A microscopic formulation for the actin-driven motion of Listeria Yuan Lin Department of Mechanical Engineering The University of Hong Kong The physics of cell functionality Kavli Institute for Theoretical Physics China, CAS, August 16,

Listeria propel themselves by actin comet tails l A typical process of infection and spread Certain bacteria like Listeria monocytogenes can hijack the actins in the host cell to propel itself. Listeria diameter: ~1 Listeria length: ~2 Typical speed: ~0.1 2

Moving Listeria 3

Why are we interested? Beads coated with ActA can move in cytoplasmic extracts via the same mechanism. Cameron et al., Current Biology, (2001) Scale bar – Cell motility itself is an important problem. 2. Actin-polymerization provides a way to convert chemical energy to mechanical energy. Can actin be utilized to deliver drugs or propel medical devices in the future? 4

Actin and actin polymerization Actin is a globular, roughly 42-kDa protein found in almost all eukaryotic cells.globularkDaproteineukaryotic cells G – actin (the monomer form) F – actin (the polymer form) 5

Proteins involved Plastic beads coated with protein ActA/N-WASP can move in cytoplasmic extracts or in a reconstituted solution consisting of only a handful of proteins. Wiesner et al., JCB, (2003) F – actin / G – actin Arp2/3 : branch/nucleate new filaments ADF/cofilin: disassembles actin filaments Gelsolin: cap filament tips far from the load surface Profilin: regulate the turnover /restructuring of actin network 6

7 Extract motility medium

8 Generation of actin-based motility Hu and Lin, work in progress

9 Beyond Biochemistry Structure of the actin comet tail Cameron et al., Current Biology, (2001) The network is very complicated Actin filament branches at 70 o.

10 Beyond Biochemistry The tail might be hollow Negative propelling force can be generated by the inner part of the tail Plastino et al., Current Biology, (2001)Upadhyaya et al., PNAS, (2003)

11 Beyond Biochemistry The tail actually attaches to the load surface The bacteria fluctuates 20 times less than a free lipid droplet Kuo and McGrath, Nature, (2000) Holding the bacteria with optical tweezer and cutting the tail with optical scissors. Gerbal et al., Eur. Biophys. J., (2000)

The mechanics of polymerization is bested summarized by the force-velocity relationship ( similar to muscle contractions ). - Total propelling force Experimentally measurable 12 Force-velocity relationship

13 Force-velocity relationship Probing the force-velocity relationship by varying medium viscosity McGrath et al., Current Biology, (2003)

14 Force-velocity relationship Probing the force-velocity relationship by micro-manipulation Marcy et al., PNAS, (2004)

15 Dancing Listeria Shenoy et al., PNAS, (2007) Moving Listeria Hu and Lin, work in progress Moving microsphere

16 Mechanics questions need to be answered Fundamental question: how can propelling forces be generated by ploymerization? Physically, what is the force generation mechanism? Mathematically, how can we formulate the process? More specific questions: what cause actin-driven cargos to move in different fashions? Force-velocity relationships. Different trajectories Smooth and jerky motions. Onset of movement ( symmetry breaking ).

17 Theory: elastic Brownian ratchet (EBR) model Mogilner and Oster, Biophys. J. (1996, 2003) Load surfaceRigid actin network actin monomer actin filament thermally deformed filament Completion of polymerization Important features need to be included in the model: The tail is attached to the load surface. New filament tips are nucleated by the branching of existing filaments. Some tips lose the ability to grow due to the binding of certain capping proteins. 17

Brownian motions of particles near a moving surface Governing equation in the moving frame ( Smoluchowski equation): p (x, t) – Density of states. U – Drifting potential. h (x, t) – Source (or sink) distribution of particles. At steady state, conservation of particles implies The particles represent the filament tips in the actin-based motility problem - Probability flux Lin, Physical Review E. (2009) 18 Theory: generalized elastic Brownian ratchet (GEBR) model

Assume the wall is actually penetratable with an energy penalty U w associated with the penetration. The rigid wall situation is approached by letting In this limiting case, p(x) in the region x<0 is essentially governed by The propelling force generated by the bombardment of a single particle is As, f takes the simple form - independent of 19 Propelling force generated on the surface

