1 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? 2 00:00:11,580 --> 00:00:17,270 Why do some animals have circulatory systems? These questions depend on random walks and 3 00:00:17,270 --> 00:00:18,450 diffusion. 4 00:00:18,450 --> 00:00:22,880 In this video, using a very simple model, you will learn the fundamental difference 5 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 6 00:00:28,029 --> 00:00:29,720 for biophysical systems. 7 00:00:29,720 --> 00:00:34,120 This video is part of the Probability and Statistics video series. 8 00:00:34,120 --> 00:00:39,559 Many events and phenomena are probabilistic. Engineers, designers, and architects often 9 00:00:39,559 --> 00:00:42,790 use probability distributions to predict system behavior. 10 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 11 00:00:48,479 --> 00:00:49,690 College. 12 00:00:49,690 --> 00:00:53,960 Before watching this video, you should be familiar with moments of distributions and 13 00:00:53,960 --> 00:00:56,010 with concentration gradients. 14 00:00:56,010 --> 00:00:58,790 After watching this video, you will be able to: 15 00:00:58,790 --> 00:01:01,659 Describe the difference between regular and random walks. 16 00:01:01,659 --> 00:01:08,659 And, explain the structure of Fick's law for flux. 17 00:01:10,230 --> 00:01:15,650 Between neurons, molecules travel by diffusion. They wander a bit, collide, change directions, 18 00:01:15,650 --> 00:01:19,980 wander back, collide again, and higgle and jiggle their way across the neural gap (the 19 00:01:19,980 --> 00:01:21,350 synaptic cleft). 20 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 21 00:01:26,550 --> 00:01:31,039 neuron, and here is the synaptic cleft in which there are molecules wandering across 22 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 23 00:01:35,300 --> 00:01:37,800 a signal in the second neuron. 24 00:01:37,800 --> 00:01:42,080 An extremely simple model of this process, which has the merit of containing all the 25 00:01:42,080 --> 00:01:47,340 essential physics, is a molecule making a random walk on a one-dimensional number line: 26 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 27 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, 28 00:01:58,810 --> 00:02:03,170 with equal probability (50 percent) of moving in each direction. 29 00:02:03,170 --> 00:02:08,970 Our molecule here, after a few ticks, has reached x=4. So the probability of finding 30 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, 31 00:02:17,060 --> 00:02:22,849 the molecule is equally likely to be at 3 or 5. That changes the probability distribution 32 00:02:22,849 --> 00:02:24,180 to the following. 33 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 34 00:02:29,019 --> 00:02:33,209 of x is still 4. 35 00:02:33,209 --> 00:02:38,060 Pause the video here, and find the expected value after one more tick -- that is, two 36 00:02:38,060 --> 00:02:45,060 ticks after it was known to be at 4. 37 00:02:47,279 --> 00:02:53,389 You should have found that the expected value is still 4. Here is the probability distribution. 38 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 39 00:02:59,279 --> 00:03:00,519 to be at 6. 40 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 41 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 42 00:03:17,019 --> 00:03:22,730 how the molecule wanders. We also need to characterize the spread in its position. 43 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 44 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 45 00:03:36,168 --> 00:03:39,819 of x^2 was just 4^2 or 16. 46 00:03:39,819 --> 00:03:46,819 What about after one clock tick? Pause the video here and work out 47 00:03:54,668 --> 00:04:01,669 You should have found that 48 00:04:16,720 --> 00:04:19,918 We find that the expected value of x-squared equals 18. 49 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 50 00:04:26,880 --> 00:04:32,950 general, no matter how many ticks you wait, or where the molecule started. Thus, for a 51 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 52 00:04:37,909 --> 00:04:43,370 of clock ticks. This equality is fascinating, because it contains 53 00:04:43,370 --> 00:04:49,030 the difference between this kind of walk, a random walk, and a regular walk. If the 54 00:04:49,030 --> 00:04:54,250 molecule did a regular walk, moving one step every clock tick, without switching directions, 55 00:04:54,250 --> 00:05:00,100 then the number of clock ticks would be the expected value of x, not x^2. 56 00:05:00,100 --> 00:05:05,060 The random walk is fundamentally different, and that fundamental difference will explain, 57 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 58 00:05:10,120 --> 00:05:13,260 some animals have circulatory systems. 59 00:05:13,260 --> 00:05:19,420 Now, instead of speaking of counting clock ticks, let's measure actual time. Instead 60 00:05:19,420 --> 00:05:26,420 of counting units left or right, let's measure actual distance. If each clock tick takes 61 00:05:26,570 --> 00:05:32,860 time tau, and each distance unit is lambda, instead of one, as before, then these relationships 62 00:05:32,860 --> 00:05:38,810 here change slightly to include the dimensions and units. For the regular walk, x is lambda 63 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 64 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 65 00:05:51,900 --> 00:05:54,230 regular walk. 66 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 67 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. 