1 00:00:13,420 --> 00:00:18,460 Welcome back to the third event of the Oxford Mathematics Public Lectures. Home Edition. My 2 00:00:18,460 --> 00:00:23,560 name is Alan Guerrieri and I'm in charge of external relations for the Mathematical Institute, 3 00:00:23,560 --> 00:00:29,080 as usual. Special thanks to the sponsors, execs, market execs market are leading 4 00:00:29,080 --> 00:00:34,660 quantitative driven electronic market maker with offices in London, Singapore and New York. 5 00:00:34,660 --> 00:00:39,700 The ongoing support is crucial in creating quality content. There has been 6 00:00:39,700 --> 00:00:44,950 a lot of talk in the last two months about modelling related to the ongoing pandemic crisis. 7 00:00:44,950 --> 00:00:49,990 I found that there is a lot of misunderstanding of what modelling is all about, and we thought it 8 00:00:49,990 --> 00:00:55,450 would be important to take a couple of steps back and give you a very broad overview of mathematical 9 00:00:55,450 --> 00:01:00,460 modelling in the context of biological system. What does it mean to model in biology where so 10 00:01:00,460 --> 00:01:06,100 much is known and so little is understood? Is the function of a model does to make predictions 11 00:01:06,100 --> 00:01:11,260 or to understand processes that we cannot observe directly or both to discuss 12 00:01:11,260 --> 00:01:16,540 these ideas? I've invited one of the most distinguished experts on the subject 13 00:01:16,540 --> 00:01:21,700 in the world, my colleague and friend, Professor Philip Me, Philip Oulds, 14 00:01:21,700 --> 00:01:26,980 the Professorship of Mathematical Biology at Oxford, and he's a fellow at St John's College. 15 00:01:26,980 --> 00:01:32,170 He has made countless contribution to both biology and medicine within the UK. 16 00:01:32,170 --> 00:01:37,740 He has also played a very significant role in fostering a strong school of mathematical biology, 17 00:01:37,740 --> 00:01:42,850 which has been a wonderful leader and a mentor to many young student and researcher. 18 00:01:42,850 --> 00:01:47,860 The breath of Professor Menees contribution is truly remarkable. Spanning developmental 19 00:01:47,860 --> 00:01:53,150 biology system biology or pathogen interaction, cancer biology, biology, 20 00:01:53,150 --> 00:01:58,270 multi scale modelling, morphogenesis and pattern formation, amongst many others 21 00:01:58,270 --> 00:02:03,700 for his contribution. He has received too many prises to list tonight, but maybe the best indicator 22 00:02:03,700 --> 00:02:08,830 of his Murty disciplinary work is his election as a fellow to both the Academy 23 00:02:08,830 --> 00:02:14,590 of Medical Sciences and the Royal Society. Finally, I want to add that Philip 24 00:02:14,590 --> 00:02:21,410 is also a wonderful lecturer and I am very keen to hear him tonight. Please start. No, Philip. 25 00:02:21,410 --> 00:02:26,480 I'd like to begin by thanking Aladdin for inviting me to give this talk. I'm going to talk about 26 00:02:26,480 --> 00:02:31,940 mathematical modelling in biology, ecology and medicine. 27 00:02:31,940 --> 00:02:37,760 So that's your first question to ask is what is a model and the best definition 28 00:02:37,760 --> 00:02:43,100 I've seen is that by Alan Turing. Who, 29 00:02:43,100 --> 00:02:48,110 talking about his model in 1952, said this model will 30 00:02:48,110 --> 00:02:54,710 be a simplification and an idealisation and consequently falsification. 31 00:02:54,710 --> 00:02:59,990 It is to be hoped that the features retained for discussion are those of greatest 32 00:02:59,990 --> 00:03:05,060 importance in the present state of knowledge. So let's see exactly what he 33 00:03:05,060 --> 00:03:11,440 said in here. First sentence model would be simplification. 34 00:03:11,440 --> 00:03:16,990 And therefore, falsification, that means the model is wrong. Because 35 00:03:16,990 --> 00:03:22,050 you're trying to understand some complex system. You 36 00:03:22,050 --> 00:03:27,180 simplify it, which means you throw. Things on it, 37 00:03:27,180 --> 00:03:32,380 and therefore, by definition, it's wrong. 38 00:03:32,380 --> 00:03:37,480 It is to be hoped smart means that modelling 39 00:03:37,480 --> 00:03:42,730 is not a science, it's an art. That the features 40 00:03:42,730 --> 00:03:49,300 are 10 for discussion are those of greatest importance in the present state of knowledge. 41 00:03:49,300 --> 00:03:54,330 So if you're working in some medical area 42 00:03:54,330 --> 00:03:59,580 or some logical area. The present state of knowledge, you probably 43 00:03:59,580 --> 00:04:04,750 don't know that she's probably going to have to work as a collaborator. 44 00:04:04,750 --> 00:04:10,750 The rule of modulated is to advance the present state of knowledge 45 00:04:10,750 --> 00:04:16,690 in the application domain to advance the state of knowledge, 46 00:04:16,690 --> 00:04:21,940 then you might have to re-evaluate your model and start 47 00:04:21,940 --> 00:04:27,930 again. Change the model, refine it and continue 48 00:04:27,930 --> 00:04:33,310 to advance the present state of knowledge. But I'm going to do this talk 49 00:04:33,310 --> 00:04:38,920 is focus really on the modelling side of things 50 00:04:38,920 --> 00:04:44,010 and the modelling side of things is where you do the substraction. And 51 00:04:44,010 --> 00:04:49,150 what I'm going to try to show is that by doing this abstraction, one can 52 00:04:49,150 --> 00:04:54,280 draw analogies between many different areas in science and transfer 53 00:04:54,280 --> 00:05:00,750 knowledge from one area of science to another area of science. 54 00:05:00,750 --> 00:05:06,840 So I'm going to begin by talking about ecology and medicine. 55 00:05:06,840 --> 00:05:12,630 So concer so roughly speaking, in our body 56 00:05:12,630 --> 00:05:18,090 every day, lots of sounds die. 57 00:05:18,090 --> 00:05:23,520 And Charles prefer it defined to compensate 58 00:05:23,520 --> 00:05:28,530 for that death. And you have a balance, 59 00:05:28,530 --> 00:05:34,720 the birth balances the death and you have something called homeostasis 60 00:05:34,720 --> 00:05:40,200 or steady state. It happens in cancer. 61 00:05:40,200 --> 00:05:46,440 Is it a group of cells through mutations? 62 00:05:46,440 --> 00:05:51,670 Their properties change so that this balance of birth and death 63 00:05:51,670 --> 00:05:57,820 is upset. And the net result of that is you get a growth. 64 00:05:57,820 --> 00:06:02,920 So the question counsellor is trying to recruit, troll this population of 65 00:06:02,920 --> 00:06:12,020 cells that are growing, that are actually invading our body. 66 00:06:12,020 --> 00:06:17,990 In ecology, there's a long history of analysing invasion. 67 00:06:17,990 --> 00:06:24,450 So if we think of cancer as a disease, invasion of our body. 68 00:06:24,450 --> 00:06:29,630 By sounds, can't we draw lessons? From 69 00:06:29,630 --> 00:06:34,710 ecology. So what are the invasions that the 70 00:06:34,710 --> 00:06:39,930 U.K. experienced and is continued to experience? It's a squirrel 71 00:06:39,930 --> 00:06:46,300 invasion. So this is the red squirrel. The indigenous squirrel. 72 00:06:46,300 --> 00:06:51,510 The host population. And last century, this character 73 00:06:51,510 --> 00:06:56,640 appeared, the grey squirrel imported from the US 74 00:06:56,640 --> 00:07:02,100 and the grey squirrel has invaded the red squirrel squirrels 75 00:07:02,100 --> 00:07:07,120 territory. So I apologise for the 76 00:07:07,120 --> 00:07:12,370 very bad quality of this very old drawing. So 77 00:07:12,370 --> 00:07:17,490 here we see the south east part of England. 