Exploring Protein Motors -- from what we can measure to what we like to know Hongyun Wang Department of Applied Mathematics and Statistics University of California, Santa Cruz Baskin School of Engineering Research Review Day October 12, 2007

The goal of mathematical studies … Chen & Berg (2000) Yasuda, et al. (1998) Visscher, Schnitzer & Block (1999) … is to decipher motor mechanism from measurements.

An example of macroscopic motor Torque of a single cylinder engine: (1) Intake(2) Compression(3) Expansion(4) Exhaust How to find the motor force of a molecular motor?

Mathematical framework for molecular motors

Mechanochemical models Chemical reaction: Mechanical motion and chemical reaction are coupled. Mechanical motion: Chen & Berg (2000) Yasuda, et al. (1998) Visscher, Schnitzer & Block (1999)

Characters of molecular motors Molecular motors Macroscopic motors Time scale of inertia << time scale of reaction cycle Instantaneous velocity >> average velocity Kinetic energy from average velocity << k B T Use thermal fluctuations to get over bumps Time scale of inertia >> time scale of reaction cycle Instantaneous velocity ≈ average velocity Kinetic energy from average velocity >> k B T Use stored kinetic energy (inertia) to get over bumps

Macroscopic motors use stored kinetic energy to get over bumps This does not work in molecular motors!

Molecular motors use thermal excitations to get over bumps The energy for accelerating a bottle of water to 100 miles/hour can only heat up the bottle of water by 0.24 degree! Thermal energy is huge!

Mathematical equations Langevin formulation (stochastic evolution of an individual motor): Fokker-Planck formulation (deterministic evolution of probability density): (mechanical motion) (chemical reaction)

Efficiencies of a molecular motor

Thermodynamic efficiency of a motor working against a conservative force Motor system External agent Visscher et al (1999). Nature 

 Viscous drag is not a conservative force: The Stokes efficiency: Stokes efficiency of a motor working against a viscous drag Hunt et al (1994). Biophys. J.  Energy output = 0 Yasuda, et al (1998). Cell.

Stokes efficiency  thermodynamic efficiency (experimental observations) Visscher et al (1999). Nature Hunt et al (1994). Biophys. J. Viscous stall load < Thermodynamic stall load Which measurement is correct?

Stokes efficiency  thermodynamic efficiency (theory) Viscous stall load: … implies that the motor force is not uniform.

Motor potential profile

A mean-field potential Fokker-Planck formulation At steady state, summing over S The motor behaves as if it were driven by a single potential  (x)  (x) is called the motor potential profile. Probability density of motor at x Motor force at x (averaged over all chemical states).

The potential profile is measurable Time series of motor positions measured in single molecule experiments … does not work well   Therefore, we only need to reconstruct PDF. A more robust formulation:

Reconstructing potential in a test problem Extracted motor potential profile A sequence of 5000 motor positions generated in a Langevin simulation  

Summary Everything depends on a proper mathematical model and analysis of the model!

Time scale of inertia Time scale of inertia: Newton’s 2nd law: For a bead of 1  m diameter: