Mathematics Is Biology's Next Microscope

The thought that in that location are fundamental laws that describe the physical universe is at present a part of everyday civilisation. We await that scientists tin predict how a physical system volition behave using these mathematical descriptions, whether information technology exist the trajectory of a football, the chain reaction in a nuclear power station or how to bounciness signals between mobile phones.

"Ya canna break the laws of physics, Captain!"
(Scotty, Star Trek)

What many people might non realise is that the same cannot be said for biological science. "Biological science is not however a predictive scientific discipline, there are essentially no key laws," says Thomas Fink, a physicist working on the DARPA Fundamental Laws of Biology program (FunBio). "But the exciting affair is that biology appears to be on the cusp of bifurcating into an experimental branch and a theoretical branch. Are in that location underlying fundamental laws? Is there an equivalent of [Newton's second law of motion], F=ma, in biology? While we can't yet say for sure, there is increasing evidence that there are deep unifying mathematical principles."

Reticulated giraffe

Beautiful mathematics

Mathematics has played a part in the development of the biological sciences and has fifty-fifty led mathematicians to push the limits of their own subjects, such as Alan Turing's pioneering work developing reaction-diffusion equations to explicate brute patterns. However, Fink says that biology, in terms of maturity, is at the stage that physics was 300 years ago — we understand the pieces, the genes and the cells, but we still can't describe how they fit together into complex systems such as organs, organisms and ecosystems, much less how they evolve or how the brain works.

Fink and his colleagues on the FunBio plan are trying to give biology a theoretical push. FunBio brings together people from many unlike fields — physics, mathematics and biological science — to endeavour to describe out the fundamental rules underpinning how biological systems work. "Is there a mathematically elegant story behind life? That's the question FunBio is trying to answer."

Are you ready to evolve?

"People who wish to analyse nature without using mathematics must settle for a reduced understanding."
(Richard Feynman, The Character of Physical Police, 1965)

One of the exciting areas they are working on is trying to empathise the intricacies of evolution. Nosotros often think of life every bit evolving from some sort of primordial soup where the complex molecules necessary for life first came together. Merely Fink believes that more than half of the story happened before that moment — only which primordial soups have the correct stuff to be able to evolve, and how did they come about?

"Everyone says the standard model for evolution is mutation, selection and inheritance. Put those ingredients together in a box and you go evolution. But the reality is, when nosotros put those things into models of development, or set up appropriate systems of artificial life, we just don't get life-like evolution — we don't detect the evolution of complex, surprising things. Some fundamental is missing. What gives a system the capacity to evolve? What makes a system evolvable?"

RNA structure

An RNA sequence folds into a specific shape determined by which parts of the sequence align.

Why are some biological systems more suited to evolution than others? Fink and his colleagues are exploring this question, and their starting point is a model of RNA folding. RNA molecules read the genetic information independent in DNA in order to produce proteins in a cell (you can read more about RNA on Plus). You can think of RNA sequences as strings consisting of the 4 messages U, G, C and A, representing the dissimilar nucleotides (the molecules that make upwards RNA and DNA). These genetic strings don't float nearly like a long piece of seaweed in the ocean: each RNA sequence folds itself into a specific shape. The letters in different stretches of the sequence line upwardly next to each other (technically speaking their base pairs are complementary) and loops and bends are formed between the aligned segments.

If you expect at a brusque segment of RNA, say 30 letters long, you have 4³⁰ possible different sequences. Each sequence folds into exactly one shape simply many other sequences might also fold into the aforementioned shape. So you lot have fewer shapes (about 1.8^thirty possible shapes for RNA sequences of length 30) than sequences.

RNA folding is an insightful framework for studying evolvability. The unlike sequences are called genotypes and the different shapes they fold into are called phenotypes. "The phenotype is the finish-product that somehow the environment notices," says Fink. "The genotype is the detail mode you made it. The surround doesn't notice the sequence; it notices the particular fold yous've got. "

To mathematically model a biological system of genotypes and phenotypes, for example a population of individual organisms and their generations of offspring, nosotros first draw a point for each item RNA sequence (or genotype) that occurs in the system. We can make a graph on these points past connecting whatever two sequences that differ by simply one betoken mutation, just one spelling error. For case, the sequences ...AACUG... and ...AACUA... are almost the aforementioned, they differ by only one alphabetic character, and so nosotros draw an edge betwixt them.

simple

A uncomplicated mutation graph. The vertices represent the different RNA sequences or genotypes that are linked by an edge if they differ by one letter. The vertices are coloured according to their pheonotype, the shape the sequence folds into.

The issue is called a mutation graph and represents the possible ways the biological system can evolve: making one signal mutation at a time traces a path through this graph. The genotypes are the points, or vertices, in the graph. At present let'south paint each point (each sequence) a colour, corresponding to which shape the sequence folds into. These colours are the phenotypes. All the sequences that fold into the same shape are represented by points of the same colour on the graph.