Descriptions suitable for actin-based motility Function form of U(x): Representing the adhesion between the tip and the wall – Depth of the attractive potential well – Width of the potential well Representing the bending energy stored in the filament 20 persistence length Tip nucleation: Tip nucleation is assumed to take place on the surface, hence can be represented by a net probability flux at x = 0, i.e., J 0

Actin polymerization and depolymerization: Capping reactions: Total source distribution: 21 Descriptions suitable for actin-based motility

General Solution Closed form solutions: Where Two constants C and s 0 can be determined by Total force generated by a single filament 22

Comparison with experimental data McGrath et al., Current Biology. (2003)Marcy et al., PNAS. (2004) Moving Listeria monocytogenes Moving beads 23

24 Comparison with experimental data The presence of Vasodilator-stimulate phosphoprotein (VASP) induces an up to 10-fold increase in the bead speed Samarin et al. (J. Cell Biol., 2003) suggested that VASP accelerates the dissociation between the tip and the surface by a factor of around100. Notice: the dissociation rate increases 50 times when C b varies from 4 to 0. Samarin et al., J. Cell Biol. (2003)

25 Remarks The GEBR model can reproduce measured force- velocity curves. The formula obtained is compact and simple. It assumes the cargo is moving along a straight line and all filaments polymerize with the same rate. It cannot predict/explain the complex trajectories of actual actin-driven cargos. To generalized the idea, the average force generated by each filament should depends on the local polymerization rate and filament growth speed, that is Shenoy et al., PNAS, (2007)

26 ActA distribution on Listeria is not uniform The distribution of ActA on Listeria surface is not uniform. Usually, the ActA density reaches the maximum at one pole and decreases gradually while moving away from that pole. However, ActA will redistribute themselves to both poles in a dividing bacteria, so both of the old poles have the highest ActA concentration after division. Rafelski et al., PLoS Comp. Biol., (2009) Rafelski and Theriot, Biophys. J. (2005) Implications: Ploymerization rate may not be spatially uniform !

Probing the consequence of non-uniform polymerization What will happen if the polymerization is not uniform. Filament force will also vary spatially. Motions of Listeria Lin et al., Biophys. J. (In press) 27 Coupling between filament growth and cargo motion

28 Numerical scheme for finding the solution Since filament growth is coupled with the cargo motion, an iterative scheme was constructed to determine the solution. The closed form solution from GEBR model is used to evaluate filament forces at different locations.

29 Symmetric perturbation in polymerization Assume Propelling force distribution among filaments arranged in a 5 X 5 array Upadhyaya et al., PNAS, (2003) Negative propelling forces can be generated by the inner part of the tail if polymerization there is much slower than that at the outer portion of the tail.

30 Asymmetric perturbation in polymerization In this case, let Filament forces also become asymmetric A net moment will be generated causing the cargo to move in circles Radius of curvature as a function of perturbation magnitude

31 Spinning of Listeria Moving Listeria actually spins along its long-axis Robbins and Theriot, Current Biology, (2003) It has been proposed that spinning is essential for the generation of different Listeria trajectories. Shenoy et al., PNAS, (2007)

32 Generation of a tangential force How can the spinning be generated? Consider a more realistic description of the tip – load surface interaction The interaction fore is along direction As such, in addition to a propelling force, a tangential force will also be induced Small deformation

33 Correlation in filament orientations leads to a torque causing Listeria to spin Notice that the direction of the tangential force depends on filament orientation RandomPartially correlatedTotally correlated A torque can be generated as Characterize the correlation in filament orientations with a single parameter The spinning angular velocity is

34 2-D Listeria motion When confined to move in a plane, say x-y plane. The z component of moment generated by polymerizing filaments is The cargo trajectory can then be determined as: Or

Comparison with experimental data 35

36 Conclusions The model can indeed reproduce almost all observed trajectories. Filaments are assumed to run into the surface with the same angle. Results are based on steady-state solutions. It did not consider the evolution of the actin comet tail during. It can not explain transient phenomena ( hopping motion, initiation of movement etc.) Need more realistic models !