Anatoly B. Kolomeisky Department of Chemistry CAN WE UNDERSTAND THE COMPLEX DYNAMICS OF MOTOR PROTEINS USING SIMPLE STOCHASTIC MODELS?
Motor Proteins Enzymes that convert the chemical energy into mechanical work Functions: cell motility, cellular transport, cell division and growth, muscles, Courtesy of Marie Curie Research Institute, Molecular Motor Group
Motor Proteins. Examples KINESINS linear processive motor proteins, move along microtubules, important for transport of vesicles and organelles, cell motility
Motor Proteins. Examples MYOSINS- linear processive or non-processive motor proteins that move along actin filaments, important for transport, cell motility and muscle functioning
Motor Proteins. Examples RNA POLYMERASES linear processive motor proteins, move along double-stranded DNA molecules, synthesize RNA molecules, important in transcription
Motor Proteins. Examples F0F1 ATP synthase rotary motor protein, membrane protein that takes part in transport of protons and ATP synthesis
Motor Proteins. Properties Non-equilibrium systems Velocities: 0.01-100 µm/s (for linear processive) Step Sizes: 0.3-40 nm Forces: 1-60 pn Fuel: hydrolysis of ATP, or related compounds, or polymerization Efficiency: 50-100% (!!!) Power like jet engine Directionality Diversity
Motor Proteins. Diversity Super family of myosin motor proteins
Motor Proteins
Motor Proteins Fundamental Problems: 2) How the chemical energy is transformed into the mechanical motion? 3) How many mechanisms of motor protein motion?
Motor Proteins. Experiments Single-Molecule Experiments: Optical trap spectrometry FRET fluorescence resonance energy transfer
Single-Molecules Experiments Optical Trap Experiment: laser microtubule bead kinesin Optical trap works like an electronic spring
Optical Trap Conservation of momentum of photons Optical gradient force
EXPERIMENTS ON KINESIN optical force clamp with a feedback-driven optical trap Visscher,Schnitzer,Block (1999) Nature 400, 184 step-size d=8.2 nm precise observations: mean velocity V(F,[ATP]) stall force F S dispersion D(F,[ATP]) mean run length L(F,[ATP])
Theoretical Modeling Microscopic (atomistic level) Currently not feasible for biological molecules Mesoscopic (molecular level) Macroscopic Our goal: Phenomenological description of motor proteins dynamics
Theoretical Problems: Description of biophysical properties of motor proteins (velocities, dispersions, stall forces, ) as functions of concentrations and external loads Detailed mechanism of motor proteins motility c) coupling between ATP hydrolysis and the protein motion d) stepping mechanism hand-over-hand versus inchworm e) conformational changes during the motion f)
Theoretical Models: Requirements for theoretical models: 2) Periodicity (molecular tracks); 3) Biochemical transitions (ATP hydrolysis); 4) Chemical reversibility; 5) Non-Equilibrium; 6) Explain experimental observations; 7) Do not contradict basic laws of Physics and Chemistry
THEORETICAL MODELING Thermal ratchet models periodic, spatially asymmetric potentials Idea: motor proteins are particles that move in periodic but asymmetric potentials, stochastically switching between them
Thermal Ratchet Models: Advantages: 1) continuum description, well developed formalism; 2) convenient for numerical calculations and simulations; 3) small number of parameters; Disadvantages: 2) mainly numerical or simulations results; 3) results depend on potentials used in calculations; 4) hard to make quantitative comparisons with experiments; 5) not flexible in description of complex biochemical systems; 6) no chemical transitions physicist s view of biology
OUR THEORETICAL APPROACH Multi-state chemical kinetic (stochastic) models =0,1,2,,N-1 intermediate biochemical states kinesin/ microtubule kinesin/ N=4 model microtubule/ ATP kinesin/ microtubule/ ADP/Pi kinesin/ microtubule/ ADP
OUR THEORETICAL APPROACH Multi-State Chemical Kinetic (Stochastic) Models 1 j N 1 Biased hopping model on 1D periodic lattice w j
1 OUR THEORETICAL APPROACH our model periodic hopping model on 1D lattice exact expressions for asymptotic (long-time) properties for any N! Derrida, J. Stat. Phys. 31 (1983) 433-450 drift velocity V V, w j lim t d dt x t, dispersion D D, w j 2 lim t d dt x 2 t x t 2 x(t) spatial displacement along the motor track