68 00:06:08,860 --> 00:06:14,240 This constant lambda squared/Tau, which has dimensions of length squared/time, is the 69 00:06:14,240 --> 00:06:16,990 diffusion constant D. 70 00:06:16,990 --> 00:06:22,400 Let's see how "fast" a random walk goes, in comparison with a regular walk. 71 00:06:22,400 --> 00:06:26,750 Suppose that the molecule has to cross the narrow gap between two neurons, a synaptic 72 00:06:26,750 --> 00:06:33,750 cleft, which has width L. If we wait long enough, until 73 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 74 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 75 00:06:50,680 --> 00:06:55,659 the distance squared divided by the diffusion constant. So this speed is the diffusion constant 76 00:06:55,659 --> 00:06:57,790 divided by distance. 77 00:06:57,790 --> 00:07:03,170 Again we see the fundamental difference between a regular and a random walk. A regular walk 78 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 79 00:07:09,320 --> 00:07:16,320 contrast, in a random walk, the speed is inversely proportional to the gap L. 80 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 81 00:07:26,050 --> 00:07:30,640 per area per time. Flux, say's Fick's law, equals the diffusion 82 00:07:30,640 --> 00:07:36,450 constant times the concentration gradient dn/dx, where n is the concentration. 83 00:07:36,450 --> 00:07:40,150 How are the flux and diffusion velocity connected? 84 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. 85 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 86 00:07:53,720 --> 00:07:55,550 and L? 87 00:07:55,550 --> 00:08:02,110 Imagine two regions. One with concentration n1 and another with concentration n2, separated 88 00:08:02,110 --> 00:08:04,480 by a distance delta x. 89 00:08:04,480 --> 00:08:09,930 So this is the concentration of neurotransmitter here at one end and concentration of neurotransmitter 90 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 91 00:08:15,870 --> 00:08:17,580 system. 92 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 93 00:08:23,430 --> 00:08:28,360 flux is n2-n1 times D/delta x. 94 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 95 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 96 00:08:42,110 --> 00:08:42,760 this is D. 97 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 98 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 99 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 100 00:09:00,700 --> 00:09:05,420 2 mm. But the winter coat may be 2 cm thick. 101 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. 102 00:09:10,450 --> 00:09:15,040 And you can stay warm just using the heat produced by your basal metabolism -- about 103 00:09:15,040 --> 00:09:16,950 100 Watts. 104 00:09:16,950 --> 00:09:21,640 For our final calculations, let's return to the neurotransmitter and then discuss circulatory 105 00:09:21,640 --> 00:09:22,920 systems. 106 00:09:22,920 --> 00:09:27,370 How long would it take a neurotransmitter molecule to diffuse across a 20 nm synaptic 107 00:09:27,370 --> 00:09:33,260 cleft? The diffusion constant for a typical neurotransmitter molecule wandering in water, 108 00:09:33,260 --> 00:09:39,340 which is mostly what's in between neurons, is about 10^-10 m^2/s. 109 00:09:39,340 --> 00:09:46,340 Pause the video here and make your estimate of the time. 110 00:09:49,330 --> 00:09:53,370 You should have found that the time is about 4 microseconds. 111 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 112 00:09:58,690 --> 00:10:04,260 of a nerve signal or the timing accuracy of nerve signals. Over the short distance of 113 00:10:04,260 --> 00:10:10,430 the synaptic cleft, diffusion is a fast and efficient way to transport molecules. 114 00:10:10,430 --> 00:10:15,980 How does this analysis apply to a circulatory system? Imagine a big organism, say you or 115 00:10:15,980 --> 00:10:18,520 me, but without a circulatory system. 116 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? 117 00:10:23,960 --> 00:10:29,550 That's where the oxygen is needed to burn glucose and produce energy. Oxygen, a small 118 00:10:29,550 --> 00:10:33,400 molecule, has a slightly higher diffusion constant than a neurotransmitter molecule 119 00:10:33,400 --> 00:10:38,560 does--D is roughly 1x10^-9 meters squared/sec. 120 00:10:38,560 --> 00:10:45,560 Pause the video and make your estimate of the diffusion time. 121 00:10:48,490 --> 00:10:53,580 You should have found that the diffusion time is roughly...10 to the ninth seconds! 122 00:10:53,580 --> 00:11:00,580 That's roughly 30 years! Over long distances, diffusion is a lousy method of transport! 123 00:11:00,670 --> 00:11:06,089 That's why we need a circulatory system. Using a dense network of capillaries, the circulatory 124 00:11:06,089 --> 00:11:11,680 system brings oxygen-rich blood near every cell...and only then, when the remaining distance 125 00:11:11,680 --> 00:11:18,680 is tiny, does it let diffusion finish the job! 126 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 127 00:11:25,720 --> 00:11:30,210 different from a regular walk, and how that difference explains the structure of Fick's 128 00:11:30,210 --> 00:11:34,120 law and allows us to estimate diffusion times. 129 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 130 00:11:39,790 --> 00:11:46,790 with a simple number line and moments of distributions.