78 00:07:17,490 --> 00:07:24,190 There is the coast there. And in 1964, these 79 00:07:24,190 --> 00:07:29,470 black. Spots are where people 80 00:07:29,470 --> 00:07:34,560 have seen red squirrels during that year. And then 81 00:07:34,560 --> 00:07:39,590 this is 64, 65, 66, 67, and 82 00:07:39,590 --> 00:07:44,730 then seventy five up to seventy nine. And you can see going from here to 83 00:07:44,730 --> 00:07:49,800 here, the red squirrels have begun to disappear. And that's because 84 00:07:49,800 --> 00:07:54,810 they've been replaced by the grey swans. So this 85 00:07:54,810 --> 00:08:00,650 is something that I worked on, actually, when I was a postdoc with Jim Marry a cute couple. 86 00:08:00,650 --> 00:08:06,180 And Mark Williams and we developed actually Lockable Care, a well known 87 00:08:06,180 --> 00:08:11,600 petition mogel to try and understand this invasion. So 88 00:08:11,600 --> 00:08:17,750 these equations here, G, is a population of grey squirrels. 89 00:08:17,750 --> 00:08:23,990 Red is a pop, art is a population of red swarms and years time. 90 00:08:23,990 --> 00:08:29,740 So what are these equations mean? Well, let's look at the red squirrel, because that's the home team. 91 00:08:29,740 --> 00:08:35,100 OK. So every author. Trumped every other 92 00:08:35,100 --> 00:08:40,140 thing. Our one K one or two K to a one the day two. These are all 93 00:08:40,140 --> 00:08:45,330 positive fixed values. OK, so suppose 94 00:08:45,330 --> 00:08:50,420 we just had this term. So our two times K. To give you something. 95 00:08:50,420 --> 00:08:55,890 No positive number multiplied by Ah. Let's consider that first of all 96 00:08:55,890 --> 00:09:02,010 we'd have Diyab ADT something positive. Times are. 97 00:09:02,010 --> 00:09:08,000 That means you have exponential growth. And in fact, exponential growth 98 00:09:08,000 --> 00:09:14,200 is a pretty good description of how a lot of population start to grow. 99 00:09:14,200 --> 00:09:19,320 But of course, populations do not grow to infinity because 100 00:09:19,320 --> 00:09:24,600 as they start to grow, there's competition for resource and you'd expect 101 00:09:24,600 --> 00:09:29,610 the growth rate to decrease. The simplest way to model that is 102 00:09:29,610 --> 00:09:34,740 to put this term here. So what this term is saying is that as 103 00:09:34,740 --> 00:09:40,560 the population increases, the growth rate instead of being 104 00:09:40,560 --> 00:09:45,810 are two times K2 becomes are two times 105 00:09:45,810 --> 00:09:51,870 K to minus R, it decreases because of the presence 106 00:09:51,870 --> 00:09:57,120 of threat. Now, here we have a competition, 107 00:09:57,120 --> 00:10:02,450 and so with the greys also decrease control 108 00:10:02,450 --> 00:10:08,220 threat. So you can think of this as competition for resources 109 00:10:08,220 --> 00:10:14,870 that decrease the growth rate. The same thing for the grace. 110 00:10:14,870 --> 00:10:20,100 And if we think of the greys as being battered competitors, then the Reds 111 00:10:20,100 --> 00:10:25,230 we'd have this is because this. OK, so then 112 00:10:25,230 --> 00:10:30,810 on top of this, we would have to fusion, which I haven't put in here, but we would have 113 00:10:30,810 --> 00:10:36,050 movement of the praise movement of the rich. 114 00:10:36,050 --> 00:10:42,720 So let's see how this system with movement in it behaves. 115 00:10:42,720 --> 00:10:48,100 So this is a video that was, in fact, generated by former graduate 116 00:10:48,100 --> 00:10:53,240 should mine a long time ago, Anthony Pollner. So what we've done 117 00:10:53,240 --> 00:10:58,310 here, we've scaled the populations to lie between zero 118 00:10:58,310 --> 00:11:03,510 and one. So here we have. The red squirrels, 119 00:11:03,510 --> 00:11:09,310 they're all a population. Grey squirrels. 120 00:11:09,310 --> 00:11:14,600 Population zero, except for some pockets here of population 121 00:11:14,600 --> 00:11:31,580 one. And then we're going to ask what happens if we run the small. 122 00:11:31,580 --> 00:11:37,650 You run the model and what you're seeing happening is. 123 00:11:37,650 --> 00:11:43,410 Waves of grey squirrels by competing threats 124 00:11:43,410 --> 00:11:48,420 to invade land, the grey squirrels. No one 125 00:11:48,420 --> 00:11:55,410 understands answer. 126 00:11:55,410 --> 00:12:00,700 So let's try to understand how it was that the GREs, a computer 127 00:12:00,700 --> 00:12:06,270 threat and the way to do this, you go back to these equations 128 00:12:06,270 --> 00:12:11,550 here and we look at something called the no fly 129 00:12:11,550 --> 00:12:16,830 and no clines or where these derivatives are zero. 130 00:12:16,830 --> 00:12:22,850 OK, so if we look at the R equation, the R by the cheese zero, 131 00:12:22,850 --> 00:12:28,910 if R zero. Which is this line here. 132 00:12:28,910 --> 00:12:35,940 Or if this is the. That's the straight line here. 133 00:12:35,940 --> 00:12:42,070 Likewise, Jean. We have either G a zero 134 00:12:42,070 --> 00:12:47,890 or this. OK. So if we look at these two lines, these are two straight lines. 135 00:12:47,890 --> 00:12:54,420 And they can. Look differently, depending on the values of Kay. 136 00:12:54,420 --> 00:12:59,820 One, a one, K two and a two. Take a look like this, or they could look like this 137 00:12:59,820 --> 00:13:05,070 or they could have other combinations. Okay, so let us look 138 00:13:05,070 --> 00:13:10,140 at this one to be less disposable sitting here. So we've got 139 00:13:10,140 --> 00:13:15,240 some grais small number of reds. The Greys have already come into the 140 00:13:15,240 --> 00:13:20,630 domain fixed on the domain and see what's going to happen now. Ah, 141 00:13:20,630 --> 00:13:26,250 not here is shorthand for Driftin. So at this point, 142 00:13:26,250 --> 00:13:33,110 we are below this line where they are guided to zero. 143 00:13:33,110 --> 00:13:39,050 So these are small, which means this is positive. 144 00:13:39,050 --> 00:13:45,660 So they are digestives positive. So are will grow. 145 00:13:45,660 --> 00:13:51,090 Also d.g by the team's roster, because we've be below 146 00:13:51,090 --> 00:13:56,510 this line. So Julie will drill. So the net result will be 147 00:13:56,510 --> 00:14:02,030 if G is growing and R is growing. We'll move like this 148 00:14:02,030 --> 00:14:07,240 now. Crucially, they crossed the red line below 149 00:14:07,240 --> 00:14:12,540 with a great line by the black line here. We cross the red 150 00:14:12,540 --> 00:14:18,300 line first when we crossed the red line. Now 151 00:14:18,300 --> 00:14:23,820 the sign off to your back. He switches NTR banditti becomes 152 00:14:23,820 --> 00:14:30,330 negative. Which means power starts to decrease. 153 00:14:30,330 --> 00:14:36,210 But we're still below the black line, so the graves are still actressy. 154 00:14:36,210 --> 00:14:41,370 So we go off like this. And we reach this point here 155 00:14:41,370 --> 00:14:46,380 where the red squirrels are zero scray squirrel. This is supposed to 156 00:14:46,380 --> 00:14:52,830 be very happy. Pretty squirrel. And we have had competitive education. 157 00:14:52,830 --> 00:14:57,890 So suppose we now think, well, let's say if the Reds and there'd 158 00:14:57,890 --> 00:15:03,250 been lots of campaigns. Throughout the century, last century 159 00:15:03,250 --> 00:15:08,680 and this century to save the Red Square. Part of those campaigns are 160 00:15:08,680 --> 00:15:14,320 let us comb the grey square. So we call the grey squirrels, 161 00:15:14,320 --> 00:15:19,830 which means we decrease the population of the grey swans. 162 00:15:19,830 --> 00:15:25,100 We've killed a lot of grey squirrels. What's going to happen? Well below 163 00:15:25,100 --> 00:15:30,730 both lines. No claims, so the populations whisstock to pro. 164 00:15:30,730 --> 00:15:35,820 And we crossed the red knoll first. And so 165 00:15:35,820 --> 00:15:40,840 we go up and we finish up. It's own race, 166 00:15:40,840 --> 00:15:45,960 so it hasn't worked. On the other hand, suppose 167 00:15:45,960 --> 00:15:51,450 these no zones look like this. Here we are 168 00:15:51,450 --> 00:15:57,820 again. Start off here with below both lines, 169 00:15:57,820 --> 00:16:03,400 both g you know, our increase. We cross the red line first. 