# Open Problems in Mathematical Biology

Sean T. Vittadello1,2 & Michael P.H. Stumpf1,2,3 1Melbourne Integrative
Genomics, University of Melbourne, Australia
2School of BioSciences, University of Melbourne, Australia
3School of Mathematics and Statistics, University of Melbourne, Australia

###### Abstract

Biology is data-rich, and it is equally rich in concepts and hypotheses. Part
of trying to understand biological processes and systems is therefore to
confront our ideas and hypotheses with data using statistical methods to
determine the extent to which our hypotheses agree with reality. But doing so
in a systematic way is becoming increasingly challenging as our hypotheses
become more detailed, and our data becomes more complex. Mathematical methods
are therefore gaining in importance across the life- and biomedical sciences.
Mathematical models allow us to test our understanding, make testable
predictions about future behaviour, and gain insights into how we can control
the behaviour of biological systems. It has been argued that mathematical
methods can be of great benefit to biologists to make sense of data. But
mathematics and mathematicians are set to benefit equally from considering the
often bewildering complexity inherent to living systems. Here we present a
small selection of open problems and challenges in mathematical biology. We
have chosen these open problems because they are of both biological and
mathematical interest.

##  1 Introduction

In 1202 Leonardo Pisano (aka Fibonacci) studied the population dynamics of
rabbits in his book Liber Abaci. By some, admittedly glib, measure
mathematical biology could be said to predate theoretical physics — which also
started in Pisa with the work of Galileo Galilei — by some 400 years. Arguably
this head start has been lost and for the past century physics and mathematics
have become deeply entwined. Biology, by contrast, was often dominated by
observational or reductionist approaches and mathematical methods have for a
long time only played a negligible role. Biologists have, however, been quick
to adopt quantitative methods and tools as soon as there was a clear use for
them. Biological problems and analyses have, for example, been important
drivers of statistical methodology development, and statistics has become a
main feature — often not well liked — of biology undergraduate and graduate
education. Mathematics, less so.

The reasons why mathematics is a useful tool for many (not all) biological
questions have been superbly laid out before [1, 2, 3]. The principal reason,
arguably, is that through the lens of mathematics we can make accessible the
structures and mechanisms underlying heterogeneous and often disparate data
and observations. If, for example, we can connect two observations or
measurements in one model (say the level of growth factor in a Petri dish or
well containing cells and the down-stream change in phosphorylation of nuclear
ERK [4, 5]) we have a non-trivial mechanistic hypothesis from which we
initiate further investigations. The analogy between mathematics and a
microscope [2] in making the unobserved or unobservable visible holds
surprisingly often.

Our focus here is therefore different: we are interested in what are the
fundamental challenges for mathematical biology right now. Any such discussion
will be subjective and incomplete. In our case it is clearly also influenced
by our respective backgrounds (pure mathematics and theoretical/statistical
physics), our past and current research interests (which are pretty broad, but
mostly concerned with cellular processes and biophysics), and the
conversations that we have had with researchers over the years. One key person
was Edmund Crampin, especially for MPHS. The selection of material here cannot
claim completeness, but we do think that all the problems touched upon here
would have been of interest to Edmund. We hope that this will also be of wider
interest.

Our discussion below is broken into four parts: (i) A brief discussion as to
the different demands on modelling and the modeller in biology and physics;
(ii) A selection of open problems in mathematical biology. These questions
were partly crowd-sourced, and we attempt to put them into the wider context
of mathematical modelling in biology and the biomedical sciences. This part
contains open problems that are concerned with making mathematical modelling
more useful and relevant to biological problems, and open problems where new
mathematics will be essential to find answers to long-standing open questions
at the core of biological research. We then (iii) provide an, arguably
opinionated, overview of the opportunities that exist for ideas and concepts
primarily from pure mathematics, but also theoretical physics to enrich the
toolbox of mathematical biologists and study aspects that are particular to
living systems.

##  2 Why is Modelling Biology so Difficult?

The question really refers to a comparison with physics (and parts of
chemistry): why is modelling biology apparently so much more difficult than
modelling physics? There are three aspects to our attempt at an answer: (i)
physicists can often rely on first principles to model systems of interest;
(ii) in biology we often have to start from heuristic models; and finally,
(iii) differences in education between physics and biology.

Our attempts at finding an answer to this question should start by pointing
out that not all of physics is easy to model. For many physical problems,
however, it is possible to write down the structure of a model, often starting
from first principles [6, 7]. While it took major intellectual feats to write
down, for example, Kepler’s laws, Newton’s laws, Maxwell’s equations,
Einstein’s theories of relativity, and the laws of quantum mechanics and
quantum field theory, the structure of a model is typically derivable from
first principles. Noether’s Theorem [8] states that every conservation law has
an associated symmetry: for example, conservation of energy is related to
invariance with respect to time. Invariance with respect to position, or
translational invariance, implies the conservation of momentum. In both a real
and an abstract sense, Noether’s theorem underlies much of what is often
perceived as the beauty of physics; and being able to rely on it (the
Ward–Takahashi identities generalise the classical picture into the domain of
quantum field theory) makes the life of the working physicist much easier.

For dissipative systems, that is for systems where energy is consumed,
Noether’s theorem does not apply in its simple form. Biology is replete with
systems that consume and turn over energy, a fundamental aspect of life. This
fact alone suffices to account for the profound difficulties that arise in
finding mathematical descriptions of biological systems and processes.

In practical terms the main difference between modelling physics and modelling
biology is that the mathematical form of the model is known in most physical
cases. It may be difficult to solve the many particle Dirac Equation, or the
Navier–Stokes equation in 3D for realistic viscosity, but the form or type of
equations is rarely uncertain. If it is, however, for example systems with
broken symmetry [9], then it is typically of utmost physical interest to
identify and impose the correct form of the model.

