charles darwin , the father of the modern theory of evolution through natural selection , once complained that  i have deeply regretted that i did not proceed far enough at least to understand something of the great leading principles of mathematics ; for men thus endowed seem to have an extra sense .  while it is well appreciated the huge contribution of mathematics to physics , its role in biology is not so generally acknowledged . however , mathematics has been of paramount importance in the understanding of life sciences , and this importance will even increase in the near future . this review provides an opinionated and biased appreciation of such a contribution from mathematics to biology and vice versa . if physics was the battlefield where mathematics was meeting the real world during the 19th century , biology became the area of greatest intellectual challenges and applications for mathematics in the 20th century . as nicely written by joel e. cohen , mathematics became to biology what the microscope represented a few centuries ago : a tool to approach and interpret a new and fascinating world . certainly , it was not until the invention of the microscope in the late 17th century when a world invisible to the naked eye was suddenly discovered . think for example in the complexity of the human brain , with its huge number of neurons and interactions , or in the thousands of species in a tropical forest interacting in complex ways . biology represents challenges into the complex , and math may help us to approach such a complexity . mathematics represents a way to extract straight forward consequences of a series of assumptions : if x is assumed , then y is derived . this does not only allow us to think clearly , but to perform mathematical experiments in systems in which real experiments are impossible . arguably , the hardy  weinberg law in population genetics is equivalent with newton 's first law : if no force is applied into a population ( eg , natural selection , migration ) , the population will remain in a genetic equilibrium , that is gene frequencies will not change in a population from one generation to another . partly this is due to the extraordinary long time scale in which evolutionary forces take place . oftentimes , what we have is a temporal slice , and from these snapshots we are to infer a process . a rich mathematical development by the fathers of the modern evolutionary synthesis allowed us to merge darwin 's theory on natural selection with quantitative genetics . the resulting theory has been a fascinating intellectual contribution to understanding what forces shape live in this planet and why there is so much genetic variation in nature . ecology has , to some extent , followed the track of population genetics with some delay . even when ecology has been mainly a descriptive science , the pioneering work by lotka , volterra , nicholson , bailey , and others first introduced mathematical representations of temporal changes in populations . their models were based on either partial differential equations or equations in difference . while the former describe time as a continuous variable , the latter use discrete representations , for example , number of generations . in any case , models of population ecology describe how a population changes the time as a function of its own density and/or the density of its predator , parasite , or a competing species . even when these are just cartoons of reality , these population models were extremely important in understanding complex real phenomena , such as population cycles . by means of these models one could first explore mechanisms contributing to the stability of populations and communities . today , ecologists face the pressing needs arising from the accelerating human influence in the biosphere . these needs require a predictive theory , the kind of theory that can guide us , for example , in understanding the consequences of habitat loss and fragmentation on biodiversity . here however , we do not have a theoretical framework to predict its consequences on populations and communities . our intuition would tell us that we would still find a population , albeit of a shrinking size , as far as some fraction of the original habitat is left . these mathematical models , however , predicted that beyond a threshold of habitat destruction , the population goes extinct . at these critical values , populations and communities are at the edge of extinction : there is no relationship anymore between the intensity of a perturbation and the magnitude of its consequences . two and two do no longer add four as robert may once noted . predicting these points of no return a particular type of extremely simplified ecosystem is that composed by the interaction between viruses and the immune system . epidemiology has been another important field in biology where mathematics has allowed a rich conceptual development . let 's consider , for example , the spread of smallpox within a human population . epidemiologic models describe the temporal dynamics in the number of hosts infected and make predictions , such as the existence of a critical population size beyond which the diseases fade away . similarly , epidemiologic models predict a critical fraction of hosts to be vaccinated for the eradication of the diseases . these thresholds are essentially identical to the extinction thresholds described above in the context of habitat loss . these rna - based viruses mutate at an incredible rate to escape the immune system . the immune system evolves to recognize the exact nature of an invader and to counterbalance it , but if the virus keeps changing , it escapes this defensive system . mathematical models of distributions of virus strains called quasi species have shown the existence of a mutation rate threshold . beyond such a threshold , called catastrophe of error , an increase of the mutation rate induces the loss of the genetic information . the same biological process that has made them so elusive to classical epidemiologic treatments , such as vaccines , can provide keys for their eradication . just push a little bit forward its mutation rates and the virus will collapse . john holland and colleagues have demonstrated experimentally this by using several mutants to increase the mutation rate of the vesicular stomatitis virus . as a result , the virus becomes non - infectious.[147 ] how can the process leading from one cell to a complex embryo be explained from basic principles ? this was a total mystery reflecting a complete divorce between the 19th century physics focusing on systems at thermodynamic equilibrium , and so showing temporal evolutions toward disorder , and the dynamics toward increasing order and complexity characteristic of life . the great mathematician alan turing provided the first theoretical approximation to solve this apparent paradox . turing is famous for at least two other contributions , namely inventing the turing machine , that is , the precursor of modern computers , and breaking the enigma code used by the german nazis to encrypt communications during the second world war . after these two previous contributions , turing had the brilliant idea in 1952 of writing down a mathematical model describing the dynamics of two chemical species called morphogens . first , one morphogen called activator produces itself at a rate proportional to its abundance . this is a type of multiplicative , nonlinear process very common in biology : the more activator , the faster is produced . second , the activator also produces a second morphogen called inhibitor , which in turn inhibits the former . third , both activator and inhibitor diffuse through space , although the inhibitor does it at a faster rate . in summary , turing showed that starting from a uniform spatial distribution of both morphogens , some random fluctuations will be amplified . at the end , there