Crowded Cells and Molecular Machines Raymond Kapral University

Crowded Cells and Molecular Machines Raymond Kapral University of Toronto collaborators A. Malevanets S. H. Lee A. Cressman C. Echeveria K. Tucci G. Rueckner K. Rohlf mesoscopic description of reaction and transport in the cell effects of hydrodynamics on conformational motions of molecular machines and self-propelled objects

crowded cells physiological fluid media contain macromolecules that occupy 7 -40% of the total volume – called crowded since no single macromolecule may be concentrated cytoplasm of Dictyostelium discodeum cell from cryoelectron tomography Medalia, et al. , Science, 298, 1209 (2002)

effects of crowding • diffusion of macromolecules is 5300 times slower in cytoplasm than in saline solution (Luby-Phelps, et al. , 1987; Gersohn; 1985) • validity of mass action and continuum approaches is questionable (Kuthan, 2001) • crowding enhances protein association, the self-association of monomers and protein folding and re-folding (Rivas et al. , 1999; van den Berg, et al, 2000; Schnell and Turner, 2004) thermodynamic aspects of crowded systems well studied -- activities and transition state rates -- little work on dynamical properties

multiparticle collision dynamics replace the net effects of many real collisions by effective collisions that occur at discrete times -- particles free stream between these effective collisions multiparticle collision dynamics collision volume molecular streaming collision Malevanets and RK, J. Chem. Phys. (1999) magnitude and direction of velocity are changed

dynamics conserves mass, momentum and energy momentum: energy: phase space volumes are preserved and the single particle equilibrium distribution is Maxwellian H theorem can be proved and shows that the one-particle distribution evolves to a Maxwellian

hydrodynamics full set of Navier-Stokes equations in hydrodynamic limit development of vortices in the flow around a sphere Re=24 Allahyarov and Gompper, Phys. Rev. E (2002) Re=76

diffusion of small molecules in crowded systems hybrid molecular dynamics-mesoscopic solvent dynamics model self diffusion coefficient in a crowded environment -- volume fraction of spherical obstacles theory: Tucci and RK, J. Chem. Phys. (2003); Echeveria, Tucci and RK, J. Phys. : Cond. Mat. (2006).

reactions of small molecules in crowded environments single catalytic sphere – no obstacles time-dependent rate coefficient

single catalytic sphere with obstacles system size is 50 x 50 MPC cells with a catalytic sphere (diameter =10) in center ~10 000 obstacles and ~700 000 reactive particles

random distribution of catalytic spheres surface reactions important simulations confirm non-analytic dependence on -- no good theoretical predictions for large Felderhof and Deutch, J. Chem. Phys. (1976); Lebenhaft and RK, J. Stat. Phys. (1979)

biochemical networks in the cell operation of biochemical networks in the cell often involve small numbers of some species – spatial effects in small cellular volumes may be important E. coli some numbers for protein synthesis: mass action kinetics not applicable under these conditions: fluctuations important spatial effects important since arrival times of RNAP at the DNA promotor site influence rate S. Fraser

spatially distributed reaction networks combine MPC dynamics with birth-death probabilistic description of reactions in cells reactions occur independently in each MPC cell with specified probabilities accounts for fluctuations and spatial effects for fast diffusion, particles are Poisson distributed reaction-diffusion on macroscopic scales after reaction step, multi-particle collision operators act to change the velocities of particles of species -- rules can account for different species particles free stream between collisions at discrete times

glycolysis oscillations on mesoscales 0. 4 0. 2 0. 05 0. 4 0. 8 1. 2 MPC dynamics simulation Selkov model for the phosfructokinase reaction in glycolsis K. Rohlf Selkov, Eur. J. Biochem. , 1968.

hydrodynamic interactions disturbances created by solute molecules are transmitted to other parts of fluid through solvent viscous modes, giving rise to longrange hydrodynamic coupling among solute molecules hydrodynamic model Oseen interactions hydrodynamic interactions play an important role in polymer dynamics: rates of conformational mesoscopic model accounts for hydrodynamic interactions

polymer collapse induced by changing solvent conditions polymer model and bead-solvent interactions • • FENE bead-spring model for polymer plus A attractive or R repulsive LJ bead-bead interactions bead-solvent particle interactions: a attractive or r repulsive LJ interactions Ra – expanded chain Ar collapsed chain Lee and RK, J. Chem. Phys. (2006)

mechanism of polymer collapse mechanism: ``blobs" separated by uncollapsed segments ``sausage-shape" that thickens and shrinks compact, aspherical collapsed chain large literature: Stockmayer (1960), de Gennes (1985), … ; elongated globules, end beads reside on surface -- Kuhn (1934), …

radius of gyration during collapse scaling of radius of gyration: collapsed chain: theory – collapsed dense globule: (de Gennes) expanded chain: theory -- swollen chain in good solvent (Le Guillou and Zinn-Justin …)

hydrodynamic interactions play an essential role in the collapse dynamics hydrodynamic interactions reduce the friction on the polymer so that collapse occurs more quickly Kikuchi, Gent and Yeomans, Eur. Phys. J. E, (2002).