In biology, by contrast, we have to start from a different level of
abstraction, and the best model structure is rarely if ever known with any
degree of certainty. We can make pragmatic assumptions to develop some
simplified models, and the canonical model of this approach is clearly the
Lotka–Volterra model

$\displaystyle\frac{dx}{dt}$ $\displaystyle=\alpha x-\beta xy,$
$\displaystyle\frac{dy}{dt}$ $\displaystyle=\gamma xy-\delta y$

where $x$ and $y$ denote a prey and a predator species respectively[1]. The
rate parameters, $\alpha$, $\beta$, $\gamma$, and $\delta$ represent, in turn,
the prey growth rate, the decline in the prey population due to predation, the
growth of the predator population, and the reduction in the predator
population due to predator death. This model has been successful in explaining
many aspects of population biology and ecology. But it is highly abstracted
and idealised, and fundamental features of reproduction, predation, and so
forth are purposefully ignored in this simple model. But this simple
Lotka–Volterra model can form the basis for more complicated descriptions,
including spatially structured populations, overlapping niches, and multiple
trophic levels. We can use these simple models to obtain qualitative insights
into fundamental aspects of biological processes and phenomena, including bet-
hedging and the evolution of virulence. The Lotka–Volterra model is
sufficiently easy to allow mathematical analysis, and simple enough to be
accessible to people with little direct modelling experience. Progress in
physics has often relied on a small canon of models that are equally relevant
and accessible to theoreticians and experimentalists: the single harmonic
oscillator, Ising Model, particle in a box, H2 molecule, etc. Biology needs
more of these [10].

Finally, the training tradition in physics is different: the typical physics
curriculum is heavy on mathematics and mathematical methods (with considerable
differences between institutions and even nations) that prepare students for
theoretical physics. Over the past generation the mathematics curricula of
physics degrees appear to have changed comparatively little in substance, but
all physicists are exposed to the mathematical techniques that are necessary
to make progress in research. The situation in biology is very different:
life-scientists often receive training in qualitative and statistical methods
required to make sense of observational and experimental data, whereas there
is generally no consideration of modelling the behaviour of biological
systems. The mathematical tools required by mathematical biology, meanwhile,
are still proliferating; the diversity of problems considered and the
complexity of data being generated pose considerable technical and conceptual
challenges which require a broad mathematical tool-set. We pick up this point
in Section 4.

##  3 Open Problems in Mathematical Biology: A Small Selection.

###  3.1 A Note on Notation

We are considering models, $f(x;\theta)$, describing the state
$x\in\Omega\subseteq\mathbb{R}^{N}$ of a biological system; the states $x$ are
confined to the state space, $\Omega$, of the system. Here
$\theta\in\Theta\subseteq\mathbb{R}^{d}$ is the $d$-dimensional vector of
model parameters (e.g. reaction rate parameters) in the parameter space
$\Theta$. We denote by $x_{i}$ and $\theta_{i}$ the $i$th components of the
state and parameter vectors, respectively.

When $f(x;\theta)$ represents a dynamical system, for example an ordinary
differential equation $dx/dt=f(x;\theta)$, we use $x_{0}\in\Omega$ to denote
the initial conditions. Alternative models will be represented by subscripts,
$j$, therefore $f_{j}(x;\theta)$.

We represent any experimental observations or data by
$\mathcal{D}=\\{d_{1},d_{2},\ldots,d_{m}\\}$; here $d_{i}$ is a vector with
number of coordinates possibly less than $N$ to account for observations that
do not capture the whole state space, therefore where $\dim(d_{i})<\dim(x)$.

###  3.2 Inverse Problems and Parameter Sensitivity

In many situations we have a candidate model, $f(x;\theta)$, or at least its
structure. What we then require are the parameters, $\theta$, and on occasion
initial conditions, $x_{0}=x(0)$. Choice of parameters is crucial if we want
to compare model output with observed behaviour or data $\mathcal{D}$. Often
parameters are chosen from the literature, but this approach is fraught with
potential problems [11]. Literature information can be imprecise about the
conditions under which parameters were determined; if, for example, ambient pH
is not comparable then the reaction rate may differ from the conditions under
consideration. Combining reaction rate parameters from different sources is
problematic if the corresponding experimental conditions were not identical or
at least comparable. So in many instances we will have to infer parameters
from data [12, 13, 14, 15, 16].

We typically have three options to delimit the range for the parameter
$\theta$. We can identify the value of $\theta$ that produces model output
that is most similar to the observed data. The likelihood,

$L(\theta)=\text{Pr}(\mathcal{D}|\theta),$ (1)

is the probability of the data for a given parameter $\theta$. The maximum-
likelihood estimate (MLE), $\hat{\theta}=\text{argmax}_{\theta}(L(\theta))$,
corresponds to the parameter that has the highest probability of producing the
observed data, and it can be obtained through optimisation. We usually use
optimisation to estimate $\hat{\theta}$. In addition to this point-estimate we
can determine confidence intervals for the value of $\theta$.

In a Bayesian framework we determine the posterior probability of a parameter,

$\text{Pr}(\theta|\mathcal{D})=\frac{\text{Pr}(\mathcal{D}|\theta)\pi(\theta)}{\text{Pr}(\mathcal{D})},$
(2)

where the prior, $\pi(\theta)$, represents our knowledge (or assumptions)
about the parameter value prior to looking at any data. The posterior is thus
determined from both prior information and the available data, $\mathcal{D}$,
via the likelihood, $\text{Pr}(\mathcal{D}|\theta)$. The probability
distribution over parameters rather than a single estimate tends to be the
focus of Bayesian inference. Both frequentist and Bayesian inference are vast
areas of research in their own right.

What really drives inference, irrespective of the framework, is the
information content of the data (assuming that the model structure, that is
the functional form of $f(x;\theta)$, is sufficiently close to the truth). One
measure of this is how much varying $\theta$ changes the probability of
observing the data, therefore the likelihood or the posterior probability.
Information geometry arguably provides the most rigorous way of assessing
certainty in an estimate, especially the so called Fisher information [17].
Under mild regularity conditions the Fisher information matrix is given by

$[I(\theta)]_{i,j}=-E\left[\frac{\partial^{2}}{\partial\theta_{i}\partial\theta_{j}}\log
L(\theta)\left|\theta\right.\right].$ (3)

The curvature of the the log-likelihood, or posterior function (or the
curvature of a different cost function), is an appropriate measure of
certainty for uni-modal likelihoods. For multi-modal likelihoods/posteriors
ad-hoc measures have been proposed, but no easy or universally satisfactory
solution exists [18, 19].