will be patches with a high concentration of activator surrounded by empty areas . imagine that this heterogeneous distribution of activator determines the formation of a head on one extreme , where the concentration of activator is beyond the average , and a tail on the other side , where the activator 's concentration is below the average . the size of the spatial domain in which these morphogens diffuse determines how many such bifurcations can be accommodated . elegant extensions of the turing model have been proposed to explain multiple examples of pattern formation in development , such as the pigmentation in the coat of some mammals . the great developmental biologist pere alberch , in collaboration with george oster , james murray , and others , used this type of mathematical formulation in combination with beautiful experiments in which the size of an amphibian extremity could be manipulated through mutagens . as this size was progressively decreased , skeletal elements were deleted in a sequential order , mirroring natural variation in related species . alberch and colleagues were able to show that evolution takes place through minor changes of a conserved developmental program self - organization plays a very important role that can not be anticipated by focusing exclusively on genes . these self - organizing spatial patterns show discrete bifurcations following a well - defined sequence , and so natural selection has only a limited set of possibilities to choose from . it seems through this brief description that the flow of ideas has always gone from mathematics to biology . although this is the case in the majority of examples , there are also some cases in which the influence is the other way around . newton 's laws were able to describe the dynamics of celestial bodies and , more spectacularly , to make powerful predictions . given an individual condition , let 's say the position and speed of a comet right now , and the deterministic law of gravity , one can predict the position of such a comet 100 or 1000 years from now . or in the past , because for that matter the system is reversible , one can move the tape recorder either backward or forward . newton started a new way to deal with nature in which deviations from what should be expected could be used to make specific claims . an example was the prediction of the planet neptune on the basis of the modification of the gravitational field of the other planets . newton 's contribution was so important , that alexander pope proposed the following epitaph for newton , who died in 1727 :  nature and nature 's laws lay hid in night : god said ,  let newton be ! and all was light . one can argue who has been the second most important scientist in history , but the first position undoubtedly belongs to isaac newton . in the previous scheme , however , there is a small caveat : we can not know perfectly an initial condition . the atmospheric temperature at a place and a time , for example , is a number with infinite decimal points . let 's take five decimal points.[147 ] this extremely small mistake remains small in systems , such as the ones studied by newton . thus , if the arrival of the comet after 1000 years is predicted with an error of five decimal points , nobody would claim the theory is not good . in this type of systems , this is because these are linear systems , that is , systems in which variables add one to another . however , lots of interesting systems , such as the weather and biological systems are nonlinear . in this case , variables do not add but multiply each other or are raised to an exponent . a very small mistake after , let 's say 10 days , will become so large that prediction just does not work for long temporal windows . the temporal dynamics of systems , such as weather show random - like behavior similar to what expect for a stochastic system , but they are generated by totally deterministic systems described by a small number of variables . some were certainly done by mathematicians , but others by climatologists , such as edward lorenz , and by the theoretical ecologists robert may . lorenz was working on a simplified model of weather and found that after running a simulation a second time he obtained a different output . since his system of three differential equations was absolutely deterministic , he thought he had made a mistake . robert may , trained as a physicist , shifted to population biology and become one of the leading theoreticians in ecology and epidemiology . may co - discovered chaos by studying a simple , deterministic model of an ecological population . it deterministically specifies the population density at the next generation given the population density at the current generation . the behavior of this model was found to be very dependent on the value of the growth rate , a measure of per - capita fertility . for low growth rates , the population evolves toward a constant value , what mathematicians would call a steady state . no big surprise so far : a simple model shows a simple behavior . if now we slightly increase the growth rate , the model evolves toward period - two cycles.[1316 ] now the population will repeat itself every other generation . although something more interesting than before , this is still a simple behavior . if this parameter is further increased , cycles with higher periods ( 4 , 8 , 16 ,  ) arise until a specific growth rate is reached . may 's work inspired mathematicians who developed a theory about the existence of some universal properties of deterministic chaos . the type of period - doubling route to chaos may had found for his logistic map , was proved to be absolutely general . similar non - linear models and experimental systems were found to have the same properties . in this case , another source of complexity in biological systems has to do with their large number of interacting elements . we already mentioned the examples of the human brain and the food webs , graphical depictions of who eats whom in an ecological community . this framework is currently quite popular , and finds important applications both in biology and in other fields . in biology , the study of networks of genetic interactions opens the path toward a better understanding of gene regulation and the treatment of some genetic disorders . in ecology , considering the network of species interactions provides a community - wide approximation to the spreading of human - induced perturbations , such as overfishing . new - generation computers , with an increasing power can eventually simulate the dynamics of these extremely rich systems . computer software based on mathematical algorithms has also been used to represent complex data , such as the human genome . a full genome is like a text with millions of letters , and one can not just plot this huge amount of gross information . to have a chance to visualize this and seek patterns , one needs intelligent software . this has the advantage of controlling all confounding factors , isolating the interesting variables , and allowing to perform experiments we would not be able to do in nature . and simple models are the  perfect crystals  of biology , a kind of benchmark or reference where one can get straight - forward insight . as we walk through the challenging complexities of brain dynamics or the consequences of global change on ecological communities
biology has become the new  physics  of mathematics , one of the areas of greatest mathematical applications . in turn , mathematics has provided powerful tools and metaphors to approach the astonishing complexity of biological systems . 
 this has allowed the development of sound theoretical frameworks . here , in this review article , some of the most significant contributions of mathematics to biology , ranging from population genetics , to developmental biology , and to networks of species interactions are summarized .