WEBVTT 00:00:05.260 --> 00:00:11.580 How do neurons send chemical signals to neighboring neurons? Why do you wear a jacket in the winter? 00:00:11.580 --> 00:00:17.270 Why do some animals have circulatory systems? These questions depend on random walks and 00:00:17.270 --> 00:00:18.450 diffusion. 00:00:18.450 --> 00:00:22.880 In this video, using a very simple model, you will learn the fundamental difference 00:00:22.880 --> 00:00:28.029 between a regular and a random walk, and be able to predict the consequences of that difference 00:00:28.029 --> 00:00:29.720 for biophysical systems. 00:00:29.720 --> 00:00:34.120 This video is part of the Probability and Statistics video series. 00:00:34.120 --> 00:00:39.559 Many events and phenomena are probabilistic. Engineers, designers, and architects often 00:00:39.559 --> 00:00:42.790 use probability distributions to predict system behavior. 00:00:42.790 --> 00:00:48.479 Hi, my name is Sanjoy Mahajan, and I'm a professor of Applied Science and Engineering at Olin 00:00:48.479 --> 00:00:49.690 College. 00:00:49.690 --> 00:00:53.960 Before watching this video, you should be familiar with moments of distributions and 00:00:53.960 --> 00:00:56.010 with concentration gradients. 00:00:56.010 --> 00:00:58.790 After watching this video, you will be able to: 00:00:58.790 --> 00:01:01.659 Describe the difference between regular and random walks. 00:01:01.659 --> 00:01:08.659 And, explain the structure of Fick's law for flux. 00:01:10.230 --> 00:01:15.650 Between neurons, molecules travel by diffusion. They wander a bit, collide, change directions, 00:01:15.650 --> 00:01:19.980 wander back, collide again, and higgle and jiggle their way across the neural gap (the 00:01:19.980 --> 00:01:21.350 synaptic cleft). 00:01:21.350 --> 00:01:26.550 Here is a diagram of it. This is the inside of one neuron, here is the inside of the other 00:01:26.550 --> 00:01:31.039 neuron, and here is the synaptic cleft in which there are molecules wandering across 00:01:31.039 --> 00:01:35.300 from the left neuron to the right where they are received and picked up and used to generate 00:01:35.300 --> 00:01:37.800 a signal in the second neuron. 00:01:37.800 --> 00:01:42.080 An extremely simple model of this process, which has the merit of containing all the 00:01:42.080 --> 00:01:47.340 essential physics, is a molecule making a random walk on a one-dimensional number line: 00:01:47.340 --> 00:01:53.780 To further simplify our life, this model molecule moves only at every clock tick, and sits peacefully 00:01:53.780 --> 00:01:58.810 waiting for the clock tick. At each clock tick, it moves left or right by one unit, 00:01:58.810 --> 00:02:03.170 with equal probability (50 percent) of moving in each direction. 00:02:03.170 --> 00:02:08.970 Our molecule here, after a few ticks, has reached x=4. So the probability of finding 00:02:08.970 --> 00:02:15.970 it at x=4 is 1. What will happen to it in the next time ticks? After the next tick, 00:02:17.060 --> 00:02:22.849 the molecule is equally likely to be at 3 or 5. That changes the probability distribution 00:02:22.849 --> 00:02:24.180 to the following. 00:02:24.180 --> 00:02:29.019 Thus, although we don't know exactly where it will be, we know that the expected value 00:02:29.019 --> 00:02:33.209 of x is still 4. 00:02:33.209 --> 00:02:38.060 Pause the video here, and find the expected value after one more tick -- that is, two 00:02:38.060 --> 00:02:45.060 ticks after it was known to be at 4. 00:02:47.279 --> 00:02:53.389 You should have found that the expected value is still 4. Here is the probability distribution. 00:02:53.389 --> 00:02:59.279 It has a one-fourth chance to be at 2, a one-half chance to be at 4, and a one-fourth chance 00:02:59.279 --> 00:03:00.519 to be at 6. 00:03:00.519 --> 00:03:07.519 Thus, the expected value of x is ¼ times 2 + ½ times 4 + ¼ times 6, which equals 00:03:08.989 --> 00:03:15.989 4. In short, the expected value never changes. Alone, it is thus not a good way of characterizing 00:03:17.019 --> 00:03:22.730 how the molecule wanders. We also need to characterize the spread in its position. 00:03:22.730 --> 00:03:29.730 Thus, we use a higher moment, the second moment, the expected value of x-squared. At first 00:03:30.340 --> 00:03:36.168 when the molecule was at x=4 right here, and it was for sure there, then the expected value 00:03:36.168 --> 00:03:39.819 of x^2 was just 4^2 or 16. 