What we do find in many practical applications is that for any given dataset
and model we can only infer some of the parameters (more generally some
combinations of parameters); the rest are simply not inferable with any
precision, for example their marginal posteriors do not differ substantially
from the posteriors. Identifying which parameter combinations can be learned
[20] from prior knowledge plus data will become a crucial part of
parameterising large models, including whole cell models [21]. In the short
term sensitivity and robustness analysis — therefore varying parameters and
assessing the resulting differences in model output — can be used to assess
which parameters can be inferred from data [22].

###  3.3 Model Selection

Parameter estimation is already a formidable problem. But generally, as we
have shown above and will argue further below, the structure and nature of the
correct model are unknown. Choosing which model $M_{i}$ from a set of models,
$\mathcal{M}=\\{M_{1},\ldots,M_{N}\\}$, does the best job of explaining data
has therefore become a central focus in some areas of mathematical biology. A
host of methods are available to enforce something akin to Ockham’s razor:
generating a model that is as complex as required but not more so. These
methods have been reviewed extensively elsewhere [23]. When parameters can be
estimated from the data using likelihood or Bayesian methods, information
criteria such as the Akaike information criterion (AIC) and the Bayesian
information criterion (BIC) are popular because they explicitly penalise
against models with more parameters:

$\displaystyle\text{AIC}_{i}$ $\displaystyle=-2\log
L(\theta_{i};M_{i})+2k_{i},$ (4) $\displaystyle\text{BIC}_{i}$
$\displaystyle=-2\log L(\theta_{i};M_{i})+k_{i}\log n,$ (5)

where $k_{i}$ is the number of parameters of model $i$ and $n$ is the size of
the dataset. They are both, however, approximations (and in the case of the
AIC biased towards overly complex models as the amount of data increases) to
the marginal likelihood of a model,

$\text{Pr}(\mathcal{D}|M_{i})=\int_{\Omega}\text{Pr}(\theta|M_{i})\text{Pr}(\mathcal{D}|\theta,M_{i})d\theta,$
(6)

and the model posterior probability,

$\text{Pr}(M_{i}|\mathcal{D})=\frac{\text{Pr}(\mathcal{D}|M_{i})\pi(M_{i})}{\text{Pr}(\mathcal{D})}.$
(7)

Here the number of model parameters is implicitly penalised.

Model selection is challenging because of the high-dimensional integration
problem in Equation (6). Any advances in constraining and guiding parameter
inference — such as by identifying subspaces of the full parameter space
$\Theta$ containing parameters that do not affect the model fit to data — will
potentially help, but these subspaces will differ between alternative models
[24]. Specifying the prior probability of models is notoriously difficult: in
principle there is no clear a priori bound on the number of models, which
could be (and arguably ought to be) infinite. This is a fundamental problem
of, and a recurring theme in, Bayesian model selection.

As a general rule, problems encountered in parameter estimation tend to be
exacerbated in the context of model selection. Computational demands obviously
increase with the number of models. Perhaps not so obvious is that many
computational schemes that apply straightforwardly in the context of parameter
estimation — such as Markov Chain Monte Carlo sampling [25], or approximate
Bayesian computation (ABC) [26, 27] — require significant modification and
adaptation for model selection.

As in the case of parameter estimation, appropriate robustness analysis can
help. This often involves extending analysis over many related models [28,
29]. In topological sensitivity analysis [30], for example, the structure of
the model is perturbed by adding, deleting, or otherwise changing one or more
terms in the equations. That is, we go from candidate model $M$ to another
model $M^{\prime}$. We then fit model $M^{\prime}$, or in reality many
alternative models with perturbed structures, to the data to see if
alternative models can explain the data. The results of such an analysis can
often be sobering, with many alternative model structures being virtually
indistinguishable. This level of topological robustness could imply two very
different things: (i) the observed behaviour is robust to variation in the
model structure; or (ii) the large number of models may simply reflect that
the data are not sufficient to discriminate between models.

###  3.4 Design Principles

Even for mathematical modellers in biology the model should be of subsidiary
interest compared with the biological phenomena under investigation. Model
comparison methods, including model selection, multi-model averaging, and
topological sensitivity analysis can be used to group models that provide
comparable descriptions of a certain type of biological behaviour, henceforth
denoted by $Q$. For biological systems, instances of a behaviour $Q$ could be
oscillations, switch-like characteristics [31], signal filtering [32], robust
perfect adaptation [33, 34], or Turing pattern behaviour [35, 36].

For some types of behaviour, $Q$, it is possible to derive precise
mathematical statements that express either the necessary or sufficient
conditions (ideally both) for a system to exhibit $Q$. For example, we know
that:  (i) multi-stability depends on the presence of positive feedback [37];
(ii) for robust perfect adaptation (RPA), algebraic conditions have been
derived that establish the existence of RPA [34];  (iii) a Turing instability
is required for spontaneous pattern formation in reaction diffusion systems
[35].  But generally we still have to validate whether a given model $M$
exhibits behaviour $Q$ or not. Further, for many behaviours $Q$ we may not
even have easy criteria to consider.

Given a model universe, $\mathcal{M}$, we can use simulations to identify a
subset, $\widetilde{\mathcal{M}}\subseteq\mathcal{M}$, of models that exhibit
a particular behaviour $Q$. From the subset $\widetilde{\mathcal{M}}$ we may
be able to distill the design principles using a comparative analysis of the
models $M\in\widetilde{\mathcal{M}}$. This process, however, is not without
difficulties: structural properties, of for example the reaction network, do
not always suffice to guarantee the behaviour $Q$; boundary conditions and
parameter regimes shape the model output, dynamics, and behaviour, so must
also be considered within the comparison methodology. Visual inspection is
prohibitive, unreliable, and unrigorous.

The necessity for more flexible, universal, and automatable methodologies to
compare models is evident. With this objective in mind, employing simplicial
complexes (which are a generalisation of combinatorial graphs) has allowed us
to develop flexible representations of both quantitative and qualitative
models that are described with any formalism [38, 39]. Comparison of the
simplicial complexes, by our notions of distance or equivalence, provides a
meaningful conceptual comparison for the corresponding models. Our model
comparison methodology allows for clustering of models according to flexible
conceptual considerations, which can lead to the discovery of novel design
principles for biological systems.