"So at present we have this interesting mathematical object," says Fink. "We can make bespeak mutations, which means we move along an edge of the graph. Some of these betoken mutations change our color, change our phenotype: others do not. If we mutate only we don't change color, we phone call that a neutral mutation. At present nosotros can start to talk virtually unlike concepts: we can think most what it means to be robust, and what it ways to be evolvable."

Continued clusters of points of the same color represent these neutral mutations. For example the isolated purple vertex in the bottom left of the graph will modify colour no matter how information technology mutates. However, the purple vertex at the correct of the graph has a probability of 1/3 that it will stay majestic.

You can define the robustness of a phenotype, say purple, by averaging the probability of staying purple, over all the purple points — how likely it is that yous will remain the same shape from mutation to mutation. (For this simple graph the robustness of the purple phenotype is 4/xv.) If y'all imagine the organism lives in an environment where any phenotype other than royal is lethal, the robustness of the system is how probable new generations are to exist the regal phenotype and therefore survive.

Merely suppose now the environment changes and it may now be more beneficial to be a dissimilar phenotype. In this example you lot are interested in your evolvability — how many unlike phenotypes you can access through mutation. In terms of the graph, the isolated purple vertex at the bottom left of the graph only has access to 1 other phenotype, ruby. Nevertheless, the cluster of 4 connected purple points beyond the middle of the graph have admission to both the red and greenish phenotypes.

"A biological system is robust if a change in genotype does not pb to a change in phenotype. The system is evolvable if a particular phenotype has admission to many other phenotypes. Until recently, these ii qualities accept appeared to be, paradoxically, opposed: if changes in genotypes don't lead to changes in phenotypes, the arrangement is robust; if they do lead to changes in phenotypes, the system is evolvable." Contempo work, notably that of the theoretical biologist Günter Wagner, suggests that actual biological systems are delicately tuned and then that robustness and evolvability are in fact correlated — the more evolvable the organisation the more robust information technology is — this has been explicitly studied in the RNA-folding model described above. "Systems which are both robust and evolvable are capable of extensive exploration of the phenotypic landscape in light of changes to the environment, and can thus try out new mechanisms without destroying cadre functionality."

"This is a simple model but there are a number of mysteries hither," says Fink. "For instance, when does increasing your evolvability correspond to increasing your robustness? Are there disquisitional phenomena hither when the organisation all of a sudden, in a spring transition, becomes very evolvable and you can access almost all other types of phenotypes?"

From his current mathematical analysis of mutation graphs, Fink suspects that for a biological system to be both robust and evolvable, information technology might not accept to be every bit delicately tuned as is presently believed. "It appears that the number of phenotypes needs to exist sufficiently minor compared to the number of genotypes, and this may modify the organization in some critical way. There may be a bound transition in the evolvability of the system, strongly related to critical phenomena in percolation theory [see below]. Is that a special belongings of living systems? Or is it actually something deeper than that? "

Can biology atomic number 82 to new theorems?

"The lack of existent contact betwixt mathematics and biology is either a tragedy, a scandal, or a challenge, information technology is difficult to decide which." (Gian-Carlo Rota, Discrete Thoughts, 1986)

It is clear that mathematical know-how is needed to transform biology into a predictive, theoretical scientific discipline. The trouble is that contributing to biological discovery alone is not plenty to draw in the best physicists and mathematicians. "I'grand not interested in biology per se," says Fink. "If I was, I'd be a biologist. I'one thousand interested in using cute mathematics to describe the world around me. If agreement life on earth needs cute mathematics then I want to get involved."

So if biologists merely turn to mathematicians and physicists to help them solve mundane mathematical problems or exercise information analysis, these theorists won't be excited. Fink jokes that when a biologist asks him to but analyse their data, he asks them to mow his backyard — to him the two tasks are equally fascinating. "It'south got to be a ii-way procedure — biological insight achieved through elegant mathematics. The culture of biological inquiry is only commencement to appreciate this — that an elegant theory is more likely to be a true theory, even in the life sciences."

One example of this two-way interaction was when Fink, Francis Brown and PhD pupil Karen Willbrand brought their mathematical perspective to a problem in medical genomics and concluded upwards proving a new result in number theory. They were looking at the data produced by a microarray study of bladder tumours washed by experimentalists at the Curie Plant. "The bones thought is that a pathologist puts twenty different neoplasm samples in gild of how avant-garde they are," explains Fink. "For each tumour sample y'all've got thirty,000 genes, and a microarray measures the concentration of each one." This gives you 30,000 curves, one curve for each gene; and each curve has twenty points, i point for each of the tumours. Most of the curves look random, but very occasionally one appears to have a recognisable blueprint: "Hey, this curve starts off high, then slowly moves down, and so shoots up again. Wow! Indicator factor! Nosotros've got a predictor for float cancer," says Fink, imagining the reaction of a biologist conducting the written report.

