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How do neurons send chemical signals to neighboring
neurons? Why do you wear a jacket in the winter?
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Why do some animals have circulatory systems?
These questions depend on random walks and
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diffusion.
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In this video, using a very simple model,
you will learn the fundamental difference
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between a regular and a random walk, and be
able to predict the consequences of that difference
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for biophysical systems.
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This video is part of the Probability and
Statistics video series.
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Many events and phenomena are probabilistic.
Engineers, designers, and architects often
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use probability distributions to predict system
behavior.
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Hi, my name is Sanjoy Mahajan, and I'm a professor
of Applied Science and Engineering at Olin
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College.
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Before watching this video, you should be
familiar with moments of distributions and
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with concentration gradients.
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After watching this video, you will be able
to:
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Describe the difference between regular and
random walks.
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And, explain the structure of Fick's law for
flux.
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Between neurons, molecules travel by diffusion.
They wander a bit, collide, change directions,
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wander back, collide again, and higgle and
jiggle their way across the neural gap (the
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synaptic cleft).
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Here is a diagram of it. This is the inside
of one neuron, here is the inside of the other
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neuron, and here is the synaptic cleft in
which there are molecules wandering across
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from the left neuron to the right where they
are received and picked up and used to generate
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a signal in the second neuron.
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An extremely simple model of this process,
which has the merit of containing all the
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essential physics, is a molecule making a
random walk on a one-dimensional number line:
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To further simplify our life, this model molecule
moves only at every clock tick, and sits peacefully
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waiting for the clock tick. At each clock
tick, it moves left or right by one unit,
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with equal probability (50 percent) of moving
in each direction.
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Our molecule here, after a few ticks, has
reached x=4. So the probability of finding
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it at x=4 is 1. What will happen to it in
the next time ticks? After the next tick,
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the molecule is equally likely to be at 3
or 5. That changes the probability distribution
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to the following.
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Thus, although we don't know exactly where
it will be, we know that the expected value
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of x is still 4.
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Pause the video here, and find the expected
value after one more tick -- that is, two
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ticks after it was known to be at 4.
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You should have found that the expected value
is still 4. Here is the probability distribution.
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It has a one-fourth chance to be at 2, a one-half
chance to be at 4, and a one-fourth chance
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to be at 6.
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Thus, the expected value of x is ¼ times
2 + ½ times 4 + ¼ times 6, which equals
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4. In short, the expected value never changes.
Alone, it is thus not a good way of characterizing
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how the molecule wanders. We also need to
characterize the spread in its position.
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Thus, we use a higher moment, the second moment,
the expected value of x-squared. At first
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when the molecule was at x=4 right here, and
it was for sure there, then the expected value
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of x^2 was just 4^2 or 16.
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What about after one clock tick? Pause the
video here and work out
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You should have found that
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We find that the expected value of x-squared
equals 18.
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Hmm, it seems like the expected value increases
by 1 with every clock tick. That's true in
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general, no matter how many ticks you wait,
or where the molecule started. Thus, for a
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molecule starting at the origin (at 0), the
expected value of x-squared is just the number
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of clock ticks.
This equality is fascinating, because it contains
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the difference between this kind of walk,
a random walk, and a regular walk. If the
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molecule did a regular walk, moving one step
every clock tick, without switching directions,
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then the number of clock ticks would be the
expected value of x, not x^2.
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The random walk is fundamentally different,
and that fundamental difference will explain,
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among a vast number of physical phenomena,
why you wear a jacket in the winter, and why
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some animals have circulatory systems.
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Now, instead of speaking of counting clock
ticks, let's measure actual time. Instead
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of counting units left or right, let's measure
actual distance. If each clock tick takes
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time tau, and each distance unit is lambda,
instead of one, as before, then these relationships
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here change slightly to include the dimensions
and units. For the regular walk, x is lambda
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times the number of ticks. The number of clock
ticks is t/tau, so the expected value of x
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squared is lambda squared times t/Tau here.
And here we have lambda times T/Tau for the
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regular walk.
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In the regular walk we can rewrite that as
Lambda/Tau times T. That lambda/tau here has
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a special name: the speed. In a random walk,
the constant of proportionality is lambda^2/tau.