###  3.5 Model Development

When we cannot rely on first principles for model development we have to use
heuristics or intuition to construct models. As we have shown above, this can
work but is in no way guaranteed. Considering large model universes,
$\mathcal{M}$, and model selection or topological sensitivity analysis offers
a potential solution. But every model $M\in\mathcal{M}$ still has to be
written down and translated into code to be executed by the computer. When
this has to happen for many models, or for very large models, the potential
for errors is considerable. If basic properties of models (number and types of
interactions, structural properties, kinetics, and so forth) can be specified
it is often possible to construct these models automatically, for example by
exhaustive enumeration. Metaprogramming provides particularly elegant ways of
automatically and on the fly developing and analysing mathematical model
structures [40]. In metaprogramming, software is considered as part of the
data and can therefore be manipulated during run-time and on the fly.

If we can guide the way in which software ‘writes and rewrites itself’, for
example by combining it with model selection or robustness analysis, we could
take steps towards developing larger and larger, including whole cell, models.
In this framework we would start from a model seed, $M^{(0)}$, and propose a
new model according to

$M^{(i)}\xlongrightarrow{G(M^{(i)})}M^{(i+1)},$ (8)

where $G(M^{(i)})$ is used to propose an update to the model that results in a
new model $M^{(i+1)}$; this could, for example, be a stochastic context-free
grammar.

In addition to being, we would argue, less prone to errors than manual
curation, this would also facilitate reproducible model development. We
already have the ability to share models between groups using file exchange
formats [41, 42]. But SBML and CellML model specifications do not allow us to
deduce how and on what basis certain modelling choices were made; even reading
the original publications rarely suffices. If models are created
algorithmically then the choices that go into a model-development pipeline can
be summarised and communicated efficiently. The development of the model thus
becomes reproducible if the updating mechanism and the initial model,
$M^{(0)}$, are known.

The procedure in Equation (8) can be interpreted as constituting a type of
random walk through model space. The process needs to be guided by model
selection after every iteration of Equation (8), or every few iterations. The
statistical challenges are considerable: (i) conventional parameter estimation
at each step would quickly become prohibitive; and (ii) the dimension of the
model space can change at each step which makes model selection — the
statistical comparison of successive models, $M^{(i)}$ and $M^{(i+1)}$ —
challenging. Techniques such as reversible jump Markov chain Monte Carlo are
required to construct appropriate samplers that can explore varying
dimensional spaces.

###  3.6 Multiscale, Hierarchical, and Spatio-Temporal Models

Most biological problems intrinsically span scales: from molecules to cells;
from cells to tissues and whole organisms; from species to ecosystems; from
genetic variants to populations. Dynamics at one scale have repercussions at
other scales, both higher and lower. Sometimes we can only observe data at a
higher level and have to draw inferences at the lower level: molecular
dynamics drive cell-fate decisions; cellular processes lead to tissue
formation; within-host dynamics drive population-level epidemics. The
literature in this field is enormous and rapidly growing; we therefore limit
the discussion in this section to two aspects: (i) mathematical and
statistical challenges related to development and analysis of multiscale
models; and (ii) a brief outline of bond graphs as a flexible framework for
thermodynamically correct models.

The joint consideration of biological processes across scales opens up new
possibilities to capture complex processes. Often we can take a pragmatic
approach and use coarse descriptions of the lower-scale dynamics to explore
the resulting behaviour at a higher scale. For example, we can use simplified
cellular dynamics to model the formation of patterns in tissues. In this
manner, agent-based models, for example, allow us to test our hypotheses about
intra-cellular dynamics by comparisons with the behaviour observed at the
tissue level. The critical evaluation of agreement between multiscale models
and biological reality requires careful consideration: just like for models at
the same scale there may be many alternative model realisations that give rise
to the same behaviour. Testing the sensitivity to model details, a multiscale
modelling equivalent of topological sensitivity analysis, does not seem to be
widely used. Agent-based models can be computationally prohibitively expensive
— this is already a problem for parameter estimation for agent-based models.
Here there is room for the development of emulation methods that replace
expensive simulation by statistical models.

One of the challenges of multi-scale and hierarchical models is to describe
the interfaces between levels or modules correctly; here thermodynamics
imposes tight constrains [43] that are generally not incorporated in
mathematical biology. Bond graphs have been conceived with that in mind [44];
they are also an area to which Edmund Crampin made major contributions [45,
46, 47, 48, 49, 50, 51].

In the bond graph formalism, energy and energy flow are tracked explicitly to
ensure that reactions are commensurate with the laws of thermodynamics. This
has two advantages: (i) resulting models incorporate stoichiometry,
conservation of energy, and molecule numbers (mass) from the outset; (ii)
thermodynamic constraints lead to models that are automatically modular, like
real biological systems. Therefore bond graphs offer a rigorous and well
defined framework to develop biophysically and thermodynamically consistent
models of molecular processes. Here we will build on this work and extend it
to develop whole cell models.

The critical point in model development is to link up sub-models of, for
example, gene regulation, metabolism, and the cell cycle. This modularity
turns out to be a natural consequence when thermodynamic conservation laws are
applied to the component models. Furthermore, the bond graph formalism
provides a straightforward pathway for modelling processes at different levels
of complexity, according to available knowledge and data. It can be combined
with both multi-physics and multiscale approaches to aid model construction.
Here multi-physics refers to models that can capture different physical
processes — such as heat transfer, dynamics of biochemical reactions over
space and time, and transport phenomena inside the cells as well as between
the cell and the extracellular environment — in a consistent manner. Bond
graphs allow for this, and therefore provide a natural framework to consider
situations where cell mechanics, electrophysiology, and metabolic processes
need to be considered jointly.

###  3.7 Stochastic Nonlinear Dynamics

The theories of stochastic processes and dynamical systems have developed
largely independently and in parallel. Many investigations in mathematical
biology have also considered deterministic dynamics with scant reference to
stochasticity. Classical work by Waddington, for example, does not consider
noise and random dynamics, with the exception of fluctuating environments.
Many important processes in ecology, population biology, epidemiology, cell
and developmental biology, and physiology, have been studied with great
success using purely deterministic descriptions. Stability analysis and
bifurcation theory can give us qualitative, even global qualitative, insights
into the complex dynamics underlying ecological dynamics or cell cycle
regulation.