170 00:16:03,400 --> 00:16:08,430 Which means are starts to decrease. And we go to here 171 00:16:08,430 --> 00:16:13,480 happy grey squirrels, red squirrels are all dead. Now, let's do the 172 00:16:13,480 --> 00:16:19,060 same experiment we did on the right hand side. Let's call 173 00:16:19,060 --> 00:16:24,310 some Grayskull. We've talked to here once again, we're 174 00:16:24,310 --> 00:16:29,380 below. Both know Clynes. So both genau will 175 00:16:29,380 --> 00:16:34,540 start to increase. Now there's a difference. We crossed 176 00:16:34,540 --> 00:16:41,560 the black line first. Which means that the grey start to decrease. 177 00:16:41,560 --> 00:16:47,620 Then we move on to here and we finish shop. Excluding the greys 178 00:16:47,620 --> 00:16:53,860 and happy Red Square. So two points to note from here. 179 00:16:53,860 --> 00:17:00,730 Same experiment. Gives different results here. 180 00:17:00,730 --> 00:17:06,650 Drays Wollman. Here, Reg., for. 181 00:17:06,650 --> 00:17:11,750 Second thing to notice is that if we are in this situation 182 00:17:11,750 --> 00:17:17,210 here, Colleen will never work. We have to do something 183 00:17:17,210 --> 00:17:22,460 else. We have to change the slope. Sloops of these 184 00:17:22,460 --> 00:17:28,040 lines so that they intersect like this. Now, the sloops 185 00:17:28,040 --> 00:17:33,110 of these mines and their intersections are determined 186 00:17:33,110 --> 00:17:38,220 by. These values here. So what we would have to 187 00:17:38,220 --> 00:17:43,890 do is we have to somehow make the red squirrels 188 00:17:43,890 --> 00:17:49,780 better competitors in the grey squirrels. Or make 189 00:17:49,780 --> 00:17:54,880 the environment such that the red squirrels are better suited 190 00:17:54,880 --> 00:17:59,950 to the environment than the grey squirrels. So that's 191 00:17:59,950 --> 00:18:06,550 the insight we get out of this. 192 00:18:06,550 --> 00:18:11,650 These insights have been used by. 193 00:18:11,650 --> 00:18:17,230 My friend and collaborator Bob Gayton B, who 194 00:18:17,230 --> 00:18:23,310 began life as a physicist and now as a cancer biologist and two radiologist 195 00:18:23,310 --> 00:18:31,590 who's at the Moffett Cancer Centre in Tampa. 196 00:18:31,590 --> 00:18:37,500 It's well known that cancer cells sometimes 197 00:18:37,500 --> 00:18:43,350 use a metabolism that is less efficient 198 00:18:43,350 --> 00:18:48,440 than normal cells. It's. This metabolism 199 00:18:48,440 --> 00:18:55,460 is called. Anaerobic metabolism, and this is called the Warburg. 200 00:18:55,460 --> 00:19:01,310 So basically, if we have lots of oxygen for ourselves, lots of oxygen, 201 00:19:01,310 --> 00:19:10,160 we we use a real big metabolism and that produces lots of energy. 202 00:19:10,160 --> 00:19:15,250 Oxygen decreases the cells, then use. 203 00:19:15,250 --> 00:19:21,650 Anaerobic metabolism, and that involves costs. 204 00:19:21,650 --> 00:19:28,170 And that is. Lower 205 00:19:28,170 --> 00:19:33,240 production of energy gives lower production of energy. And what Warburton 206 00:19:33,240 --> 00:19:39,460 noted was that in some. Situations 207 00:19:39,460 --> 00:19:44,860 where there is enough oxygen. Tumour cells use 208 00:19:44,860 --> 00:19:51,220 this anaerobic tablets. That doesn't seem to make sense. 209 00:19:51,220 --> 00:19:56,440 Why would you use a lower. Energy 210 00:19:56,440 --> 00:20:02,030 generating system in the competition. 211 00:20:02,030 --> 00:20:07,490 So Hope's theory was the following are his hypothesis was the following. 212 00:20:07,490 --> 00:20:14,110 He said that, OK, you've got these invaders. 213 00:20:14,110 --> 00:20:19,360 By using the clitic pathway, one of the by-products of the black Lytic 214 00:20:19,360 --> 00:20:24,780 pathway in the Fotopoulos and lactic acid. And that makes 215 00:20:24,780 --> 00:20:31,390 the environment sediq. And the civic environment. 216 00:20:31,390 --> 00:20:37,020 Favours. Chuba Cell Division found it's more toxic 217 00:20:37,020 --> 00:20:43,170 to normal cells than to the tumour cells. So what 218 00:20:43,170 --> 00:20:49,290 the cancer cell is doing is it's changing the parameter values 219 00:20:49,290 --> 00:20:54,610 to make the environment more suitable for it. 220 00:20:54,610 --> 00:20:59,730 And for the normal cells and to help turn fit. So he 221 00:20:59,730 --> 00:21:04,800 wrote down a system of equations. Bit different to the ones we just 222 00:21:04,800 --> 00:21:10,200 looked at. But the essence is the same. A competition 223 00:21:10,200 --> 00:21:15,420 model. And he solved this competition model. And so what we see 224 00:21:15,420 --> 00:21:21,150 here is this is in space. 225 00:21:21,150 --> 00:21:26,770 This is his model where this. This black line 226 00:21:26,770 --> 00:21:31,990 are the tumour cells that are invading. So they're forming travelling wave. It's moving 227 00:21:31,990 --> 00:21:38,340 along like this. This is the acid that they're producing 228 00:21:38,340 --> 00:21:43,600 and these are the normal cells. So the normal cells 229 00:21:43,600 --> 00:21:49,220 initially were at one everywhere. And they're still at one here 230 00:21:49,220 --> 00:21:54,350 because there's no not no tumour cells, but the tumour cells have started 231 00:21:54,350 --> 00:21:59,360 to invade. They're producing the acid. The acid 232 00:21:59,360 --> 00:22:04,640 is changing the environment, making it more suitable for the tumour 233 00:22:04,640 --> 00:22:09,890 cells and for the normal cells. And so this will grow. You see what's 234 00:22:09,890 --> 00:22:14,910 happened here? The normal cells. It worked. One contact point 235 00:22:14,910 --> 00:22:19,930 for. In other PROMETA regimes, it was 236 00:22:19,930 --> 00:22:25,240 even more dramatic. What you find here, you say started off 237 00:22:25,240 --> 00:22:30,580 again with normal cells everywhere. Then he introduced 238 00:22:30,580 --> 00:22:36,310 some tumour cells. And now these tumour cells are invading. 239 00:22:36,310 --> 00:22:42,840 And they have pushed the normal cells to extinction. 240 00:22:42,840 --> 00:22:47,990 So this is competitive exclusion. Now, what you find, which 241 00:22:47,990 --> 00:22:53,440 was a novel thing, was this strange business here. Here, 242 00:22:53,440 --> 00:22:58,630 there are no. So the prediction of this 243 00:22:58,630 --> 00:23:03,710 model was. If 244 00:23:03,710 --> 00:23:09,670 tumours are using acid to mediate the invasion, 245 00:23:09,670 --> 00:23:14,920 there should be a gap with no cells between the invading front 246 00:23:14,920 --> 00:23:19,930 of tumour cells and the regress in front of 247 00:23:19,930 --> 00:23:24,940 the normal cells. So then Bob went 248 00:23:24,940 --> 00:23:31,350 and looked at some human head and neck cancers 249 00:23:31,350 --> 00:23:41,510 and had several of these head and neck cancers. He found his. 250 00:23:41,510 --> 00:23:46,830 So. This leads to various 251 00:23:46,830 --> 00:23:51,950 obvious next experiment. If Schumer 252 00:23:51,950 --> 00:23:58,650 is changing the environment, making it sitting so that it can invade. 253 00:23:58,650 --> 00:24:03,660 Then what would happen if we neutralise the. This 254 00:24:03,660 --> 00:24:08,730 is an experiment by a real B and various people. 255 00:24:08,730 --> 00:24:14,310 This is done on mice. Two mice 256 00:24:14,310 --> 00:24:19,890 were injected with tumours. The control makes. 