00:03:39.819 --> 00:03:46.819 What about after one clock tick? Pause the video here and work out 00:03:54.668 --> 00:04:01.669 You should have found that 00:04:16.720 --> 00:04:19.918 We find that the expected value of x-squared equals 18. 00:04:19.918 --> 00:04:26.880 Hmm, it seems like the expected value increases by 1 with every clock tick. That's true in 00:04:26.880 --> 00:04:32.950 general, no matter how many ticks you wait, or where the molecule started. Thus, for a 00:04:32.950 --> 00:04:37.909 molecule starting at the origin (at 0), the expected value of x-squared is just the number 00:04:37.909 --> 00:04:43.370 of clock ticks. This equality is fascinating, because it contains 00:04:43.370 --> 00:04:49.030 the difference between this kind of walk, a random walk, and a regular walk. If the 00:04:49.030 --> 00:04:54.250 molecule did a regular walk, moving one step every clock tick, without switching directions, 00:04:54.250 --> 00:05:00.100 then the number of clock ticks would be the expected value of x, not x^2. 00:05:00.100 --> 00:05:05.060 The random walk is fundamentally different, and that fundamental difference will explain, 00:05:05.060 --> 00:05:10.120 among a vast number of physical phenomena, why you wear a jacket in the winter, and why 00:05:10.120 --> 00:05:13.260 some animals have circulatory systems. 00:05:13.260 --> 00:05:19.420 Now, instead of speaking of counting clock ticks, let's measure actual time. Instead 00:05:19.420 --> 00:05:26.420 of counting units left or right, let's measure actual distance. If each clock tick takes 00:05:26.570 --> 00:05:32.860 time tau, and each distance unit is lambda, instead of one, as before, then these relationships 00:05:32.860 --> 00:05:38.810 here change slightly to include the dimensions and units. For the regular walk, x is lambda 00:05:38.810 --> 00:05:45.280 times the number of ticks. The number of clock ticks is t/tau, so the expected value of x 00:05:45.280 --> 00:05:51.900 squared is lambda squared times t/Tau here. And here we have lambda times T/Tau for the 00:05:51.900 --> 00:05:54.230 regular walk. 00:05:54.230 --> 00:06:00.650 In the regular walk we can rewrite that as Lambda/Tau times T. That lambda/tau here has 00:06:00.650 --> 00:06:07.650 a special name: the speed. In a random walk, the constant of proportionality is lambda^2/tau. 00:06:08.860 --> 00:06:14.240 This constant lambda squared/Tau, which has dimensions of length squared/time, is the 00:06:14.240 --> 00:06:16.990 diffusion constant D. 00:06:16.990 --> 00:06:22.400 Let's see how "fast" a random walk goes, in comparison with a regular walk. 00:06:22.400 --> 00:06:26.750 Suppose that the molecule has to cross the narrow gap between two neurons, a synaptic 00:06:26.750 --> 00:06:33.750 cleft, which has width L. If we wait long enough, until 00:06:38.100 --> 00:06:45.100 How long do we wait on average? Until "t" here is about L^2 / D. Thus, the "speed" of 00:06:45.100 --> 00:06:50.680 the random walk is something like the distance divided by this time t, and that time t is 00:06:50.680 --> 00:06:55.659 the distance squared divided by the diffusion constant. So this speed is the diffusion constant 00:06:55.659 --> 00:06:57.790 divided by distance. 00:06:57.790 --> 00:07:03.170 Again we see the fundamental difference between a regular and a random walk. A regular walk 00:07:03.170 --> 00:07:09.320 has a constant speed here of lambda over tau, as long as lambda and tau don't change. In 00:07:09.320 --> 00:07:16.320 contrast, in a random walk, the speed is inversely proportional to the gap L. 00:07:20.240 --> 00:07:26.050 This result explains the structure of Fick's Law for the flux of stuff. Flux is particles 00:07:26.050 --> 00:07:30.640 per area per time. Flux, say's Fick's law, equals the diffusion 00:07:30.640 --> 00:07:36.450 constant times the concentration gradient dn/dx, where n is the concentration. 00:07:36.450 --> 00:07:40.150 How are the flux and diffusion velocity connected? 