Often the average of a stochastic process can be described adequately using
deterministic models. Diffusion and reaction-diffusion systems are good
examples for which the underlying stochastic dynamics are sufficiently simple
and deterministic partial differential equations can model the system
sufficiently: solutions to the diffusion equation have been used, for example,
to describe ecological migration, tumour growth, and biological pattern
formation.

In population and evolutionary genetics, by contrast, stochasticity — random
sampling of alleles/individuals — has generally provided the accepted null
model. The neutral model of evolution, for example, assumes that most genetic
variation has no effect on fitness, the expected number of offspring.
Historically, the development of statistics, probability theory, and
population genetics are tightly linked. But in ecology and cell biology, for
example, stochastic analyses have sometimes been conducted as ad hoc additions
to deterministic analyses.

In many biological problems chance plays an important role, as do nonlinearity
and feedback. The analysis of these systems is fraught with challenges. For
illustration let us consider stochastic differential equations. Because random
processes are not differentiable, we write this in the form,

$dX=f(X;\theta)dt+g(X;\eta)dW_{t},$ (9)

where the first term, $f(X;\theta)$, captures the deterministic dynamics and
the second term, $g(X;\eta)$, the stochastic contributions to the dynamics;
$dW_{t}$ denotes the Wiener process increment; and $\theta$ and $\eta$ are the
parameters of deterministic and noisy dynamics, respectively. For most
biological systems, especially biochemical reaction systems, $f(X;\theta)$ and
$g(X;\eta)$ are not independent but related. This relationship can be derived
from the microscopic dynamics; frequently, however, $g(X;\eta)$ is chosen for
convenience alone, and then typically in the form of white noise, therefore
$g(X;\eta)=\sigma^{2}=\text{constant}$; this type of noise, which is
independent of the state variable $X$ is often referred to as “additive
noise”, in contrast to noise that depends explicitly on the state, which we
refer to as “multiplicative noise”.

The extent to which qualitative features of the deterministic dynamics affect
the behaviour of a model once noise enters the equations requires careful
consideration and depends a lot on the type and form of the noise function,
$g(X;\eta)$. For example, the fact that the deterministic system is bi-stable
in no way guarantees that the corresponding stochastic system has a bi-modal
distribution over state space. This lack of conservation of qualitative
features of dynamical systems — including number and nature of stationary
states, existence of bifurcations, etc, — as noise is introduced, or more
precisely our inability to predict a priori if qualitative features are
conserved upon the introduction of noise, is becoming a limiting factor in
many analyses. In the 1970, for example, René Thom and colleagues introduced
catastrophe theory[52], which describes the types of sudden change
(catastrophes) that can be encountered in dynamical systems. This found
applications in population and developmental biology, but fell quickly out of
favour [53]. The past two years have experienced renewed interest in
catastrophe theory[54, 55, 56] and stochastic dynamics, if accounted for
correctly, can change the qualitative behaviour [57].

In short, we do not understand anywhere near enough how to model, analyse, and
understand complex dynamical systems that are subject to stochastic effects.
Conceptually there are many problems that need to be considered: for example,
we would really like to understand how the effects of noise are mitigated or
attenuated in order to give rise to highly stable and predictive phenotypes.
Computationally, there are also a host of important and worthwhile problems to
consider. Simulating stochastic dynamics is challenging and costly. Powerful
approximations to exact simulation methods exist, and continue to proliferate
[58], but ultimately we lack reliable approximations that continue to work
when dynamics become nonlinear.

###  3.8 Natural Selection

Evolution provides the organising principle for the life sciences.
Dobzhansky’s statement “Nothing in biology makes sense except in the light of
evolution” has become a rallying cry for biologists. The more recent version
by Michael Lynch [59], “Nothing in evolution makes sense except in light of
population genetics”, makes this more precise as genetic change and selection
on new variants occurs at the population level. Where, when and how selection
operates has been at the core of evolutionary analyses and discussions [60].
Here, as in the previous section, there is a tension between deterministic and
stochastic drivers, which need to be considered jointly to understand. The
deterministic dynamics reflect the effects of natural selection, and the
stochastic dynamics are due to the inherent randomness in the demographic and
genealogical dynamics [61].

Probabilistic arguments have been dominant in the population genetics
literature [61, 62]: random sampling of gametes appears to be remarkably
effective at describing many real population genetic diversity patterns. Even
the neutral model, which assumes that most genetic variants are not subject to
natural selection (or, in its modified form, that most genetic variation is
neutral or mildly deleterious), has underpinned the bulk of population genetic
analysis. Neutrality makes the problem of studying evolutionary processes and
data much easier. Most importantly, neutrality is a powerful null model
against which we can test our hypotheses on experimental data.

In population dynamics, deterministic approaches have been used to study, for
example, pathogen evolution and early pre-biotic evolutionary processes. One
argument that has been put forward is that for sufficiently large population
sizes the random evolutionary (demographic sampling) effects become less
important. The effects of finite population size have also been largely
ignored in game theoretic approaches to evolution.

The discussion of the previous section is, of course, relevant here as well.
The tension between, and combined effects of, deterministic and stochastic
dynamics are clearly important in understanding evolutionary dynamics. Simple
models are useful, but not always enough [62]; as evolutionary arguments
continue to increase in importance in very applied and data-rich areas, such
as oncology and host-pathogen evolution, there is a need for both nuance and
increased precision. In cancer biology the evolutionary characteristics and
the disease aetiology have been shown to be tightly linked [63, 64, 65, 66,
67]. These studies have shown how complicated evolutionary processes are, and
there are some salient lessons for evolutionary analysis more generally. Other
evidence for the diverse molecular factors affecting evolution come from long
term evolution experiments [68, 69, 70, 71, 72].