"The trouble is, to be able to know what is interesting, one needs to know what is dull," explains Fink. "Is it a likely thing or an unlikely matter to detect a curve that starts of going downwardly, down, down, then finishes going upward, upward, upwardly? What if we discover i among 30,000 curves, is that a surprise?" A theoretical agreement of how a random curve typically behaves will not only tell u.s. whether a curve that looks interesting really is interesting, but too helps pinpoint curves that don't look interesting just in fact are.

Instead of trying to understand the biology, Fink and his collaborators focused on the maths. Imagine you accept a curve of 5 information points, say, 0.77, 0.84, 0.51, 0.30, 0.26. If you connect the points with line segments you'll see that the first segment is increasing, and the adjacent three segments are decreasing. So the curve goes upwardly, down, down, downwards, and the bend's and then-called up-down signature is + - - -. The question is, is this curve unusual and therefore something to shout about if information technology shows up in an experiment?

Table of probabilities of finding a random curve with a given up–down signature for 2, 3 and 4 data points.

Probability P(σ) of finding a random curve with up–down signature σ, for 2, 3 and 4 data points. You tin can recollect of curves of, say, three points, as permutations of the set ane, 2 and three. The permutation three two 1 has the up-down signature or - -, while the permutations 3 ane two and 2 one 3 both have the same up-down signature of - +.

To reply this question, Fink and his collaborators calculated the probability of a randomly generated permutation (say a random organisation of the numbers 1, 2, 3, 4 and 5) having the same upward-down signature. (The theory of up-downward signatures is the same for random curves and random permutations.) If a permutation is very unlikely to accept that shape by chance solitary, so the bend, and hence the gene, is likely to exist biologically meaning. All the same, the distribution of upwards-downwardly signatures over random permutations is an unsolved problem in mathematics. "It turns out to be a nice combinatorics problem that people started looking at about 130 years ago," says Fink. The problem was first studied by D. André in 1881, when he calculated the probability that a permutation has an alternating up-down signature (+ - + - + - …). Fink and collaborators generalised this consequence for arbitrary signatures, which led to their number theory results.

Dorsum in the original biological context, their technique provides a way of blindly identifying biologically important genes from microarray experiments, without whatever prior assumptions almost what sort of behaviour to look. As a benchmark, they tested their technique on well-studied yeast cell bicycle experiments (you tin read more than in their paper). It has since been used in electric current inquiry including studies on other forms of cancer.

On the brink of new mathematics

Fink believes the mathematical discoveries from theoretical biology are only but kickoff and there is a lot more even so to exist constitute. "The view amidst many physicists is that biology today is like quantum mechanics in the twentieth century. In that location is huge virgin territory, and people are racing in to make discoveries. At that place'southward a lot of low lying fruit, whereas in more mature fields like particle physics you've got to climb up high."

Tell united states what you think!

Practice you call up that biological science can become a theoretical predictive science? You can too leave a comment at the end of this article to voice your thoughts.

Charting this new mathematical territory of theoretical biology is starting to excite the mathematical community. Bernd Sturmfels, also role of the FunBio plan, has asked in his 2008 Clay lecture: "Volition a theoretical biologist ever win a Fields medal [1 of the highest hounours in mathematics]?" Finks says "Many theoretical physicists have won Fields medals and the boundary between physics and mathematics is almost ephemeral. Is the lack of real contact between mathematics and biological science a matter of the by?"

I encouraging sign is that the work of the 2010 Field's medallist, Stanislav Smirnov , is already linked, indirectly, to the mutation graphs and evolvability described earlier. Understanding the continued clusters of phenotypes on mutation graphs tin can exist posed as a problem in percolation theory — the expanse of statistical physics in which Smirnov works . "In a randomly coloured mutation graph, if the number of phenotypes is small enough, the typical size of a continued phenotype cluster will be sufficiently large to have access to lots of other phenotypes," says Fink. "Then if you are in a shifting environment yous can move into that phenotype if you need to. And percolation theory asks: practise you have lots of isolated piffling components or is there one giant component that spans the whole system?"

So perhaps the mathematical world will kickoff to recognise the rich pickings in theoretical biology, and it's merely a thing of time earlier we report on a theoretical biologist winning a Fields medal. The fundamental laws of biology may not be equally far off as they seem.

About this article

Thomas Fink

Thomas Fink is a theoretical physicist at the Curie Constitute and the London Institute for Mathematical Sciences. He uses statistical mechanics to report complex systems in physics and interdisciplinary fields. His research interests include detached dynamics, complex networks and primal laws of biology. Thomas has also written 2 popular books: The Man'south Volume, an almanac for men; and The 85 Ways to Tie a Tie, a volume near ties and tie knots.

Rachel Thomas is co-editor of Plus. She interviewed Thomas Fink in London in September 2010.