257 00:24:19,890 --> 00:24:24,950 We're just left to do their own thing. The treated 258 00:24:24,950 --> 00:24:30,290 mice were given water that had 259 00:24:30,290 --> 00:24:35,320 carbonate. And then what they looked 260 00:24:35,320 --> 00:24:40,810 at was how many metastases 261 00:24:40,810 --> 00:24:46,150 reformed, in other words, from this initial tumour? How many groups of tumours 262 00:24:46,150 --> 00:24:51,610 broke off and invaded? And what we see here 263 00:24:51,610 --> 00:24:58,080 in the old treated here is the average. 264 00:24:58,080 --> 00:25:03,800 I didn't. Treated. Here's the average. So the treatment 265 00:25:03,800 --> 00:25:09,360 didn't stop. Totally, totally stop the invasion. 266 00:25:09,360 --> 00:25:15,160 Could have significantly reduced the invasion. 267 00:25:15,160 --> 00:25:20,830 So the mathematical model. Which tested and 268 00:25:20,830 --> 00:25:26,920 validated the hypotheses led to these experiments. 269 00:25:26,920 --> 00:25:32,200 That, again, would be more. Now, 270 00:25:32,200 --> 00:25:37,330 this is a nice Melhuish is a model. We're 271 00:25:37,330 --> 00:25:42,370 humans, we're not mice. And then the question is, just because 272 00:25:42,370 --> 00:25:47,560 it works in the mice cannot work in the human. We've got 273 00:25:47,560 --> 00:25:52,750 a completely different metabolism, completely different immune system. Lots 274 00:25:52,750 --> 00:25:58,180 of things are different between us and the mice. And so this but this 275 00:25:58,180 --> 00:26:03,840 work is actually instigated. A whole series of experiments 276 00:26:03,840 --> 00:26:08,860 don't work that. No, I'm modelling that this group at the Moffitt 277 00:26:08,860 --> 00:26:14,830 Cancer Centre are doing and they continue to work on the rule of metabolism 278 00:26:14,830 --> 00:26:19,960 in cancer progression and can be altered metabolism 279 00:26:19,960 --> 00:26:25,100 in order to stop cancer invasion. They work 280 00:26:25,100 --> 00:26:30,550 the cancer at the Moffitt Cancer Centre and in fact, there are 281 00:26:30,550 --> 00:26:35,860 a group of seven or eight full time mathematicians 282 00:26:35,860 --> 00:26:40,990 working with clinicians and the experimentalists headed 283 00:26:40,990 --> 00:26:46,620 by Sandy Anderson. So 284 00:26:46,620 --> 00:26:52,990 I'm not going to move on to something to do with. 285 00:26:52,990 --> 00:26:58,300 Slime mould, which is developmental biology and fact tax, 286 00:26:58,300 --> 00:27:03,330 which is physiology. So 287 00:27:03,330 --> 00:27:08,440 but before we do, that is just the chemistry. 288 00:27:08,440 --> 00:27:13,760 In the 1950s, the loose of. 289 00:27:13,760 --> 00:27:18,780 And then later dropped, Pensky worked on this 290 00:27:18,780 --> 00:27:25,110 chemical problem. And this was a chemical reaction 291 00:27:25,110 --> 00:27:30,210 where the chemical here. This is time. 292 00:27:30,210 --> 00:27:36,090 And it is. 293 00:27:36,090 --> 00:27:44,510 And then it goes black. It was back to the head. 294 00:27:44,510 --> 00:27:49,600 And this child happily. So what's going on 295 00:27:49,600 --> 00:27:54,600 here? These this chemical that they're looking at, this chemical 296 00:27:54,600 --> 00:27:59,760 reaction is a redox reaction where the chemical, when it 297 00:27:59,760 --> 00:28:05,310 is oxidised, it's in certain colour. 298 00:28:05,310 --> 00:28:10,410 And when it is reduced to deoxygenated, it's a different colour. 299 00:28:10,410 --> 00:28:16,130 So what we're seeing here is oxidation reduction, oxidation reduction, 300 00:28:16,130 --> 00:28:21,130 etc cetera. So why is this happening? And in fact, if you look 301 00:28:21,130 --> 00:28:26,450 at this in. A dish. 302 00:28:26,450 --> 00:28:32,180 You get all you get waves like you did for the squirrels. But 303 00:28:32,180 --> 00:28:37,340 these are much more interesting. You get these what are called target patterns 304 00:28:37,340 --> 00:28:42,650 because they look like targets. You can get here's target pattern. And here 305 00:28:42,650 --> 00:28:48,510 are. Cheryl Pratchett's. So what's happening here? 306 00:28:48,510 --> 00:28:53,550 These patterns propagate and they break up and they form spines. 307 00:28:53,550 --> 00:29:00,050 So go to the Internet and you can see videos of these. 308 00:29:00,050 --> 00:29:05,540 OK, so what can we understand? So lots of people have developed models for these. 309 00:29:05,540 --> 00:29:10,550 And this is the sort of the field noise model. 310 00:29:10,550 --> 00:29:15,950 And a very, very simplified version of it. So we I think 311 00:29:15,950 --> 00:29:21,400 in this setup, X and Y to the chemical players 312 00:29:21,400 --> 00:29:26,950 and we told the no fly. No Clines. 313 00:29:26,950 --> 00:29:32,180 For X. And here's one adult by. 314 00:29:32,180 --> 00:29:37,450 It's a Kubic. And the no fine 315 00:29:37,450 --> 00:29:42,870 for why. I've just drawn it by simply 316 00:29:42,870 --> 00:29:48,230 here as a straight line. The important thing is that the no plan 317 00:29:48,230 --> 00:29:53,570 only it intersects along this branch. So there's no climate 318 00:29:53,570 --> 00:29:59,010 could be like this. Doesn't really matter. The key thing is it intersects 319 00:29:59,010 --> 00:30:04,120 along here. OK. And it turns out in 320 00:30:04,120 --> 00:30:09,820 this system, when you are above the black, 321 00:30:09,820 --> 00:30:15,220 no climb. Not only is DXP by positive, 322 00:30:15,220 --> 00:30:20,590 it's very possible the number is low. Crime is where the derivative is here. 323 00:30:20,590 --> 00:30:25,920 So one part of the no crime, the derivative would be positive on the other part 324 00:30:25,920 --> 00:30:31,060 would be negative diff. One of the differences between this and the previous 325 00:30:31,060 --> 00:30:37,710 model is that this x2 of a diff is very big. 326 00:30:37,710 --> 00:30:43,010 On this side, positive, very big on this side, Nick. 327 00:30:43,010 --> 00:30:48,010 So suppose we start at this point. P. So at this point, p p 328 00:30:48,010 --> 00:30:53,120 y by two zero. The X by DTL zero. This is a steady 329 00:30:53,120 --> 00:30:58,170 state. So the system would stay there. Nothing's changing. But in 330 00:30:58,170 --> 00:31:03,390 biological system, there will be small fluctuations. 331 00:31:03,390 --> 00:31:08,510 So suppose there's a small fluctuation that pushes us to clear. So 332 00:31:08,510 --> 00:31:13,550 if we look at to why by detainees, negative, so why will 333 00:31:13,550 --> 00:31:18,740 decrease? To expire, duty is positive, 334 00:31:18,740 --> 00:31:25,020 so X will increase. But it's very big. So it'll increase rapidly. 335 00:31:25,020 --> 00:31:30,280 So the net result will be we should to hear. And we crossed 336 00:31:30,280 --> 00:31:35,980 the no fly zone and nobody explained t is negative. 337 00:31:35,980 --> 00:31:41,090 So we start to go down. And the thing is that we stick 338 00:31:41,090 --> 00:31:46,420 to the no climb, because if we try to fall off. No. To explain to T. 339 00:31:46,420 --> 00:31:51,750 is very large negative. It'll push us back. So we go time 340 00:31:51,750 --> 00:31:57,150 and then we reach the point B. So ideally, we would like to 341 00:31:57,150 --> 00:32:02,960 stick with no climb. We'd like to move up. 342 00:32:02,960 --> 00:32:08,080 But that would mean why increasing? But look, why 343 00:32:08,080 --> 00:32:13,790 does it increase? Why decreases so we fall off the loped fine 344 00:32:13,790 --> 00:32:19,260 and we should approach us to this point. And not at this point, 345 00:32:19,260 --> 00:32:24,530 do what I buy duty is my positive. So we increase. 346 00:32:24,530 --> 00:32:30,460 Go up to here. Reach this point and then at this point. 347 00:32:30,460 --> 00:32:35,560 We fall off the Noel Klein again, shoot across to a cheap on an 348 00:32:35,560 --> 00:32:40,630 ongoing. What does the X 349 00:32:40,630 --> 00:32:45,640 look like as a function of T? Let's look down here. Say, 350 00:32:45,640 --> 00:32:50,800 Andy X increases very quickly to a it 351 00:32:50,800 --> 00:32:56,130 off to a. Slowly decreases to be 352 00:32:56,130 --> 00:33:01,530 slowly goes dark, to be quickly decreases time to see and slowly 353 00:33:01,530 --> 00:33:09,120 moves up to tea and keeps on going. And that's the oscillation. 