The upshot of the wealth of data, and the complicated interplay between
deterministic and stochastic effects in driving evolution, is that we need
more sophisticated maps from genotype to phenotype, especially if the view of
omnigenic inheritance persists [73, 74, 75]. Having better understanding of
how genes, their expression, and their interactions affect phenotypes may help
in resolving long standing questions in both fundamental biology, but also in
understanding the causes and mechanisms of complex diseases. Developing
mathematical models for whole cells, and linking these in multi-scale models
of organisms, is likely to form an important step in that direction.

###  3.9 Hybrid and Data Driven Modelling

Uncertainty about the correct model structure has been a common theme to many
of the instances touched upon above. We often lack the knowledge to construct
meaningful mathematical models; model selection can help to rule out some
models. Automated model development and multi-scale models are opening up new
opportunities to increase our universe of mathematical models for biological
systems. Hybrid models [76, 77], see Figure 1, offer another route towards
building better models.

Figure 1: (A) We are used to modelling biological systems as, for example,
systems of coupled ordinary differential equations. (B) Hybrid models combine
statistical models with mechanistic models, and allow investigation of complex
systems where only part of the mechanistic relationship is sufficiently well
developed to be modelled mechanistcially.

These models contain mechanistic elements that interface with purely data
driven, statistical, or black-box models, to combine the benefits of both
approaches. Purely data driven statistical models can provide a baseline
against which we compare — using for example information-theoretic measures
such as the Kullback–Leibler divergence [78] — mechanistic and hybrid models.
Further, we can develop data driven (or deductive) modelling approaches as
potential emulators to feed back into, for example, the statistical inference
approaches where computationally costly simulation is replaced by statistical
(“black box”) models [79].

Flexibility in choosing between mechanistic and data-driven modelling
approaches, and the ability to combine them can open up powerful new ways to
study, explain, and predict the behaviour of biological systems. Statistical
“black-box” and machine learning models, by themselves, provide little direct
insight into the underlying causal mechanisms that link genetic components to
organismic behaviour. Analysing the resulting models with “white box”
approaches including saliency maps or input-masked gradients provide important
mechanistic, testable, and predictive insights into the critical regulatory
regions and their interactions. Knowing the statistical dependencies should
ultimately serve as a guide towards generating causal hypotheses for
mechanistic/mathematical/functional relationships. We can use hybrid models to
test the robustness of learned design principles: the mechanistic model embeds
the inferred design principle, and the statistical model the physiological or
environmental context (akin to, for example, extrinsic noise in gene
expression).

The concept of neural differential equations [80, 81], where the right-hand
side of the differential equation, $dx/dt=f(x;\theta)$ is replaced by a neural
network,

$\frac{dx}{dt}=\text{NN},$ (10)

holds the potential to develop new forms of hybrid models, all from within the
same dynamical systems perspective.

Both mechanistic modelling and data-driven/machine learning modelling are well
developed in their own right. Hybrid modelling, by contrast, is still in its
infancy and the particular combination of hypothesis-driven and data-driven
modelling required to develop useful models is only beginning to be studied.
There are, however, exciting opportunities to bring methods from, for example,
control engineering [82, 83, 84] to bear on hybrid modelling attempts. The use
of a hybrid-Ansatz in modelling may also resolve some of the outstanding
issues in parameter and model identifiability [13, 85], where the non-
identifiable parts could, or should, perhaps be modelled using a machine
learning approach.

##  4 A New Mathematical Biology

In this section we change pace and consider some broad mathematical and
theoretical ideas for new approaches to mathematical biology. While this
discussion is largely from the perspective of pure mathematics, it provides a
reflective analysis of the possibilities for the role of mathematics in
biology.

###  4.1 Overview

Mathematical biology has predominantly avoided the explicit consideration of
life. This paradoxical situation reflects a fundamental deficiency not only of
mathematical biology but also, though perhaps to a lesser extent, of
theoretical and experimental biology. Of primary concern here is the absence
of a general theoretical definition of life and consequent realisation within
an appropriate mathematical framework. While progress has been made, including
the _relational biology_ by theoretical physicist Nicolas Rashevsky [86] and
theoretical biologist Robert Rosen [87], the _hypercycle_ by biophysical
chemist Manfred Eigen [88], and the _chemoton_ by theoretical biologist Tibor
Gánti [89], such a key concern is inexplicably the province of only a select
few.

Definitions serve to identify and provide the foundation for explanation and
knowledge. That a general definition of life is wanting is attendant to the
complexity of the emergent phenomenon of life, though arguably more so to the
methodological reductionism that has dominated scientific practice,
particularly within biology and related fields, in recent decades.
Reductionism has been preeminent in experimental biology, and consequently
biologists collect enormous quantities of experimental data often without an
integrated perspective of the living organism, thus without progression
towards knowledge of the phenomenon of life [90].

While the ultimate ambition of biology is a general theory of life, this
remains an open problem: even general concepts and principles are absent [91].
Mathematical biology, along with theoretical biology, can assume a fundamental
role in developing a theory of life. Much of current mathematical biology,
however, is subject to the same shortcomings as experimental biology. To make
progress mathematical biology may need to consider three main aspects, namely
emergent behaviour, mathematical maturity, and unification, which together
form a basis for a more functional and effective approach to the study of
living organisms with mathematics.

###  4.2 More is different

In 1972, Philip W. Anderson111In 1977 P.W. Anderson shared the Nobel Prize in
Physics with Edmund’s maternal grand father, Sir Nevill F. Mott, and J.H. Van
Vleck. published his article “More is different” [9], which became a highly
influential exposition on the limitations of methodological reductionism and
the importance of emergence in understanding complex phenomena. Anderson
relates the hierarchical progression from simplicity to complexity with the
progression from reduction to emergence [92, Chapter II, page 110]. New levels
of system complexity are then associated with the emergence of novel
phenomena, concepts, principles, and functions that are consistent with,
though not readily explained by, the lower levels of the system. While
emergent properties are often associated with the hierarchical, or multiscale,
structure of a system [92, Chapter III, page 139], there are alternative
perspectives that attempt to avoid any potential circularity arising from this
association [93].