00:07:40.150 --> 00:07:47.150 Well flux is also equal to the concentration n times the speed. And here the speed is D/L. 00:07:48.120 --> 00:07:53.720 But where did the dx here and the dn here come from? What do those have to do with n 00:07:53.720 --> 00:07:55.550 and L? 00:07:55.550 --> 00:08:02.110 Imagine two regions. One with concentration n1 and another with concentration n2, separated 00:08:02.110 --> 00:08:04.480 by a distance delta x. 00:08:04.480 --> 00:08:09.930 So this is the concentration of neurotransmitter here at one end and concentration of neurotransmitter 00:08:09.930 --> 00:08:15.870 here at the other end of say, a gap. We could use this same model for oxygen in a circulatory 00:08:15.870 --> 00:08:17.580 system. 00:08:17.580 --> 00:08:23.430 Then the flux in one direction is this and in the reverse direction, it's this. The net 00:08:23.430 --> 00:08:28.360 flux is n2-n1 times D/delta x. 00:08:28.360 --> 00:08:34.979 So we've explained the d and the delta x in Fick's law over here. What about the dn? Well 00:08:34.979 --> 00:08:41.979 n2-n1 is the difference in n, or just dn, so this piece here is dn. This is dx, and 00:08:42.110 --> 00:08:42.760 this is D. 00:08:42.760 --> 00:08:49.320 So we arrive at Fick's Law based on the realization that flux is concentration times speed, and 00:08:49.320 --> 00:08:55.029 the speed here in a random walk is the diffusion constant divided by L. And that's why you 00:08:55.029 --> 00:09:00.700 wear a coat, rather than a thin shirt, in the winter. The thin shirt has a dx of maybe 00:09:00.700 --> 00:09:05.420 2 mm. But the winter coat may be 2 cm thick. 00:09:05.420 --> 00:09:10.450 That reduces the heat flux by a factor of 10 through your coat compared to the shirt. 00:09:10.450 --> 00:09:15.040 And you can stay warm just using the heat produced by your basal metabolism -- about 00:09:15.040 --> 00:09:16.950 100 Watts. 00:09:16.950 --> 00:09:21.640 For our final calculations, let's return to the neurotransmitter and then discuss circulatory 00:09:21.640 --> 00:09:22.920 systems. 00:09:22.920 --> 00:09:27.370 How long would it take a neurotransmitter molecule to diffuse across a 20 nm synaptic 00:09:27.370 --> 00:09:33.260 cleft? The diffusion constant for a typical neurotransmitter molecule wandering in water, 00:09:33.260 --> 00:09:39.340 which is mostly what's in between neurons, is about 10^-10 m^2/s. 00:09:39.340 --> 00:09:46.340 Pause the video here and make your estimate of the time. 00:09:49.330 --> 00:09:53.370 You should have found that the time is about 4 microseconds. 00:09:53.370 --> 00:09:58.690 Is that time short or long? It's short, because it's much smaller than, say, the rise time 00:09:58.690 --> 00:10:04.260 of a nerve signal or the timing accuracy of nerve signals. Over the short distance of 00:10:04.260 --> 00:10:10.430 the synaptic cleft, diffusion is a fast and efficient way to transport molecules. 00:10:10.430 --> 00:10:15.980 How does this analysis apply to a circulatory system? Imagine a big organism, say you or 00:10:15.980 --> 00:10:18.520 me, but without a circulatory system. 00:10:18.520 --> 00:10:23.960 How long would oxygen need to diffuse from the lung to a leg muscle say, one meter away? 00:10:23.960 --> 00:10:29.550 That's where the oxygen is needed to burn glucose and produce energy. Oxygen, a small 00:10:29.550 --> 00:10:33.400 molecule, has a slightly higher diffusion constant than a neurotransmitter molecule 00:10:33.400 --> 00:10:38.560 does--D is roughly 1x10^-9 meters squared/sec. 00:10:38.560 --> 00:10:45.560 Pause the video and make your estimate of the diffusion time. 00:10:48.490 --> 00:10:53.580 You should have found that the diffusion time is roughly...10 to the ninth seconds! 00:10:53.580 --> 00:11:00.580 That's roughly 30 years! Over long distances, diffusion is a lousy method of transport! 00:11:00.670 --> 00:11:06.089 That's why we need a circulatory system. Using a dense network of capillaries, the circulatory 00:11:06.089 --> 00:11:11.680 system brings oxygen-rich blood near every cell...and only then, when the remaining distance 00:11:11.680 --> 00:11:18.680 is tiny, does it let diffusion finish the job! 00:11:20.120 --> 00:11:25.720 In this video, we saw how a random walk, which is the process underlying diffusion, is fundamentally 00:11:25.720 --> 00:11:30.210 different from a regular walk, and how that difference explains the structure of Fick's 00:11:30.210 --> 00:11:34.120 law and allows us to estimate diffusion times. 00:11:34.120 --> 00:11:39.790 The moral is that we live and breathe based on the random walk, whose physics we can understand 00:11:39.790 --> 00:11:46.790 with a simple number line and moments of distributions.