354 00:33:09,120 --> 00:33:14,250 So there'll be other chemicals that will be oscillating either fingers 355 00:33:14,250 --> 00:33:19,450 with this. So with this one is high. And you've got, 356 00:33:19,450 --> 00:33:25,480 say, an oxidised version and the reduced version is low to one colour 357 00:33:25,480 --> 00:33:31,140 and then they swap. Then the. Reduced one is high. 358 00:33:31,140 --> 00:33:36,230 The oxidised one is low. It's a different colour. And this keeps on happening. And this is 359 00:33:36,230 --> 00:33:42,560 called the vexation relaxation. Also. 360 00:33:42,560 --> 00:33:47,670 Now, what would happen instead of this no fine intersecting 361 00:33:47,670 --> 00:33:52,790 here. Suppose it intersected here. 362 00:33:52,790 --> 00:33:58,050 On this branch near this maximum. 363 00:33:58,050 --> 00:34:03,120 Now, I suppose we do the perturbation again from P 364 00:34:03,120 --> 00:34:08,190 and we move to Q so a Q does X by detail is 365 00:34:08,190 --> 00:34:13,470 negative because we're below the X, no client. So X 366 00:34:13,470 --> 00:34:19,670 will deepfreeze. Why is negative? Why will decrease 367 00:34:19,670 --> 00:34:24,700 and will should die? But suppose instead we 368 00:34:24,700 --> 00:34:30,020 moved to here in a perturbation. Now to explain, to tease 369 00:34:30,020 --> 00:34:35,270 positive. And we should in the opposite direction. 370 00:34:35,270 --> 00:34:40,340 Right across to here come time to hear. Right 371 00:34:40,340 --> 00:34:45,400 across to hear again. Up to here. And then we stop at this 372 00:34:45,400 --> 00:34:50,410 steady state. So crucially, here we have what's called the 373 00:34:50,410 --> 00:34:55,580 threshold for Nordgren. If we make a small perturbation. 374 00:34:55,580 --> 00:35:01,010 We go back to where we started. But if we breach the threshold, 375 00:35:01,010 --> 00:35:06,450 we then have a big excursion before we go back. 376 00:35:06,450 --> 00:35:12,090 And that is this type of behaviour here. 377 00:35:12,090 --> 00:35:17,270 That's called an excitable system. Now, one 378 00:35:17,270 --> 00:35:22,400 of the amazing examples where we see an excitable system is something called the slime mould. 379 00:35:22,400 --> 00:35:27,620 And again, I would urge you to go look at movies of this on the Internet to 380 00:35:27,620 --> 00:35:32,900 slime mould our single celled creatures who 381 00:35:32,900 --> 00:35:38,360 live around you enjoy life if he'd be producing, but then times 382 00:35:38,360 --> 00:35:44,850 become bad and they run out of food. So what did he do? 383 00:35:44,850 --> 00:35:51,770 Well, some of them start to secrete a chemical called Cycler GameSpy. 384 00:35:51,770 --> 00:35:57,050 This chemical diffuses and the 385 00:35:57,050 --> 00:36:02,150 chemical and cell combination is 386 00:36:02,150 --> 00:36:09,390 an excitable system, so that when this chemical. 387 00:36:09,390 --> 00:36:14,980 Close to the next cell. It perturbs at. 388 00:36:14,980 --> 00:36:20,410 And so it then releases a chemical. 389 00:36:20,410 --> 00:36:25,530 And this isn't in time, but in space, as we saw here, 390 00:36:25,530 --> 00:36:30,940 moving back here, we get travelling waves. So this would be a travelling 391 00:36:30,940 --> 00:36:36,570 cross. So in this way, the signal 392 00:36:36,570 --> 00:36:41,780 gets relayed. And these. Cells 393 00:36:41,780 --> 00:36:47,930 move up the chemical gradient and form the streams, 394 00:36:47,930 --> 00:36:53,130 which eventually aggregate and form. 395 00:36:53,130 --> 00:36:58,390 A slok, a mind and then a stroke. So basically this idea, 396 00:36:58,390 --> 00:37:03,490 you sound the chemical and the reason to send up the chemical is to signal to 397 00:37:03,490 --> 00:37:09,770 all your friends. We've got to get together and form a asla. 398 00:37:09,770 --> 00:37:14,780 The importance of this slug formation is that when the cells come 399 00:37:14,780 --> 00:37:19,940 close to each other, they can exchange certain key 400 00:37:19,940 --> 00:37:27,560 proteins that allow some of them to be common. 401 00:37:27,560 --> 00:37:32,680 Stock sales and so on become spoors. 402 00:37:32,680 --> 00:37:37,920 So what you get here is a fruiting body where. 403 00:37:37,920 --> 00:37:43,080 These things grew up and then you have spores at the top and stop 404 00:37:43,080 --> 00:37:48,120 at the bottom and the spore cells are the ones that survive. So 405 00:37:48,120 --> 00:37:53,960 an animal comes. Pushes them away or the wind comes 406 00:37:53,960 --> 00:38:00,030 blows the stock cells of the sport out away and then hopefully 407 00:38:00,030 --> 00:38:07,070 blows them away to a better environment where this has to. 408 00:38:07,070 --> 00:38:13,110 Now, in the 1980s and 90s, this was studied a lot 409 00:38:13,110 --> 00:38:18,430 because. It had processes that are 410 00:38:18,430 --> 00:38:23,560 vital in higher organisms, signal transduction, which 411 00:38:23,560 --> 00:38:29,020 is how you really signal. Table taxes. The movement 412 00:38:29,020 --> 00:38:34,030 up, a chemical grading, which is very important in higher blood vessels form 413 00:38:34,030 --> 00:38:40,040 in higher immune system, responds to to disease and differentiation 414 00:38:40,040 --> 00:38:45,160 of how these cells change from one cell type to another. These are all 415 00:38:45,160 --> 00:38:50,520 key processes in higher organisms. But the 416 00:38:50,520 --> 00:38:55,520 slime mould is a really good study. A really good model 417 00:38:55,520 --> 00:39:01,470 system to study. 418 00:39:01,470 --> 00:39:06,750 And in fact, if you look at this in two days, here are cyclic 419 00:39:06,750 --> 00:39:11,880 empey waves and you can see they form spirals and target 420 00:39:11,880 --> 00:39:17,210 patterns. They much like the. 421 00:39:17,210 --> 00:39:23,740 The list of top 10 scary action you can see here we've got. 422 00:39:23,740 --> 00:39:30,750 A biological system. There we had a chemical system. 423 00:39:30,750 --> 00:39:35,790 They both exhibit similar patterns from a mathematical point 424 00:39:35,790 --> 00:39:40,960 of view. We see that there are several provinces between. 425 00:39:40,960 --> 00:39:46,460 And then here are the cells responding and forming streams. 426 00:39:46,460 --> 00:39:51,660 So we see the equivalence there between a. 427 00:39:51,660 --> 00:39:56,950 Chemical system and a biological system. So excitable systems 428 00:39:56,950 --> 00:40:02,040 are very important. Here's another 429 00:40:02,040 --> 00:40:07,520 example of an excitable system. Flushing toilet. 430 00:40:07,520 --> 00:40:12,560 The one with a handle, not the one that you wave. OK. 431 00:40:12,560 --> 00:40:17,710 Think about a flushing toilet. If you go along and you push the handle 432 00:40:17,710 --> 00:40:23,310 just a little bit, what happens? Little bit of water dribbles out. 433 00:40:23,310 --> 00:40:28,330 You perturb a little bit from the steady state and you go by. Pull 434 00:40:28,330 --> 00:40:35,200 the handle a little bit more when you breach the threshold, then wish. 435 00:40:35,200 --> 00:40:41,300 A whole pile of water comes down and then you relax back to the stand. 436 00:40:41,300 --> 00:40:46,680 So that is an excitable system. 437 00:40:46,680 --> 00:40:52,850 So moving on quickly. In 1952, 438 00:40:52,850 --> 00:40:58,400 Hodgkin and Huxley wrote a series of papers for which they won the Nobel Prise. 439 00:40:58,400 --> 00:41:03,470 Some years later, they were looking at the giant 440 00:41:03,470 --> 00:41:09,080 squid giant axon. So I nearly said it wrong. Their giant squid oxen. That's quite scary. 