OUR THEORETICAL APPROACH Simultaneous knowledge of velocity and dispersion Microscopic information and mechanisms randomness r 2D dv bound! r >1/N d motor protein s step size; d=8.2 nm for kinesins, d=36 nm for myosins V and VI stall force F S k B T d ln N j 0 1 w j 0 0 V F F S 0
"! "! #!! % $ % $ $ $ ( ) ) ' ', +, + + + 0 2 2 / /, + + 0 2 2 / / 5 5 9 OUR THEORETICAL APPROACH Dynamics of the system is described by Master equation d dt P j P j l, t l, t 1 P j 1 l, t w j 1 P j 1 l, t -the probability of finding the particle at site l in state j at time t Derrida s idea: d dt B j d dt C j t t %&,.- 1 B j 1 B j 1 C j 1 1 1 t B j t t t %(, 3,10 w j l w j w j P j 1 B j 1 B j 1 C j 1 l, t 1 1 t t t ; C j t $ %* +,43 l w j w j w j j B j C j P j t Nl t l, t P j l, t Ansatz at large times B j t 687 b j, C j t 6 7 a j t T j
=? = ; = > ; = ;? @ @ OUR THEORETICAL APPROACH discrete-state sequential stochastic model Advantages: V u 0 u 1 u 0 u 1 2) exact solutions 3) extensions u 0 u 1 ; : d? w 0 w 1 < w 0 w 1 < 2 V D d ; => 2 ]/ @ @ :1 2 d 2 u 0? w 0 u 1? w 1? w 0 for N=2? w 1
THEORY: EXTENSIONS 1) Periodic Stochastic Model with Irreversible Detachments important for kinesins, which can irreversibly dissociate from the track 2) Periodic Stochastic Model with Branched States important for RNA-, DNApolymerases, e.g., transcription pauses
THEORY: EXTENSIONS 3) Parallel-Chain Stochastic Models backsliding in RNA polymerases 4) Stochastic Models with General Waiting-Time Distributions
D Basic concept of stochastic models General Waiting Times number of events time intervals are distributed exponentially according to Poisson statistics u State j j State j+1 exp ACB t exponential waiting-time distributions time intervals consider stochastic models with general (non-exponential) waiting-time distributions
P ON M G F L K J G F F F E Q \] W T S R _ ` b d g e OUR THEORETICAL APPROACH Effect of an external load F: F GIH 0 e j Fd k B T, w j E w j F G H w j 0 e j Fd k B T j and j Uload distribution factors j N W 1X V 0 j Y[Z j ^ 1 F=0 activation barrier E a F >0 j j+1 j Fd j j+1 j c a Fd 1 e f E a k B T
q p o m p n k i j h k i l l l RESULTS FOR KINESINS stall force depends on [ATP] F S k B T d ln N j 0 1 w j 0 0 q Michaelis-Menten plots N=2 model F=3.59 pn F=1.05 pn V d u 0 u 0 u 1 u 1 w 0 w 1 w 0 w 1
RESULTS FOR KINESINS force-velocity curves randomness
Mechanochemical Coupling in Kinesins How many molecules of ATP are consumed per kinesin step? Is ATP hydrolysis coupled to forward and/or backward steps? Nature Cell Biology, 4, 790-797 (2002)
Mechanochemical Coupling Kinesin molecules hydrolyze a single ATP molecule per 8-nm advance Schnitzer and Block, Nature, 388, 386-390 (1997) Hua et al., Nature, 388, 390-394 (1997) Coy et al., J. Biol. Chem., 274, 3667-3671 (1999) Problem: back steps ignored in the analysis The hydrolysis of ATP molecule is coupled to either the forward or the backward movement (!!!!!!!!!!) Nishiyama et al., Nature Cell Biology, 4, 790-797 (2002) Backward steps are taken into account
Mechanochemical Coupling Investigation of kinesin motor proteins motion using optical trapping nanometry system Nishiyama et al., Nature Cell Biology, 4, 790-797 (2002)
Mechanochemical Coupling Fraction of 8-nm forward and backward steps, and detachments as a function of the force at different ATP concentrations circles - forward steps; triangles - backward steps; squares detachments Stall force when the ratio of forward to backward steps =1 Nishiyama et al., Nature Cell Biology, 4, 790-797 (2002)
Mechanochemical Coupling Dwell times between the adjacent stepwise movements Dwell times of the backward steps+detachments are the same as for the forward 8-nm steps Both forward and backward movements of kinesin molecules are coupled to ATP hydrolysis Nishiyama et al., Nature Cell Biology, 4, 790-797 (2002)
u t s u t Mechanochemical Coupling Branched kinetic pathway model with asymmetric potential of the activation energy Idea: barrier to the forward motion is lower than for the backward motion s1 r1 k 1 s1 k 2 k 3b F k 3f F Conclusion: kinesin hydrolyses ATP at any forward or backward step Nishiyama et al., Nature Cell Biology, 4, 790-797 (2002)
Mechanochemical Coupling PROBLEMS: 2) Backward biochemical reactions are not taken into account 3) Asymmetric potential violates the periodic symmetry of the system and the principle of microscopic reversibility 4) Detachments are not explained Nishiyama et al., Nature Cell Biology, 4, 790-797 (2002)
Mechanochemical Coupling Periodicity is violated!
