Lecture 10Background for cell propulsionFluid dynamicsEnzyme kinetics How do animals swim?: 1. pushing fluid backward by limb action; 2. pushing fluid forward by resistance of body. I.e fish starting from release will accelerate until the backward & forward momentum (of the fluid) balance. Viscosity is only significant at the boundary layer.

Cell PropulsionSmall scale phenomenon: slow velocities driven by surface forces: pressure and viscous stress. Fluid resistance is significant, and balances propulsive force.Motion of a body depends on the ratio of viscous and inertial effects: Reynolds number: Small for cells, large for almost all animals. Cellular world is ruled by friction.

Reynolds number quantifies the relative magnitudes of frictional and inertial forces

Cellular MotorsMolecular motors must move (swim) in fluids, where most of the work is dissipatedWhat forces must they overcome?Where do the motors get their fuel? How do they exhaust spent fuel?What is the efficiency?

Oscillatory musclesSynchronous AsynchronousStretch activation

Stretch- activated currents

Sliding filamentds

Myosin5.3 pN for each myosin molecule100 molecules per filament.Each filament has c.s.a. of 1.8 X 10 15 m2 in the relaxed muscle.

Strain in solids and fluids fAd

Sample fluid propertiesWhen f > fcrit- inertial forces dominate

Fluid

m (kg m-3)

Pa-S

fcrit (N)

Air

1

2 X 10-5

4 X 10-10

Water

1000

0.0009

8 X 10-10

Olive Oil

900

0.08

7 X 10-6

Glycerine

1300

1

0.0008

Corn Syrup

1000

5

0.03

Swimming: is it worth it?Cilium with velocity, v, length, d, time scale:

Diffusion time scale :

Swimming time, ts should be < tD

Viscous flowNewtonian fluids are isotropicWhat is a viscous fluid?When f< fcrit ShearPlanar geometry

I.e., 1 mm cilium, D = 10-5 cm2/sec, so v> 103 mm /sec: stirring and swimming is not energetically favorable for nutrition.

Comparative motors

Rotary Cellular MotorsThe rotary mechanism of ATP synthase , Stock D, Gibbons C, Arechaga I, Leslie AGW, Walker JE CURRENT OPINION IN STRUCTURAL BIOLOGY ,10 (6): 672-679 DEC 20002. ATP synthase - A marvellous rotary engine of the cell, Yoshida M, Muneyuki E, Hisabori T NATURE REVIEWS MOLECULAR CELL BIOLOGY 2 (9): 669-677 SEP 20013. The gamma subunit in chloroplast F-1-ATPase can rotate in a unidirectional and counter-clockwise manner Hisabori T, Kondoh A, Yoshida M FEBS LETTERS 463 (1-2): 35-38 DEC 10 19994. Constructing nanomechanical devices powered by biomolecular motors.C. Montemagno, G Bachand, Nanotechnology 10: 225-2312, 1999.

F1 ATPase: A rotary motorCan either make or break ATP, hence is reversibleTorque of 40 pN-nM; work in 1/3 rev. is 80 pn-nM (40 * 2p/3) equivalent to free energy from ATP hydrolysisCan see rotation by attaching an actin filament

ElasticityNano versus macro elasticityBehaviour relative to kT: Stretch a rubber band and a string of paper clips. Significant for The nanometer-scale monomers of a macromolecule, but not for a string of paper clips. The retracting force exerted by a stretched rubber band is entropic. It increases disorder.

Do most polymers have persistence lengths longer than their total (contour) length?

When L>> x, the chain has many bends and is always crumpled in solution the FJC model applies, with each link approximated as 2 x, and perfectly flexible joints.To count all possible curved states in a smooth-bending rod in solution- its a WLC- supercoiling is possible.

Promoters have different abilities to uncoilTwisting DNA torsional buckling instability Unwinding and causes local denaturation Many motors are needed: RNA plymerase, DNA polymerase: 100 nucleotides/sec.Forces (pN) can stop transcription

Mechano - regulationGrowth, proliferation, protein synthesis, gene expression, homeostasis.Transduction process- how?Single cells do not provide enough material. MTC can perturb ~ 30,000 cells and is limited.MTS is more versatile- more cells, longer periods, varied waveforms..

Markov ChainsA dynamic model describing random movement over time of some activityFuture state can be predicted based on current probability and the transition matrix

Transition ProbabilitiesTodays Game OutcomeTomorrowsGame OutcomeNeed a P forTodays game

WinLoseWin3/41/2Lose1/41/2Sum11

Grades Transition Matrix This SemesterNext SemesterGrade TendenciesTo predict future:Start with now:What are the gradeprobabilities for thissemester?

Markov ChainIntial ProbabilitySet independently

Win

Lose

1/4

1/2

1/2

3/4

Computing Markov Chains% A is the transition probabilityA= [.75 .5 .25 .5]% P is starting ProbabilityP=[.1 .9]for i = 1:20P(:,i+1)=A*P(:,i)end