441 00:41:09,080 --> 00:41:14,180 This is the squid giant Daxam. So what does that mean? Well, basically 442 00:41:14,180 --> 00:41:19,580 act so he's looking at the nerve cells. And 443 00:41:19,580 --> 00:41:25,440 the nerve cells. Communicate with each other via long protrusions 444 00:41:25,440 --> 00:41:32,360 that are called accents, and the communication is the electrical communication. 445 00:41:32,360 --> 00:41:38,770 And basically what happens is you have an electrical pulse 446 00:41:38,770 --> 00:41:45,140 that bleeps in your cell. And then that propagates 447 00:41:45,140 --> 00:41:50,750 it kickstarts. The next show. 448 00:41:50,750 --> 00:41:56,050 The policies and pick stocks, the next step it touches. 449 00:41:56,050 --> 00:42:01,110 And in that way, you propagate the signal. And 450 00:42:01,110 --> 00:42:06,150 this work, they're there on paper for this mathematical 451 00:42:06,150 --> 00:42:11,250 model, for this is a very complicated model of four equations. And it was 452 00:42:11,250 --> 00:42:16,280 later reduced to the equations by fortune. You know, this showed 453 00:42:16,280 --> 00:42:21,540 that this behaviour was identical to the behaviour we've just discussed in terms 454 00:42:21,540 --> 00:42:29,600 of relaxation, oscillator and the acceptable. 455 00:42:29,600 --> 00:42:35,480 And in fact, there's been a lot of work on marketing of the heart, 456 00:42:35,480 --> 00:42:40,490 and in particular people like Dennis Gilbert and Peter Hunter have done a huge amount 457 00:42:40,490 --> 00:42:46,130 of work on developing models of the heart because these excite so into the heart. 458 00:42:46,130 --> 00:42:51,210 You basically have an electrical pulse that sweeps time the heart. 459 00:42:51,210 --> 00:42:57,240 That triggers a mechanical response that causes the heart to beat. 460 00:42:57,240 --> 00:43:03,870 Now, as we've seen these excitable systems propagate 461 00:43:03,870 --> 00:43:10,110 this. Signal. But if there is 462 00:43:10,110 --> 00:43:15,300 a bit of the heart that maybe has died, maybe there wasn't 463 00:43:15,300 --> 00:43:20,740 enough. There was a blood clot was not enough 464 00:43:20,740 --> 00:43:25,990 uterine getting to that vet. Then also six hours this wave comes to that 465 00:43:25,990 --> 00:43:31,330 point. That point is going to have different properties to the rest. 466 00:43:31,330 --> 00:43:36,730 Therefore, the flow of this wave through thought part of the heart 467 00:43:36,730 --> 00:43:42,450 is going to be a different speed and the wave of Break-Up. 468 00:43:42,450 --> 00:43:48,850 And as we saw with the slime mould and with the 469 00:43:48,850 --> 00:43:54,000 Robert Pinsky reaction, once these things break up, they form spa, 470 00:43:54,000 --> 00:43:59,550 which. And that means your heart would then try to beat 471 00:43:59,550 --> 00:44:04,990 not as a coherent beat, mechanical signal, coherently 472 00:44:04,990 --> 00:44:10,740 beating the heart, but the signal would be a spiral signal 473 00:44:10,740 --> 00:44:16,270 causing the heart crew to act. And so that can't be properly. That's called 474 00:44:16,270 --> 00:44:21,720 fibrillation. And in fact, these researchers 475 00:44:21,720 --> 00:44:27,890 and many other researchers have actually developed very, very sophisticated models 476 00:44:27,890 --> 00:44:33,330 of four heart models where they take into account a lot of them 477 00:44:33,330 --> 00:44:38,570 the detail, anatomical detail of the heart 478 00:44:38,570 --> 00:44:43,680 and the lettre physiology and mechanical aspects of the heart 479 00:44:43,680 --> 00:44:49,590 to see how you get these beats and how it is then that 480 00:44:49,590 --> 00:44:54,840 you can get fibrillation, etc., and then you can use 481 00:44:54,840 --> 00:44:59,870 these if you're going to give a drug. Some drop 482 00:44:59,870 --> 00:45:05,340 and you want to make sure that that drug doesn't lead to heart problems. 483 00:45:05,340 --> 00:45:10,920 Then you can say, well, if that drug affects the ion channels 484 00:45:10,920 --> 00:45:15,930 and therefore the electrical system and excitable system, can 485 00:45:15,930 --> 00:45:22,160 we make sure that it doesn't lead to these instabilities 486 00:45:22,160 --> 00:45:27,270 that lead to different relation to fibrillation? So what 487 00:45:27,270 --> 00:45:32,890 we're seeing here, then, is Phenomenon Romila, a phenomenon 488 00:45:32,890 --> 00:45:38,270 that occurs in. Chemistry, 489 00:45:38,270 --> 00:45:44,460 developmental biology, physiology and plumbing. 490 00:45:44,460 --> 00:45:50,780 The last thing I'm going to look at is animal poop markets. 491 00:45:50,780 --> 00:45:57,160 And football shirts. OK, so here's an animal. 492 00:45:57,160 --> 00:46:02,200 It is a cheater and it's sleeping. OK? So I didn't do 493 00:46:02,200 --> 00:46:07,240 anything back to it is just having a nap. I'm just talking and this is 494 00:46:07,240 --> 00:46:13,780 what I mean by pattern. She only spots. It was a package. 495 00:46:13,780 --> 00:46:18,850 Stripey tale. Here's another powter digits, 496 00:46:18,850 --> 00:46:25,370 five digits pattern of Gore's. 497 00:46:25,370 --> 00:46:30,440 Question is, how do these patterns arise? So I mentioned the beginning to talk 498 00:46:30,440 --> 00:46:36,800 Alan Turing. He starts to look at this problem in the early 1950s 499 00:46:36,800 --> 00:46:41,900 before he tried to. So 500 00:46:41,900 --> 00:46:47,070 he had this notion here, think of this tree. And the tree is growing. 501 00:46:47,070 --> 00:46:52,120 Suppose you were to look at this tree at this point here. What do you would say 502 00:46:52,120 --> 00:46:58,520 if you look down on it? It be a circle? And the tree is growing. 503 00:46:58,520 --> 00:47:03,950 And then at some point here. That's circular symmetry is broken 504 00:47:03,950 --> 00:47:08,990 by this branch cheering had the following idea. There must be some 505 00:47:08,990 --> 00:47:14,450 growth hormone that's causing this to grow. 506 00:47:14,450 --> 00:47:19,970 And then it's. And so if the growth is symmetric, that growth hormone 507 00:47:19,970 --> 00:47:25,200 must be symmetrically. 508 00:47:25,200 --> 00:47:30,270 Then at a certain point, that cemetery must break, must become 509 00:47:30,270 --> 00:47:38,100 destabilised. And then you have a. 510 00:47:38,100 --> 00:47:44,120 Asymmetric. 511 00:47:44,120 --> 00:47:49,130 Growth through asymmetric concentration profile. Of 512 00:47:49,130 --> 00:47:55,660 the growth hormone. And that leads to an asymmetry in the drugs. 513 00:47:55,660 --> 00:48:00,850 So you can think about taking this even further and you could say that 514 00:48:00,850 --> 00:48:06,340 instead of that asymmetry in the chemical resulting in a symmetry 515 00:48:06,340 --> 00:48:12,220 and growth, it could result in an asymmetry in Fed determination. 516 00:48:12,220 --> 00:48:17,350 So, for example, where the chemical concentration is high, that could result 517 00:48:17,350 --> 00:48:22,420 in cells adopting a certain type of fat where the chemical concentration is 518 00:48:22,420 --> 00:48:27,550 low. The cells do something different. So to form 519 00:48:27,550 --> 00:48:33,600 bone with the chemical concentrations high do form ball. It's chemical castration. 520 00:48:33,600 --> 00:48:41,960 And he called these chemicals morphogenesis. 521 00:48:41,960 --> 00:48:47,450 That's what he said. It is suggested that system of chemicals called morphogenesis reacting 522 00:48:47,450 --> 00:48:52,570 together and diffusing. Through tissue is adequate 523 00:48:52,570 --> 00:48:59,590 to account for the main phenomena with morphogenesis, meaning after information. 524 00:48:59,590 --> 00:49:05,110 And in fact, many, many years later, people actually find more fidgets. 