Our Approach The protein molecule moves from one binding site to another one through the sequence of discrete biochemical states, i.e., only forward motions are coupled with ATP hydrolysis Random walker hopping on a periodic infinite 1D lattice Dwell times mean first-passage times; Fractions splitting probabilities
v x y w w z z { Our Approach π N,j the probability that N is reached before N, starting from the site j N, j w j N, j ww j 1 w j N, j 1 Boundary conditions: N, N 1, N, N 0 N.G. van Kampen, Stochastic Processes in Physics and Chemistry, Elseiver, 1992
~ } Our Approach N, 0 -splitting probability to go to site N, starting from site 0, fraction of forward steps N, 0 1 N, 0 -fraction of backward steps N, 0 1 1 N j 0 1 w j
Š ˆ ƒ Š Œ Ž Our Approach T N,j mean first-passage time to reach N, starting from j T N,0 dwell time for the forward motion; T -N,0 dwell time for the backward motion T N, 0 u eff ƒn, 0 u eff, T ŒN, 0 N, 0 N N, 0 w eff 1 with u eff 1 N j 1 0 r j, r j 1 1 N k 1 1 i j j k 1 w i u i w eff N, 0 j 0 w j
Our Approach T N, 0 T N, 0 N, 0 u eff T Drift velocity, T N, 0 N, 0, but N, 0 V d u eff N, 0 w eff w eff N, 0 Important observation: Dwell times for the forward and backward steps are the same, probabilities are different
š œ Our Approach With irreversible detachments δ j N, j, j -probability to dissociate before reaching N or -N, starting from j 1 - fractions of steps forward, backward and N, j, j detachments N, j w j j N, j 1 w j w j j N, j 1
ž ž Ÿ ž ž Ÿ Ÿ «ª ««Our Approach With irreversible detachments δ j Define new parameters: N, j ϕ j the solution of matrix equation 1,,...,,...,, 1 1 j N 1 vector N matrix elements M ij ŸN, j j j w i u i, T N, j 1, w j M w j 0 1, for j 1, for j j T N, j j j, for i i i 1 1 ; 1 ;, j ;
² ± ± ± ± ± ± ² ± Our Approach With irreversible detachments δ j Model with detachments, w j, N, j, T N, j Model without detachments, w j, N, j, T N, j N=1 case: 1,0 T 1,0 u u T w 1,0, T, 0 1,0 1 u w u w w ±,, 0 u w,
¹ ³ ³» º ³ ³ ³ ¼ Our Approach With irreversible detachments δ j Description of experimental data using N=2 model; reasonable for kinesins Fisher and Kolomeisky, PNAS USA, 98, 7748 (2001). w j F µ F µ 0 0 exp exp ³ j Fd k B T Fd j k B T Load dependence of rates
Comparison with Experiments Fractions of forward and backward steps, and detachments [ATP]=10µM [ATP]=1mM
Comparison with Experiments Dwell times before forward and backward steps, and before the detachments at different ATP concentrations
Á À À À ¾ ½ Á À APPLICATION FOR MYOSIN-V N=2 model mean forward-step first-passage time u 0 u 1 u 0 u 1 w 0 w 0 w 1 w 1 Kolomeisky and Fisher, Biophys. J., 84, 1642 (2003)
APPLICATION FOR MYOSIN-V Our prediction: Substep d 1 =13-14 nm Kolomeisky and Fisher, Biophys. J., 84, 1642 (2003) Uemura et al., Nature Struct. Mol. Biol., 11, 877 (2004)
Future Directions: Motor protein 2 interacting particles Mechanisms of motility
PUBLICATIONS: 1) J. Stat. Phys., 93, 633 (1998). 2) PNAS USA, 96, 6597 (1999). 3) Physica A, 274, 241 (1999). 4) Physica A, 279, 1 (2000). 5) J. Chem. Phys., 113, 10867 (2000). 6) PNAS USA, 98, 7748 (2001). 7) J. Chem. Phys., 115, 7253 (2001). 8) PNAS USA, 98, 7748 (2001). 9) Biophys. J., 84, 1642 (2003).
CONCLUSIONS Multi-State Chemical Kinetic (Stochastic) models of motor protein dynamics are developed All available experimental observations can be explained by this approach Multi-State Stochastic Models might serve as a framework for atomistic description of motility mechanisms in motor proteins