mit.edu

Myosin walks hand-over-hand Hand over hand Inchworm Catalytic Cargo binding domain domain Light chain domain 74 nm x 37 nm 37 nm 37 nm — 2x 37 nm 74 nm 37 nm 37 nm 37 nm + 2x 37 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 (2003) Science dunkel@math.mit.edu

Bacteria-driven motor Di Leonardo (2010) PNAS dunkel@math.mit.edu

Feynman-Smoluchowski ratchet dunkel@math.mit.edu

generic model of a micro-motor dunkel@math.mit.edu

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 external bias, energy input, periodic forcing, memory, etc. dunkel@math.mit.edu

Eukaryotic motors Sketch: dynein molecule carrying cargo down a microtubule http://www.plantphysiol.org/content/127/4/1500/F4.expansion.html Yildiz lab, Berkeley dunkel@math.mit.edu

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 p dX ( t ) = � U 0 ( X ) dt + F ( t ) dt + 2 D ( t ) ⇤ dB ( t ) . (1.116)

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 p dX ( t ) = � U 0 ( X ) dt + F ( t ) dt + 2 D ( t ) ⇤ dB ( t ) . (1.116) Here, U is a periodic potential U ( x ) = U ( x + L ) (1.117a) with broken reflection symmetry, i.e., there is no δ x such that U ( − x ) = U ( x + δ x ) . (1.117b)

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 p dX ( t ) = � U 0 ( X ) dt + F ( t ) dt + 2 D ( t ) ⇤ dB ( t ) . (1.116) Here, U is a periodic potential U ( x ) = U ( x + L ) (1.117a) with broken reflection symmetry, i.e., there is no δ x such that U ( − x ) = U ( x + δ x ) . (1.117b) A typical example is U = U 0 [sin(2 π x/L ) + 1 4 sin(4 π x/L )] . (1.117c) The function F ( t ) is a deterministic driving force, and the noise amplitude D ( t ) can be time-dependent as well.

P. Reimann / Physics Reports 361 (2002) 57–265 66 2 1 0 V ( x ) / V 0 -1 -2 -1 -0.5 0 0.5 1 x /L Fig. 2.2. Typical example of a ratchet-potential V ( x ), periodic in space with period L and with broken spatial symmetry. Plotted is the example from (2.3) in dimensionless units. dunkel@math.mit.edu

time-dependent as well. The corresponding FPE for the associated PDF p ( t, x ) reads j ( t, x ) = − { [ U 0 − F ( t )] p + D ( t ) ∂ x p } , ∂ t p = − ∂ x j , (1.118) and we assume that p is normalized to the total number of particles, i.e. Z L N L ( t ) = dx p ( t, x ) (1.119) 0 gives the number of particles in [0 , L ]. The quantity of interest is the mean particle velocity v L per period defined by Z L 1 v L ( t ) := dx j ( t, x ) . (1.120) N L ( t ) 0

time-dependent as well. The corresponding FPE for the associated PDF p ( t, x ) reads j ( t, x ) = − { [ U 0 − F ( t )] p + D ( t ) ∂ x p } , ∂ t p = − ∂ x j , (1.118) and we assume that p is normalized to the total number of particles, i.e. Z L N L ( t ) = dx p ( t, x ) (1.119) 0 gives the number of particles in [0 , L ]. The quantity of interest is the mean particle velocity v L per period defined by Z L 1 v L ( t ) := dx j ( t, x ) . (1.120) N L ( t ) 0 Inserting the expression for j , we find for spatially periodic solutions with p ( t, x ) = p ( t, x + L ) that Z L 1 dx [ F ( t ) − U 0 ( x )] p ( t, x ) . v L = (1.121) N L ( t ) 0

1.6.1 Tilted Smoluchowski-Feynman ratchet As a first example, 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 ≥ 0 accounting for the polarity. We would like to determine the mean 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 ( x ) of the corresponding FPE (1.118) must yield a constant current-density j 1 , i.e., j 1 = − [( ∂ x Φ ) p 1 + D ∂ x p 1 ] (1.122) where Φ ( x ) = U ( x ) − xF (1.123) 2 1 eff (x) 0 V -1 -2 -1 -0.5 0 0.5 1 x P. Reimann / Physics Reports 361 (2002) 57–265

1.6.1 Tilted Smoluchowski-Feynman ratchet As a first example, 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 ≥ 0 accounting for the polarity. We would like to determine the mean 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 ( x ) of the corresponding FPE (1.118) must yield a constant current-density j 1 , i.e., j 1 = − [( ∂ x Φ ) p 1 + D ∂ x p 1 ] (1.122) where Φ ( x ) = U ( x ) − xF (1.123) is the full e ff ective potential acting on the walker. By comparing with (1.85), one finds that the desired constant-current solution is given by Z x + L p ∞ ( x ) = 1 Z e − Φ ( x ) /D dy e Φ ( y ) /D . (1.124) x

Constant current solution Z L Z L 1 1 j 1 = − [( ∂ x Φ ) p 1 + D ∂ x p 1 ] ( dx [ F ( t ) − U 0 ( x )] p ( t, x ) v L ( t ) := dx j ( t, x ) = N L ( t ) N L ( t ) 0 0 Z x + L p ∞ ( x ) = 1 Z e − Φ ( x ) /D dy e Φ ( y ) /D . (1.124) x This solution is spatially periodic, as can be seen from Z x +2 L 1 Z e − [ U ( x + L ) − ( x + L ) F ] /D dy e [ U ( y ) − yF ] /D p ∞ ( x + L ) = x + L Z x + L 1 Z e − [ U ( x ) − ( x + L ) F ] /D dz e [ U ( z + L ) − ( z + L ) F ] /D = x Z x + L 1 Z e − [ U ( x ) − ( x + L ) F ] /D dz e [ U ( z ) − ( z + L ) F ] /D = x = p ∞ ( x ) , (1.125) where we have used the coordinate transformation z = y − L ∈ [ x, x + L ] after the first line. Inserting p ∞ ( x ) into Eq. (1.121) gives

Z L Z L 1 1 j 1 = − [( ∂ x Φ ) p 1 + D ∂ x p 1 ] ( dx [ F ( t ) − U 0 ( x )] p ( t, x ) v L ( t ) := dx j ( t, x ) = N L ( t ) N L ( t ) 0 0 where we have used the coordinate transformation z = y − L ∈ [ x, x + L ] after the first line. Inserting p ∞ ( x ) into Eq. (1.121) gives Z L − 1 = dx ( ∂ x Φ ) p ∞ v L N L 0 Z L Z x + L 1 dx ( ∂ x Φ ) e − Φ ( x ) /D dy e Φ ( y ) /D = − ZN L 0 x Z L ∂ x e − Φ ( x ) /D ⇤ Z x + L D dy e Φ ( y ) /D . ⇥ = (1.126) dx ZN L 0 x