525 00:49:05,110 --> 00:49:12,320 These were predicted by Alan Turing. Thirty years before it was. 526 00:49:12,320 --> 00:49:17,420 So the question is, how do you get this? 527 00:49:17,420 --> 00:49:22,810 Symmetry breaking or instability tertrais. And what Turing 528 00:49:22,810 --> 00:49:30,240 said was. Diffusion costs the instable. 529 00:49:30,240 --> 00:49:35,590 Now, this seems really counterintuitive because think schools, you start 530 00:49:35,590 --> 00:49:40,690 off here, here's some water and you put a globe think into the water. 531 00:49:40,690 --> 00:49:46,280 You have a pattern here, ink here. Nothing else here. 532 00:49:46,280 --> 00:49:51,590 Come back a little while later, the patterns disappeared. There's no spatial 533 00:49:51,590 --> 00:49:56,720 structure here. It's all the same. That's because the diffusion, the random 534 00:49:56,720 --> 00:50:03,570 movement of the molecules around has ended up meaning they distribute uniformly. 535 00:50:03,570 --> 00:50:10,790 And yet Turing was saying. That diffusion causes Patrick. 536 00:50:10,790 --> 00:50:16,200 That seems really counterintuitive. 537 00:50:16,200 --> 00:50:21,250 So he will turn to a system of two equations. Two 538 00:50:21,250 --> 00:50:26,950 chemicals, you and V. So and there's a diffusion term here. 539 00:50:26,950 --> 00:50:32,600 And here's the. We action terms and what he said 540 00:50:32,600 --> 00:50:38,140 was for this to happen. You need a number of key 541 00:50:38,140 --> 00:50:44,850 things. You need one chemical to produce. 542 00:50:44,850 --> 00:50:50,640 No other chemical. Outproduced itself. 543 00:50:50,640 --> 00:50:55,710 This chemical inhibits this chemical. So this is what he 544 00:50:55,710 --> 00:51:00,900 called the activator, because it activates inhibitor 545 00:51:00,900 --> 00:51:06,100 and inhibitor inhibits the activity. So let's think about this 546 00:51:06,100 --> 00:51:11,210 spoon's. You've got a pile of a activator. It's catalysed 547 00:51:11,210 --> 00:51:18,510 in itself, producing more of itself, but it's also producing the inhibitor. 548 00:51:18,510 --> 00:51:25,470 Then the inhibitor inhibits the activate. So there's more activator. 549 00:51:25,470 --> 00:51:30,540 There's more inhibitor and that pulls activator die, so you can imagine getting 550 00:51:30,540 --> 00:51:35,540 to the equilibrium. Well, you've got a level of activity 551 00:51:35,540 --> 00:51:41,010 and a level of inhibitor, and that's stable because if you added 552 00:51:41,010 --> 00:51:46,690 more and more activator. That would activate the inhibitor, 553 00:51:46,690 --> 00:51:52,070 which would make more inhibitor, which would pull the activator packed on 554 00:51:52,070 --> 00:51:57,130 less activator Shlash inhibitor inhibitor. We'll come back and you get back to the 555 00:51:57,130 --> 00:52:02,200 steady state. So this is still. Now, I suppose we have 556 00:52:02,200 --> 00:52:07,890 to fusion. And suppose that the inhibitor 557 00:52:07,890 --> 00:52:13,550 diffuses faster than the activator. 558 00:52:13,550 --> 00:52:18,770 Then what will happen if you do that prohibition again? The 559 00:52:18,770 --> 00:52:23,960 activator Stotch produce itself starts to grow, produces 560 00:52:23,960 --> 00:52:29,110 the inhibitor. Now, the inhibitor, some of it 561 00:52:29,110 --> 00:52:34,600 inhibits the activator, but a lot of it gets out of the way. So 562 00:52:34,600 --> 00:52:39,640 there's not enough inhibitor to inhibit the activity you the activator 563 00:52:39,640 --> 00:52:45,440 grows even more. Produces more inhibitor. It goes away. 564 00:52:45,440 --> 00:52:50,950 So what you'll end up getting will be lots of activator, surrounded by 565 00:52:50,950 --> 00:52:56,080 lots of inhibitor, a spatial pattern. So our 566 00:52:56,080 --> 00:53:01,100 intuition, that diffusion. Gets rid of 567 00:53:01,100 --> 00:53:06,940 pattern is based on one thing diffusing. 568 00:53:06,940 --> 00:53:12,470 And that's the singular case. As soon as you have more than one 569 00:53:12,470 --> 00:53:17,800 species, chemical fusing diffusion can drive 570 00:53:17,800 --> 00:53:23,510 Patrick. So this is a nice example of mathematics, pointing out 571 00:53:23,510 --> 00:53:29,560 where our intuition is wrong and building our intuition. 572 00:53:29,560 --> 00:53:34,900 And Cherry used mathematical analysis to prove 573 00:53:34,900 --> 00:53:40,750 this. And this idea 574 00:53:40,750 --> 00:53:46,120 of reaction to fusion forming a pre pattern. 575 00:53:46,120 --> 00:53:51,130 For animal, quote, markings, patterns and various other 576 00:53:51,130 --> 00:53:56,230 things has been used a lot. And so Jim Murray, who is my supervisor, nice 577 00:53:56,230 --> 00:54:01,690 graduate student here in Oxford. Very famous paper of his here, where he 578 00:54:01,690 --> 00:54:06,850 shows how you can get patterns that look like the spots 579 00:54:06,850 --> 00:54:12,150 on a leopard. And you can get those reactions. Fusion system. 580 00:54:12,150 --> 00:54:17,220 And here's another idea. I here hear this often looks like an animal 581 00:54:17,220 --> 00:54:23,170 put marking. And what's happening here so you can think of right here. There's no pattern. 582 00:54:23,170 --> 00:54:29,350 The chemical doesn't form any pattern. And then as you move here, 583 00:54:29,350 --> 00:54:35,290 the demand is got bigger. But he scaled the domain box, which fits 584 00:54:35,290 --> 00:54:40,630 on the graph, on another piece paper. Otherwise the pictures would get too big. But this 585 00:54:40,630 --> 00:54:46,980 is. For larger to mean now you start getting some pattern 586 00:54:46,980 --> 00:54:52,430 as you increase the domain further, you get more pattern. Further, you could more 587 00:54:52,430 --> 00:54:57,640 you get more and more. So we can sort of think 588 00:54:57,640 --> 00:55:02,990 about these. Is it basically these patterns, it turns out if you look at them, 589 00:55:02,990 --> 00:55:08,410 they are what are called Igen functions of the class in. And on these sorts of 590 00:55:08,410 --> 00:55:13,710 diagrams here, they're essentially mixtures of signs and cosine. 591 00:55:13,710 --> 00:55:18,730 And what you can think of is also demand gets larger. You can fit more 592 00:55:18,730 --> 00:55:23,970 signs and put signs in and get more complicated patterns. 593 00:55:23,970 --> 00:55:29,110 So therefore, the prediction is that there should be nominal dips like this one, the analytics like 594 00:55:29,110 --> 00:55:34,580 this, you might have seen an animal that looks like this. Maybe not seen an animal 595 00:55:34,580 --> 00:55:39,810 like this. But here we are. These are not photo shopped. This is Volly 596 00:55:39,810 --> 00:55:45,810 Goat. This is about the goat. 597 00:55:45,810 --> 00:55:52,420 So as I mentioned, these Igen functions, Pacien. 598 00:55:52,420 --> 00:55:57,550 Form the pattern. So are there other places 599 00:55:57,550 --> 00:56:02,580 in mathematics or in science where we see Igen functions of the class? Well, 600 00:56:02,580 --> 00:56:09,920 if you vibrate a plate or a drum or strong a guitar. 601 00:56:09,920 --> 00:56:17,050 The vibrating modes are Igen function plasty. 602 00:56:17,050 --> 00:56:22,090 So these authors here. Had the idea that if 603 00:56:22,090 --> 00:56:27,530 you vibrated a plate, it looked like an animal put marking. 604 00:56:27,530 --> 00:56:33,820 And you could visualise the vibration then, since the vibration. 605 00:56:33,820 --> 00:56:39,010 Or Igen functions of the reclass in. And they're 606 00:56:39,010 --> 00:56:44,020 proposing that animal coop markings or Igen functions as a claassen, then 607 00:56:44,020 --> 00:56:49,240 you should see animal markings. So this is the vibrating plate. 608 00:56:49,240 --> 00:56:54,570 Look. These are very like, um, local blockage. 609 00:56:54,570 --> 00:57:03,410 Look at here. You got that sort of thing here. Here's a separate fossas. 