Reductionism, where the intention is to explain phenomena via upward causation
in relation to underlying processes, has been very effective in science. The
limitations of reductionism, however, are readily apparent for both simple and
complex systems. For example, copper metal has the properties of being red,
shiny, malleable, ductile, and conductive, while individual copper atoms
exhibit none of these properties that emerge when many copper atoms form the
metal structure [92, Chapter III, page 143]. Superconductivity provides an
example of emergence in a more complex system: detailed computation of the
motions of all electrons and ions in a superconducting material could describe
the known phenomenology of superconductivity, however the computation cannot
reveal the cause of superconductivity, namely the phenomenon of symmetry
breaking. This is simply not captured by the framework at this lower physical
level [92, Chapter III, page 136]. At much greater complexity, the behaviour
of a living cell is an emergent property of the enormous number of molecules
of which it consists [94]. Cells are regarded as the smallest functional unit
of life, a property not associated with, and not readily predictable from,
their constituent molecules.

Different types of emergence have been defined, including pragmatic, nominal,
weak, and strong emergence. Strong emergence describes the appearance of an
emergent phenomenon that cannot be described with a reductive account, and may
be associated with downward causation and emergent forces or physical laws.
Weaker forms of emergence are not necessarily associated with novel forces or
physical laws. In particular, weak emergence describes a phenomenon that
arises in an unexpected or very complicated manner at a higher level of
complexity, and would not likely be evident without direct observation, even
though a reductive understanding of the physics of the lower levels is in
principle sufficient to describe the phenomenon [95]. A proposition for the
existence of an emergent phenomenon must be accompanied by a rigorous
explanation, via a detailed physical mechanism at the level of emergence, for
the qualitative difference between the emergent phenomenon and the underlying
levels. In Figure 2 we illustrate the complementary perspectives of
methodological reductionism and weak emergence.

Figure 2: Illustration of methodological reductionism and weak emergence.
Reductionism assumes that all complex behaviour can be described from
knowledge of the underlying constituents. While weak emergence is consistent
with the properties of the underlying constituents the emergent behaviour,
which may correspond to novel principles of structure, organisation, or
function, is not readily described from knowledge of the individual
constituents.

A fundamentally important reason for identifying weak emergence is that it may
allow for the explanation of a phenomenon without the need to consider the
reductive understanding of the lower levels, thereby allowing for a much
simplified mathematical description or simulation. Identifying emergence
thereby provides for potential flexibility and simplification in describing a
complex system. For example, we can consider the motility of a living cell
without knowing the movements of all constituent molecules.

While the concept of emergence has been incorporated into many areas of
physics, the same cannot be said for biology and related fields, notably
mathematical biology. Biologists generally believe that, given a sufficient
understanding of chemistry, molecular biology, and cell biology, a complete
explanation of living organisms will be achieved through reductive
investigation. This is particularly evident in genetics and related areas
where the tendency is to believe that understanding gene or protein sequences
will lead to a complete explanation of life. To the contrary, the biologist
Lynn J. Rothschild illustrates the numerous examples of form and function in
biological systems for which the concept of emergence may, if not should, be
employed [96, Chapter 6, page 151]. Moreover, the evolutionary biologist Ernst
Mayr recognised the ubiquity of emergent behaviour in biological systems, and
therefore included emergence as one of six reasons underlying the difficulties
in predicting the behaviour of biological systems [97, Chapter 2, pages
58–59].

While reductionism has been very successful in yielding knowledge of many
biochemical and cellular processes underlying biological systems, the
limitations of reductionism are now apparent from the seemingly underestimated
complexity of these systems [98]. By identifying emergent phenomena we can
explain the properties of a biological system without the limitations of
accounting for the complexity of the underlying constituents. Whether or not
living organisms can be explained in principle from biochemistry, and thereby
the laws of physics, is not of concern from the pragmatic perspective of
utilising emergence to simplify and expedite the process of understanding
biological systems.

Embracing emergence in biology does not dispel the need for much of current
mathematical biology, but requires the additional consideration of identifying
emergent phenomena at higher levels of system complexity. The latter is a very
difficult program that requires new theoretical and mathematical ideas, some
of which we have outlined above. The result, however, will be a much more
sophisticated understanding and description of biological systems that yields
mathematical models which avert prohibitively complicated computations. For
this, rigorous mathematical definitions of emergence and emergent properties
are required, and while there are some initial considerations [99, 93],
further work is required.

To summarise with the words of Anderson on the “triumph of emergence over
reductionism”: “… large objects such as ourselves are, in myriad ways, the
product of principles of organization and of collective behavior which in no
meaningful sense can be reduced to the behavior of our elementary
constituents. Large objects are often more constrained by those principles
than by what the principles act upon.” [92, Chapter 4, Pages 195–196].

###  4.3 Mathematical Maturity

The list of articles in any issue of the Journal of Mathematical Physics
illustrates the mathematical maturity enjoyed by theoretical physics, notably
the tendency for mathematical rigour including the use of the definition-
theorem-proof style of pure mathematics [100], and the use of sophisticated
mathematical theory. Currently, only a relatively small proportion of the
research literature in mathematical biology exhibits such maturity.

Nicolas Rashevsky, one of the pioneers of mathematical biology, envisioned
that the relationship between mathematical biology and experimental biology
would correspond to that between mathematical physics and experimental physics
[101]. That the vision of Rashevsky has been realised in such limited ways is
probably due to several reasons, which we have already touched on above:
biological systems are generally more complex than inanimate systems;
avoidance by mathematicians who regard biology as unsophisticated and lacking
rigorous definitions; strictly reductionist approaches, possibly influenced by
experimental biology; insufficiency of deep theorising about living organisms,
and in particular the formation of rigorous definitions; imprudent importation
of mathematical models from other fields, such as physics and engineering;
complacency with modelling simple biological systems with simple models and
simple mathematics; indifference, or inability, toward applying more
sophisticated mathematics or developing new mathematics.

The complexity of living organisms relative to physical systems suggests the
need for the application of mathematical theory that is even more
sophisticated, as well as the need for new mathematical ideas leading to new
areas of mathematics [2, 102, 103]. For this, rigorous definitions of
biological concepts are a necessity, and without such rigour the progression
of mathematical biology, and how it supports experimental biology, is clearly
limited. Mathematical biology requires new mathematical theory [103], and the
passive importation of mathematical models from other fields, particularly
physics and engineering, and their application in superficial ways to
biological systems runs counter to the constructive progression of
mathematical and theoretical biology. Capturing the intricacies of living
systems, especially their emergent properties, will require but also inspire
such new mathematics.