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This constant lambda squared/Tau, which has
dimensions of length squared/time, is the
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diffusion constant D.
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Let's see how "fast" a random walk goes, in
comparison with a regular walk.
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Suppose that the molecule has to cross the
narrow gap between two neurons, a synaptic
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cleft, which has width L. If we wait long
enough, until
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How long do we wait on average? Until "t"
here is about L^2 / D. Thus, the "speed" of
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the random walk is something like the distance
divided by this time t, and that time t is
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the distance squared divided by the diffusion
constant. So this speed is the diffusion constant
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divided by distance.
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Again we see the fundamental difference between
a regular and a random walk. A regular walk
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has a constant speed here of lambda over tau,
as long as lambda and tau don't change. In
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contrast, in a random walk, the speed is inversely
proportional to the gap L.
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This result explains the structure of Fick's
Law for the flux of stuff. Flux is particles
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per area per time.
Flux, say's Fick's law, equals the diffusion
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constant times the concentration gradient
dn/dx, where n is the concentration.
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How are the flux and diffusion velocity connected?
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Well flux is also equal to the concentration
n times the speed. And here the speed is D/L.
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But where did the dx here and the dn here
come from? What do those have to do with n
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and L?
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Imagine two regions. One with concentration
n1 and another with concentration n2, separated
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by a distance delta x.
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So this is the concentration of neurotransmitter
here at one end and concentration of neurotransmitter
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here at the other end of say, a gap. We could
use this same model for oxygen in a circulatory
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system.
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Then the flux in one direction is this and
in the reverse direction, it's this. The net
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flux is n2-n1 times D/delta x.
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So we've explained the d and the delta x in
Fick's law over here. What about the dn? Well
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n2-n1 is the difference in n, or just dn,
so this piece here is dn. This is dx, and
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this is D.
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So we arrive at Fick's Law based on the realization
that flux is concentration times speed, and
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the speed here in a random walk is the diffusion
constant divided by L. And that's why you
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wear a coat, rather than a thin shirt, in
the winter. The thin shirt has a dx of maybe
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2 mm. But the winter coat may be 2 cm thick.
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That reduces the heat flux by a factor of
10 through your coat compared to the shirt.
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And you can stay warm just using the heat
produced by your basal metabolism -- about
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100 Watts.
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For our final calculations, let's return to
the neurotransmitter and then discuss circulatory
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systems.
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How long would it take a neurotransmitter
molecule to diffuse across a 20 nm synaptic
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cleft? The diffusion constant for a typical
neurotransmitter molecule wandering in water,
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which is mostly what's in between neurons,
is about 10^-10 m^2/s.
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Pause the video here and make your estimate
of the time.
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You should have found that the time is about
4 microseconds.
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Is that time short or long? It's short, because
it's much smaller than, say, the rise time
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of a nerve signal or the timing accuracy of
nerve signals. Over the short distance of
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the synaptic cleft, diffusion is a fast and
efficient way to transport molecules.
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How does this analysis apply to a circulatory
system? Imagine a big organism, say you or
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me, but without a circulatory system.
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How long would oxygen need to diffuse from
the lung to a leg muscle say, one meter away?
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That's where the oxygen is needed to burn
glucose and produce energy. Oxygen, a small
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molecule, has a slightly higher diffusion
constant than a neurotransmitter molecule
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does--D is roughly 1x10^-9 meters squared/sec.
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Pause the video and make your estimate of
the diffusion time.
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You should have found that the diffusion time
is roughly...10 to the ninth seconds!
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That's roughly 30 years! Over long distances,
diffusion is a lousy method of transport!
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That's why we need a circulatory system. Using
a dense network of capillaries, the circulatory
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system brings oxygen-rich blood near every
cell...and only then, when the remaining distance
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is tiny, does it let diffusion finish the
job!
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In this video, we saw how a random walk, which
is the process underlying diffusion, is fundamentally
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different from a regular walk, and how that
difference explains the structure of Fick's
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law and allows us to estimate diffusion times.
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The moral is that we live and breathe based
on the random walk, whose physics we can understand
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with a simple number line and moments of distributions.