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?