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Interaction of motor proteins with obstacles Helicase unwinding of DNA M. D. Betterton Department of Applied Mathematics University of Colorado at Boulder joint work with Frank Jülicher MPIPKS, Dresden http://www.mpipks-dresden.mpg.de/mpi-doc/julichergruppe/ Not all motors move on an infinite periodic track The Polymerization Ratchet Growth of a polymer near a wall wall F polymer given enough space, next monomer can bind Peskin, Odell, and Oster, Biophys J 65, 316 (1993) The Polymerization Ratchet Dogterom and Yurke, Science 278, 856 (1997) Mogilner and Oster, Eur Biophys J 28, 235 (1999) Carlsson, Phys Rev E 62, 7082 (2000) Kolomeisky and Fisher, Biophys J 80, 149 (2001) and important applications to cell motility, … Kinesin-related ATPase Localizes to microtubule ends Hunter et al. Mol. Cell 11 445, (2003) http://www.mpi-cbg.de/research/groups/howard/projects.html MCAK Accelerates MT depolymerization 100x Appears to processively depolymerize MCAK off rate 0.054 s-1 Tubulin dimer off rate 1 s-1 Hunter et al. Mol. Cell 11 445, (2003) http://www.mpi-cbg.de/research/groups/howard/projects.html MCAK http://fajerpc.magnet.fsu.edu/Education/2010/Lectures/26_DNA_Transcription.htm Nucleic-acid motors RNA Polymerase http://ntri.tamuk.edu/cell/ribosomes.html NA-based motors Ribosome http://www.stanford.edu/group/blocklab/Exo2.gif NA-based motors Exonuclease NA-based motors Helicase Interacting Hopping Model Hopping rates: k Position: n k m 1D lattice Two fluctuating degrees of freedom Hypothesize interaction potential k Position: n k m 0 m U ( m n) m n n Interaction changes rates Detailed balance n ,mn ,m 1 e n ,m 1n ,m k k n , m n 1, m n 1, m n , m [U ( m n 1) U ( m n )] kT k e k [U ( m n 1) U ( m n )] kT Simplest interaction Exclusion interaction Steric inhibition n=m forbidden k n=m-1 Steric Inhibition k n=m-1 [U (0) U (1)] kT mm1 e m 1m 0 when U (0) Hard-wall Potential 1 e 0 1 0 [U (0) U (1)] kT k k e k k [U (0) U (1)] kT 1 0 U (0) k1 0 Questions How does changing the interaction potential change the motion of the complex? Is there an optimal potential for fastest motion? Helicase opens dsNA Motor protein – fueled by ATP hydrolysis Can open duplexes of DNA-DNA, DNA-RNA, or RNA-RNA +ATP +ATP (Assumes strands don’t re-anneal) Cellular Role of Helicases All cellular processes involving nucleic acids Replication Transcription Translation RNA processing DNA repair Important for Genome Stability helicase binds 3’ 5’ single strand double strand helicase translocates helicase moves junction displaces strand Bird et al. Nucl Acids Res 26, 2686 (1998) Dillingham et al. Biochemistry 39, 205 (2000) Dillingham et al. Biochemistry 41, 643 (2002) Mechanism Passive Doesn’t interact directly with duplex Waits for fluctuation to advance Inhibits closing Active Interacts with duplex Destabilizes duplex Increases opening rate Hard wall Lohman and Bjornson, Ann Rev Biochem 65, 169 (1996) Singleton and Wigley, J. Bacteriol 184, 1819 (2002) Mutation Studies support idea of an active mechanism Mutate PcrA residues which touch duplex Unwinding rates decrease 10–30x Soultanas et al. EMBO Journal 19, 3799 (2000) Helicase motion k3’ k+ 5’ Helicase motion k3’ k+ 5’ If out of equilibrium can have k+>k- PcrA: k+-k- =80 bases/s Dillingham et al. Biochemistry 41, 643 (2002) DNA ss-ds junction motion Junction motion G Junction at Junction at base m base m+1 Closing lowers energy e G kT 1 e 7 2 Effects