610 00:57:03,410 --> 00:57:08,480 And hands my heart. And this is from taken from his book, The Algorithmic 611 00:57:08,480 --> 00:57:13,610 Beauty of Seashells. A beautiful book where this is 612 00:57:13,610 --> 00:57:19,550 a real seashell and this is a pattern of sea shell generated 613 00:57:19,550 --> 00:57:24,650 by system of reaction fusion equations. And the book is full of 614 00:57:24,650 --> 00:57:30,810 beautiful pictures like this. 615 00:57:30,810 --> 00:57:35,910 The Igen functions of the PACIEN have certain properties. One of their properties 616 00:57:35,910 --> 00:57:41,070 is that. Ask them to man the geometry 617 00:57:41,070 --> 00:57:46,380 changes, the pattern changes in particular 618 00:57:46,380 --> 00:57:52,690 as the domain becomes Nardwuar and more one dimensional. 619 00:57:52,690 --> 00:57:58,080 The pattern changes from being intrinsically two dimensional exports 620 00:57:58,080 --> 00:58:03,630 to becoming one dimensional like. So here is the model. 621 00:58:03,630 --> 00:58:12,270 And here is an. Michael and. 622 00:58:12,270 --> 00:58:17,290 So if we take this a bit further, what this town, us? Is 623 00:58:17,290 --> 00:58:22,950 that if you have a domain and you see a pattern on 624 00:58:22,950 --> 00:58:29,080 this. The only thing you change. Is the demand 625 00:58:29,080 --> 00:58:34,190 geometry. So, for example, you go from a fact being 626 00:58:34,190 --> 00:58:40,860 to in to a small to man. Then the pattern. 627 00:58:40,860 --> 00:58:46,590 Cannot get more complicated because the complication of the pattern 628 00:58:46,590 --> 00:58:51,840 increases with domain size. So 629 00:58:51,840 --> 00:58:57,090 if you have an animal that's got spots 630 00:58:57,090 --> 00:59:03,000 on its body. And then you look at the tail, which has got 631 00:59:03,000 --> 00:59:08,220 a narrower, more one dimensional geometry. You will either 632 00:59:08,220 --> 00:59:13,590 see spots or stripes. More likely 633 00:59:13,590 --> 00:59:18,600 strikes, on the other hand, if the animal 634 00:59:18,600 --> 00:59:24,360 body you see stripes, then I should knock with the to man. 635 00:59:24,360 --> 00:59:29,850 You will either see stripes or no pattern. You will not see spots 636 00:59:29,850 --> 00:59:34,940 because that would mean the pattern getting more complicated. 637 00:59:34,940 --> 00:59:40,670 This is their idea of developmental constraints. This is a paper where two theoreticians, 638 00:59:40,670 --> 00:59:47,120 OSTREM Murray, came up with this idea from the mathematics. 639 00:59:47,120 --> 00:59:52,730 And these are two biologists who came up with the idea for actually looking 640 00:59:52,730 --> 00:59:58,340 at skeletons, looking at put markings, etc. So 641 00:59:58,340 --> 01:00:03,380 these were data driven. These were mechanistically driven 642 01:00:03,380 --> 01:00:08,590 results. So the idea is 643 01:00:08,590 --> 01:00:13,870 that a spotted animal can have a stripy tail, a striped animal 644 01:00:13,870 --> 01:00:19,030 Congleton spotted. OK, so now 645 01:00:19,030 --> 01:00:24,300 let's move to. Premier 646 01:00:24,300 --> 01:00:29,370 League football shirts or football shirts? The Premier League 647 01:00:29,370 --> 01:00:34,430 here, stripey body. 648 01:00:34,430 --> 01:00:40,190 Similar pattern on the art spots. Simple pattern, 649 01:00:40,190 --> 01:00:45,830 stripes, stripes, stripes, no pattern. 650 01:00:45,830 --> 01:00:51,440 So these will be Turing's model. 651 01:00:51,440 --> 01:00:56,660 Pattern on the sleeves is not more complicated than the pattern 652 01:00:56,660 --> 01:01:02,830 on the body. Here, season 2016, 2017, 653 01:01:02,830 --> 01:01:09,100 the Premier League and virtually all of these satisfy cheerleaders 654 01:01:09,100 --> 01:01:14,260 theory, a pattern on the sleeves is not more 655 01:01:14,260 --> 01:01:20,460 complicated than the pattern on the body. Rugby shirts. 656 01:01:20,460 --> 01:01:28,340 Same thing, pattern on the body. 657 01:01:28,340 --> 01:01:35,140 As complicated or more complicated and part on the streets. 658 01:01:35,140 --> 01:01:40,520 But there are contradictions. This is the daughter of a former graduate 659 01:01:40,520 --> 01:01:46,730 student of mine. And he sent me this photograph. And instead of said, look. 660 01:01:46,730 --> 01:01:51,740 How beautiful is my daughter? He said, look, contradiction 661 01:01:51,740 --> 01:01:59,810 to cheering small. Stripey body spotted arms. 662 01:01:59,810 --> 01:02:04,950 Here are some cuddly examples, too. You see these animals, you should do them because they don't 663 01:02:04,950 --> 01:02:11,130 satisfy tributes, theory, stripey body, spotted tail, 664 01:02:11,130 --> 01:02:16,700 no pattern pattern. These her 665 01:02:16,700 --> 01:02:22,290 country examples. So what do you think is going on here? Is that not only 666 01:02:22,290 --> 01:02:29,900 is the domain geometry changing, but something else is changing. 667 01:02:29,900 --> 01:02:34,910 Now, the question is, do these occur in nature, 668 01:02:34,910 --> 01:02:40,520 in the actual electoral system? We know that morphogenesis prayer 669 01:02:40,520 --> 01:02:46,070 is still a huge controversy has to do with these patterns to occur 670 01:02:46,070 --> 01:02:52,040 because of. The way to insert the action diffusion. 671 01:02:52,040 --> 01:02:57,170 They've been shown to a crowd chemistry. So in chemistry, these were cheering 672 01:02:57,170 --> 01:03:02,810 patterns. They've found in chemistry. And here is the simulation. 673 01:03:02,810 --> 01:03:07,880 This is actually just taken from this paper showing that the model 674 01:03:07,880 --> 01:03:12,980 predicts you can get these types of patterns like this. You can get Spaull hexagonal 675 01:03:12,980 --> 01:03:18,370 patterns like this. In fact, I haven't got the picture here, but you can also get these lab 676 01:03:18,370 --> 01:03:23,620 redesigned. So hopefully 677 01:03:23,620 --> 01:03:28,870 what I've shown here is that by understanding the mathematics in one 678 01:03:28,870 --> 01:03:35,870 area of biology. On chemistry. 679 01:03:35,870 --> 01:03:41,340 Mechanics, you can. Understand? Other areas 680 01:03:41,340 --> 01:03:47,130 of science. And that is the power of mathematics 681 01:03:47,130 --> 01:03:53,750 because by abstracting by the art of modelling, which means you abstract. 682 01:03:53,750 --> 01:03:58,910 Processes, many different processes. How fundamentally 683 01:03:58,910 --> 01:04:06,570 the same principles. So you could argue that mathematics. 684 01:04:06,570 --> 01:04:11,900 Is what unifies science. A drinking 685 01:04:11,900 --> 01:04:17,670 night. These. 686 01:04:17,670 --> 01:04:23,220 Processes that are common across different areas so they can transfer 687 01:04:23,220 --> 01:04:30,430 knowledge from one area to site to another. 688 01:04:30,430 --> 01:04:35,780 Shows that mathematics, really, Prof. But remember, 689 01:04:35,780 --> 01:04:40,850 with great power comes great responsibility. And you have to 690 01:04:40,850 --> 01:04:46,220 be careful not to push your luck, not to look at some thing, 691 01:04:46,220 --> 01:04:51,410 some scientific phenomenon and say, oh, that patriot or that structure 692 01:04:51,410 --> 01:04:57,020 looks like this other area of science. So since I understand that area of science, 693 01:04:57,020 --> 01:05:03,440 I understand that that is not science. 694 01:05:03,440 --> 01:05:09,380 He's got to know the signs. You've got to know the biology. Know 695 01:05:09,380 --> 01:05:14,600 the medicine, the physiology. Work with the key 696 01:05:14,600 --> 01:05:19,700 people in the field. The experts in the field to make sure that your 697 01:05:19,700 --> 01:05:25,960 mathematics is relevant and realistic in that area. 698 01:05:25,960 --> 01:05:31,090 In other words, you've got Cropper's. So thank you for 699 01:05:31,090 --> 01:05:51,440 your attention.