Cellular Biophysics

The world you live in An inertial world- objects that are

moving tend to keep moving even after force is removed- inertia

This is the basis of motion in our world Fi=ma

The Viscous World In fluids, viscosity becomes important The force imparted by the fluid is

dependent upon its viscosity

Force

distance, l

velocity, vArea, S

Viscous Force

Things get weird when viscosity increases Consider a cylinder containing corn syrup Add a dot of dye in corn syrup Stir the syrup/dye in one direction Reverse the direction of stirring The dot reforms Viscous fluids do not flow or mix No turbulence, no inertia

Now consider density and viscosity

Fluid Density Viscosity

Air 1 2x10-5

Water 1000 .0009

Olive oil 900 .08

Glycerine 1300 1

Corn syrup 1000 5

Reynolds Number-the ratio of inertal to viscous forces

Reynolds number=inertial force/viscous force =density (of medium), l=length, S=area,

v/t=velocity over time=acceleration µ=viscosity (of medium)

Inertial is densityxvolume=mass x accel (v/t) Viscous is Force/area= viscosity x the velocity

gradient.

Inertial Force F=ma= (l3v2/l Viscous Force= l3v/R2

Re= Inertial Force = vR/ Viscous Force

What does it mean?

As size goes down, Re goes down As viscosity goes up, Re goes down At high Reynold’s numbers- inertial forces

dominate At low Reynold’s numbers- viscous forces

dominate Small objects in fluids are affected by the

frictional drag of the media to a great extent

Sample Reynolds’ numbers Bacterium swimming (organelle) 10-6

Sperm swimming 10-2

Fruit fly in flight 100 Small bird flying 105

Whale swimming2x108

What does it mean The forces associated with molecules of water interacting with each

other and solutes become relevant To a small molecule (bacteria) moving through a fluid is like you trying to

move in a highly viscous liquid. Imagine yourself living in asphalt (Berg experiment)

Being small is equivalent to being in a very viscous environment Water is highly ordered around you-you are the boundary layer surfaces nearby create boundary effects that alter the properties of

water significant distances away There is no inertia- if a bacteria stops swimming, it glides about the

distance of a hydrogen atom drag is irrelevant (shape is irrelevant) so streamlining is irrelevant

What does it mean to Cell Biology?

A small predator cannot catch a prey by swimming at it, because it pushes the prey away as it swims

A bacteria cannot swim by waving a flagella or cilia- it would return to the same place after a cycle of motion

Diffusion What is diffusion?

The random movement of molecules due to thermal energy

The fundamental principle underlying all life processes! Determines the rate of enzyme reactions Determines the size and shape of cells Determines the speed of signal transduction

History Until the early 1900’s, the idea of molecules

was controversial In 1828, Robert Brown observed movement

of pollen particles in suspension (Brownian motion)

What was driving the motion? Hypothesis 1- they were alive

But they never stopped! Lifeless particles (soot) did the same

Hypothesis 2 (1860’s)- movement was caused by collisions of water molecules with the pollen

At higher temperatures, they moved faster! But- particles are much larger than water

molecules- how can water move particles? The speed of water molecules is 103m/s and there

would be about 1012 collisions/sec. Too fast for the eye to see

How to resolve this???

Einstein strikes again

Clarified the stochastic nature of molecular motion- there are many events happening very rapidly

If you take the look at the probabilities, then with that many collisions with water molecules with a range of velocities, then periodically you will get a displacement of the particle by many more collisions on one side than another

The process will lead to a 3D random walk of the particle: Diffusion

The Diffusion Law

mean square displacement x2=6Dt This is stochastic, not the behavior of a individual molecule

Any molecule might not move at all Others may move a great distance

There is no “rate” of diffusion x/t=v=6D/x or the rate gets slower the farther you are away

So if you follow a certain concentration of molecules, that concentration will move rapidly away from a source, and the farther you get from the source, the slower that concentration will move

If the source only produces a limited number of molecules, then at some distance, you will never reach that concentration

Diffusion of Biological Molecules

Substance M D (cm2/sec) time to diffuse 1µ diffuse 10µm

bacterium 5x10-9 1 sec 100sec TMV 4x107 3x10-8 0.1 sec 10 sec albumin 7x104 6x10-7 10 msec 1 sec sucrose 3x102 5x10-6 1msec

100msec

Diffusion and Signalling If you want to send a signal inside a cell, how

do you do it? IP3 or Ca release at the membrane You assume there is a threshold for the

effect- ie. you need above a certain concentration of the signal molecule to activate the effectors

Do you want the response to be local or general?

Do you want it to continue or terminate?

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Diffusion of Pulse vs. Continuous signal

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Diffusion of Pulse vs. Continuous signal

Signaling in large cells (multicellular organisms) If you release a signal in a cell (Ca ions) and they

diffuse from the site of release, it will take time for signal to reach other parts of the cell, and the concentration will be lower, the farther you get from the site of release

If there is a threshold for action, you might not exceed it at distant sites- allows for local action

It would take about 10 minutes for a Ca wave to get across a 1mm Xenopus egg and it would never reach as hi a concentration because it would be diluted

Reaction diffusion waves- you relay the signal so that the size of the signal remains constant

What is cytoplasm like Cells are about 20mg/ml protein You can’t dissolve 20mg/ml of most proteins How do you do it in a cell? Based upon this, it was hypothesized that

most of the cellular water was tied up in coating proteins, and thus the cytoplasm had limited water

This would affect diffusion

FRAP of cytoplasm Introduce a fluorescent molecule into the

cytoplasm of the cell Microinject fluorescein dextran Shine a very bright light source as a small

spot onto a stained region to bleach the dye Produces a dark spot on a light background Now measure the fluorescence intensity of

the spot over time as fluorescence recovers (Fluorescence Recovery After Photobleaching)

FRAP analysis

Figure 2 JCB 138:131Figure 2 JCB 138:131

Conclusions Dc/Dw is constant over a range of sizes and

locations in the cell The ratio is about 0.25: diffusion in cytoplasm is

about 4x slower than in water for macromolecules At these rates it would take a large macromolecule

about 7 seconds to diffuse across a cell For large macromolecules, there is little diffusion Reason is controversial

Immobile obstacles? Cytoskeletal mesh?