neglected Helicase binding/unbinding DNA flexibility Different biochemical states of helicase DNA sequence variability Effects of randomness on unzipping Lubensky and Nelson Phys Rev E 2002 Effects of randomness on motor protein motion Kafri, Lubensky and Nelson cond-mat 2003 Computing Unwinding Rate P (n, m, t ) probability of finding helicase at n, junction at m, time t dP(n, m, t ) ( mm 1 mm1 knn1 knn1 ) P(n, m, t ) dt kn 1n P(n 1, m, t ) m 1m P (n, m 1, t ) kn1n P(n 1, m, t ) m 1m P(n, m 1, t ) Simulation of Full Equations Junction Closing rate = 0.1/time step Opening rate 0.1/7 Helicase Forward hop rate = Closing rate/100 Backward hop rate = Forward rate/40 Start with uniform junction position Run for 25 closing times Junction position m Thanks to Alex Barnett Helicase position n Simulation of Full Equations NA closes quickly compared to helicase hop Speed up movie 500x Junction position m Thanks to Alex Barnett Helicase position n Separate dynamics j m n difference l m n midpoint Rates depend on j only U ( m n) U ( j ) j,j,k ,k j j P j (t ) P( j, l , t ) l Difference-variable equation dP j dt (k k j j )P j j j j 1P j 1 j 1P j 1 DNA j 1 j 1 k P j 1 j 1 k P Helicase Difference-variable equation dP j dt (k k j j )P j j j ( j 1 k ) P j 1 Increase j j 1 ( j 1 k j 1 )P j 1 Decrease j Difference-variable dynamics equilibrate quickly compared to midpoint motion Boundary conditions: zero-current solution j k P j 1 j 1 k j j 1 P j Steady-State Unwinding Rate Find current the average current in l Unwinding velocity 1 v ( j k j j k j )P j 2 j Forward rate at j Backward Prob rate at j at j Hard-Wall Opening v HW k k v HW k k 0 when 0.14 k Effective chemical potential of opening must be larger than the free energy change of DNA closing Numbers: upper bound Assume k 0 k v HW 1 k 11 bp/s Bp at junction is open 1/7 of the time When helicase tries to move forward, it succeeds with probability 1/7 Varying step size Hard-wall unwinding velocity drops rapidly with increasing step size Questions How does changing the interaction potential change the unwinding rate? Is there an optimal potential for fastest opening? The Step Energetic Cost Uo for dsDNA and helicase to overlap by one base 0 U e 1 Relatively faster opening slower forward hop 0 1 o k k Uo e k k Determining Rates A degree of freedom remains 1 e 0 e fU o ( f 1)U o 0 f 1 f 0 f 1 1 0 eU Increase opening o 1 e U 0 o Decrease closing One-Step Unwinding Rate v1 c (1 c)e fU o 1 U o v HW c (1 c)e k c k One-step active opening faster than passive Multi-step Staircase Each step height Uo For larger number of steps n, v increases more rapidly n vn v HW c n (1 c)e ( f 1)U o c n j e jU o j 1 n c (1 c) c n j 1 n j jU o e Multi-step Staircase For large n, maximum at U max ln c 2 Optimal potential cancels base pairing energy Opening neutral Helicase crystal structures suggeset n of 5-10 Thanks to Seth Fraden Interaction potential Worse than Hard Opening: Negative Step Height -1 0 1 2 Difference variable m-n Well depth of 2kT decreases velocity to 0.2 of hard-wall velocity Force-velocity curves Soft opening, one step Force changes base-pairing energy G Go Fl Result strongly depends on step height Summary Simple model for motor protein-obstacle interactions Comparison of active and passive helicase unwinding Predict changes if vary k+/kBase-pair free energy Speed decrease from active to passive Model: factor of 7 (hard to soft) Model: factor of 35 (soft to well) Experiment: factor of 10-30