There is, we feel, increasing need of employing the definition-theorem-proof
style of mathematics in mathematical biology. The general lack of rigorous
definitions in biology is of fundamental concern for mathematical biology
since all mathematical assertions originate from definitions, and the
requirement to state clear definitions in mathematical biology would stimulate
much needed theoretical thought regarding biological concepts. Further, a
theorem asserts a general principle, and formulating a mathematical result as
a theorem encourages the discovery of more general ideas rather than
individual cases, demonstrates rigour, allows for immediate connection with
other mathematical theory, simplifies the exposition, and may occasion a
deeper understanding of the mathematics and the corresponding system. Rigorous
mathematics, consisting of the definition-theorem-proof style is indubitably
of importance for the success of mathematical biology [104].

###  4.4 Unification

Given the lack of general theoretical frameworks in biology [105] it is of no
surprise that biological knowledge is fragmented, even to the extent that
different subfields routinely use distinct terms for the same concept [106,
107]. In principle, mathematical and theoretical biology should be placed to
remedy this situation, however they are arguably in a similar position which
may only serve to further this fragmentation. The fundamental need for
unification, or integration, within mathematical biology has long been
realised, however little progress has been made. Here we discuss three
particular areas associated with mathematical biology for which unification is
imperative: biological data, mathematical models, and system perspective.

In his 2002 Nobel lecture, molecular biologist Sydney Brenner said that “We
are all conscious today that we are drowning in a sea of data and starving for
knowledge” [108]. Rather than mere descriptions and routine analysis of
biological data, deep theorising and integration of the often heterogeneous
data is required to yield knowledge [109]. Mathematical modelling can be of
much assistance here, particularly regarding the assessment of consistency
between a hypothesis and the experimental data, or for the identification of
design principles. Mathematical models are fundamental for developing
knowledge of an observed biological phenomenon, such as the identification of
design principles. Given a causative hypothesis for a phenomenon we can
construct a mathematical model that represents the hypothesis explicitly, and
then compare model predictions with experimental data to determine whether the
hypothesis is consistent, or not, with the evidence. In opposition to much of
current practice in mathematical biology, however, integration of
heterogeneous data requires the consideration of multiple insightful
hypotheses, and therefore multiple models, or the consideration of a single
hypothesis for consistency with a broad range of experimental data.
Consideration of a single hypothesis/model within the context of a single data
set, or multiple data sets that are closely related, achieves little toward
data integration and novel biological theory. The objective is the development
of new theoretical knowledge: being able to simulate a phenomenon does not in
itself correspond to an explanation of the phenomenon.

The unification of mathematical models is at the frontier of mathematical
biology, and general methodologies for comparing the underlying conceptual
structure of models, irrespective of mathematical formalism, provide essential
facilitation for the process [38, 39]. Model comparison can partition a set of
models into categories where similar models, regarded as sharing an essential
qualitative characteristic, belong to the same category. Similar models may
simply be incremental variations of each other, or may originally appear
distinct and unrelated. Discovering similarities between seemingly distinct
models can lead to novel theoretical insights into biological phenomena [38],
and ultimately to a unified mathematical description of a biological system. A
perspective of unification can encourage the avoidance of excessive
incrementalism in model development and, along with a rigorous definition-
theorem-proof style, yield general mathematical frameworks for biological
systems, rather than the essentially algorithmic approach of developing simple
models of simple systems.

It is widely recognised that to understand a biological phenomenon it must be
perceived as an integrated whole [86]. For example, a biological function has
meaning for an organism at the level at which the function is integrated, and
below this level the function does not exist [110, 111, 112]. Ecology has, of
course, long recognised this. Comprehending the integration is particularly
difficult since at a given level of organisation in a biological system the
biological constituents are generally heterogeneous [2], and naturally span
several scales. Most of current mathematical biology is either concerned with
“life-less” systems such as biochemical networks, or considers living
organisms (such as a cell) as inanimate objects. The latter is a pragmatic and
justified choice when the aim is to develop models that replicate, rather than
explain, the behaviour of living systems, for example cell migration in
response to gradients. But it falls way short of describing essential
hallmarks of life. Here there appears to be genuine scope for a new
mathematical biology that captures such hallmarks, including life as an
emergent phenomenon.

##  5 Conclusion

Open problems in research can be interpreted as gaping holes in our
understanding, or as exciting opportunities for future research. Most of the
problems discussed here have aspects of both. There are considerable technical
and computational challenges, but also some conceptual challenges of a
mathematical nature, related to model development and model calibration. There
is a definite appeal of using and combining methods from dynamical systems
theory, computer science, statistical inference and machine learning to
develop better, more reproducible methods of model development. Multiscale
modelling, hybrid models, model selection, and automated generation of models
individually, but especially when taken together, offer enormous potential to
expand the scope of mathematical biology. We believe that the ability to
explain, predict and control living systems requires a broadening of the
toolbox of mathematical biologists.

In our view the appeal of combining hypothesis-driven and data-driven
modelling frameworks is considerable. The technical challenges in doing so are
also considerable. Data-driven modelling has seen a rapid rise over the past
decade and deep neural networks have become powerful tools in biology [113,
114, 115]; and their predictive performance is perhaps more immediately
persuasive than are the mechanistic insights that modelling can provide. But
both are needed [109].

We believe that a new mathematical biology will be required to understand life
at the fundamental level we discussed in the previous section. Since this
requires the substantial development of new biological and mathematical
theory, we have little concrete idea of what this will look like. Anderson’s
work [9], maybe coupled to thermodynamic considerations [43], provides a
starting perspective: emergent phenomena, including life itself, are not
readily explained from principles that are detected by reductionist
approaches.

## Acknowledgements

The authors thank the members of the Theoretical Systems Biology group, past
and present, for stimulating discussions that have shaped our view. Much of
the conceptual work has been the result of our ongoing collaborations,
especially in MACSYS. Both authors acknowledge the financial support from the
Volkswagen Foundation through a Life? program grant.

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