molecular machines convert chemical energy into directed motion in the presence of strong fluctuations elastic network (Tirion) model of a molecular machine that undergoes cyclic conformational changes upon binding and unbinding of a ligand Togashi and Mikhailov enzyme catalysis

mesoscopic dynamics of molecular machines previous studies used deterministic overdamped dynamics with no specific solvent effects explicitly include solvent in the dynamics –solvent-network interactions described by Lennard-Jones forces mesoscopic simulation of machine dynamics: 64 molecules in elastic network; 1, 000 solvent molecules solvent penetrates the network A. Cressman

hydrodynamic interactions influence machine dynamics no hydrodynamic interactions with hydrodynamic interactions hydrodynamics can couple the operations of many molecular machines

mesoscale dynamics of self-propelled objects dynamics at low Reynolds numbers swimming E Coli Purcell, Am. J. Phys. (1977) for small Reynolds numbers inertial effects are not important – for small lengths, fluctuations are important

chemically powered nanorods oxygen produced at the Pt end by the reaction proposed mechanisms for motion: surface tension gradients or nano-bubbles Paxton, et al. , JACS, (2004); Fournier-Bidoz, et al. , Chem. Commun. , 441 (2005)

experimental observations of nanorod motion translational nanorod motion; tethered nanorods undergo rotational motion movie from Ozin group (University of Toronto)

chemically powered nano-dimers particle-level mesoscopic model for motion – as in experiments, catalytic and non-catalytic sites in the dimer solvent-dumbbell interactions – repulsive Lennard-Jones potentials A B A A C irreversible chemical reaction N at the catalytic sphere B B nonequilibrium concentration gradient in combination with potential asymmetry provides mechanism for motion

simulations of nano-dimer motion -- forward drive -- reverse drive with no reaction and all A or B (or any mixture), no directed motion strong fluctuations – directed motion is masked by fluctuations and averaging is needed to observe motion

origin of driving force average force on dimer center of mass along dimer bond z in steady state this force is balanced by the frictional force: in equilibrium and average force is zero

nonequilibrium density field subject to boundary condition approximate nonequilibrium density field due to reaction

nanodimer velocity early time quasi-steady state

some numbers velocity can be estimated from 0. 5 1. 0 1. 5 0. 0033 0. 0 -0. 0016 0. 003 0. 0 -0. 0013 5. 0 -0. 0067 -0. 0048 more interesting numbers: Reynolds number: Peclet number:

fluctuations and nanodimer motion on nanoscales fluctuations compete strongly with directed motion probability distribution function for velocity along the dimer axis fluctuations are very important in this regime the dimer tumbles as it moves dimer travels 2 -3 times its length before it reorients, scales as so reorientation can be made less important

self-propelled enzymes? model for self-propelled motion by asymmetric production of reaction products in enzyme catalysis is a characteristic length that depends on the mechanism for motion – diffusiophoretic or osmiophoretic Golestanian, Liverpool and Ajdari, PRL (2005)

other types of swimmers a simple three-bead swimmer (Najafi and Golestanian, PRE, (2004)) swimmer moves by extending and shortening arms in a periodic, time irreversible manner result is a net translation of the swimmer by a distance along the internuclear axis these and other similar swimmers have been studied using MPC and lattice Boltzmann dynamics (Earl, et al. , cond-mat preprint (2007))

remarks • dynamics of small molecules in crowded environments can lead to nontrivial collective effects on both diffusion and reaction • hydrodynamic interactions strongly influence conformational changes and modify how molecular machines operate • model can be generalized to treat more general open reacting systems displaced far from equilibrium at the mesoscopic level -- applications to reactions in small physical and biochemical systems • hydrodynamic coupling and flows in such environments can also be studied

hydrodynamics and dynamics of biopolymers hinge motion of DNA Polymerase Beta hydrodynamic coupling will influence large-scale molecular motions conformational motions of large molecular groups will perturb the solvent flow field leading to long range coupling among the molecular groups in the molecule – the local friction is difficult to compute for such a complex system

polymer and solvent structure Ar interactions – solvent is excluded from the collapsed polymer Aa interactions – solvent penetrates the collapsed polymer

scaling of collapse time varies linearly with number of beads collapse model involving fast formation of pearls along chain (especially from ends), followed by slower compaction of chain Buguin, Brouchard-Wyart and de Gennes(1996)