Barry Cooper, S

Maini, P. K.

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S. Barry Cooper

Philip K. Maini

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Interface Focus

The mathematics of nature at the Alan Turing centenary

Mathematical Institute Eprints Archive

doi: 10.1098/rsfs.2012.0018
, 393-396 first published online 4 April 20122 2012 Interface Focus
 
S. Barry Cooper and Philip K. Maini
 
The mathematics of nature at the Alan Turing centenary
 
 
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Interface Focus (2012) 2, 393–396
 on June 18, 2012rsfs.royalsocietypublishing.orgDownloaded from doi:10.1098/rsfs.2012.0018
Published online 4 April 2012INTRODUCTION*Author for c
One contribu
Turing centenThe mathematics of nature at the Alan
Turing centenary
S. Barry Cooper* and Philip K. Maini
School of Mathematics, University of Leeds, Leeds LS2 9JT, UKKeywords: Alan Turing; morphogenesisAlan Turing inhabited a very individual world between
the abstract and the concrete—see Mrs Turing’s sketch
of her son playing hockey in 1923 (ﬁgure 1). His 1936 uni-
versal Turing machine detached computation from the
actuality of its physical host, and established a powerful
computational paradigm, dominated by logical structure.
The grip of this model on our thinking and practice con-
tinues to this day. At other times, Turing immersed
himself in the embodying engineering that enabled the
ﬁrst real-life computers. And in Manchester in the early
1950s, in one of the more remarkable switches of focus
in scientiﬁc history, he turned his attention from man-
made computing machines to the hidden mathematics
of emergent form in nature. The one published paper aris-
ing from this interest became one of the most cited papers
in science, and potentially challenges the primacy of his
own universal machine.
Turing’s 1952 paper, ‘The chemical basis of morpho-
genesis’, was ground-breaking as it proposed for the ﬁrst
time that biological patterning arose as a self-organized
emergent phenomenon in which, counterintuitively,
instabilities leading to spatial pattern arise via the inter-
action of stablizing processes. The model, in its simplest
form of a coupled system of reaction–diffusion equations,
assumes that a chemical pre-pattern is set up to which
cells respond by differentiating in a concentration-
dependent manner. This paper has led to a rich literature
in analyses (both mathematical and computational) of
the model and variants thereof—together with bio-
logical/chemical experiments aiming to lend support to
the model by identifying the chemicals (termed morpho-
gens by Turing) that may be involved. The Turing
centenary selection of papers here illustrate how the
model has been applied in biology and how, as experimen-
tal techniques have become more sophisticated increasing
biological data, the models have been updated through
identiﬁcation of possible morphogens, and extensions of
the basic model to include other mechanisms.orrespondence (pmt6sbc@leeds.ac.uk).
tion of 13 to a Theme Issue ‘Computability and the
ary’.
393Our introductory paper is by James Murray [1], one of
the formative inﬂuences in the area. It puts in context
the work of Turing by providing a broad overview
of the development of the ﬁeld of mathematical biology
since the time of Turing. It then considers two exam-
ples in depth. The ﬁrst applies Turing’s idea to animal
coat markings. The second is in the emerging ﬁeld of
mathematical applications in medicine, with a reaction–
diffusion model for the extensive invasion of tumour
cells into regions of the brain during glioblastoma. This
modelling, now being used to help neuroscientists, is
based on the formation of travelling waves. In fact,
Turing identiﬁed such behaviour as a possible emergent
property of his system, but the model here uses a dif-
ferent approach, more appropriate for the particular
biological application.
Turing’s approach was really brought to the attention
of thewider readership in the 1970s by a number of impor-
tant articles. Among these, the body of work of Hans
Meinhardt [2] played a crucial role. In his paper
Meinhardt shows how the initially biologically unrealistic
kinetics used by Turing can be expanded into biologically
relevant kinetics that still preserve the fundamen-
tal principles of his theory. This approach leads to the
patterning principle of ‘short-range-activation, long-
range-inhibition’ which has been instrumental in helping
experimentalists to identify possible Turing morphogen
pairs. Meinhardt’s paper considers a number of examples,
including regulation and formation of tentacles in Hydra,
establishment of polarity and shell pigmentation pattern
formation.
Szalai and co-authors [3] present a semi-empirical
method for determining how stationary patterns
may arise owing to a Turing mechanism. They verify
the model and approach by demonstrating statio-
nary patterns in two examples of halogen-free redox
reaction systems. They also show how to obtain
localized structures.
The article by Painter et al. [4] presents a review of
models based on Turing’s paper, paying particu-
lar attention to morphogenesis of skin organs (hair
follicles and feather germs). The model is brought upThis journal is q 2012 The Royal Society
Figure 1. Drawing of Alan Turing by his mother, 1923. Courtesy of Sherborne School.
394 Introduction. Alan turing centenary S. B. Cooper and P. K. Maini
 on June 18, 2012rsfs.royalsocietypublishing.orgDownloaded from to date by considering key pathways that have
been identiﬁed as having appropriate activator–
inhibitor kinetics for a Turing-type patterning insta-
bility. It is shown that the model captures core
experimentally observed features and the test–predict–
reﬁne cycle of iteration between experiment and theory
reveals that there may be spatial heterogeneity already
present in the domain which the model reﬁnes. As
Turing himself said, recognizing the biological limit-
ations of his de novo patterning theory, ‘Most of an
organism, most of the time, is developing from one pat-
tern into another, rather than from homogeneity into
a pattern’.
Yoshimoto & Kondo [5] consider patterning in
insects. While it has been shown that patterning
in the early Drosophila is most likely not from a
Turing mechanism, but from a complex gradient
model, the authors argue that models of the latterInterface Focus (2012)type are unable to explain vein patterns in certain
insects. They show how a Turing-type model coupled
with a gradient model can yield patterns consistent
with experimental observations.
The paper by Chen et al. [6] on ‘Patterns of perio-
dic holes created by increased cell motility’ extends
the Turing framework to pattern formation in cul-
tured vascular mesenchymal cells (VMCs). Their
reaction–diffusion model (the Gierer–Meinhardt
model) is applied to the morphogens BMP-2 and
MGP and many of the model parameters are
determined experimentally. They validate key nonli-
nearities in this model by experiment and then
consider a model in which the VMC density is coupled
to the reaction–diffusion model via chemotaxis. An
experimental assay is set up to determine the diffusion
coefﬁcient for the cell density and analysis of the
coupled model then shows how altering cell motility
Introduction. Alan turing centenary S. B. Cooper and P. K. Maini 395
 on June 18, 2012rsfs.royalsocietypublishing.orgDownloaded from can profoundly affect the form of the patterns
exhibited. This research could have important
implications for tissue engineering.
One of the major controversial issues regarding
Turing’s original model is its inability to produce pat-
terns that are robust to natural biological variations
present during development. Two papers look at dif-
ferent aspects of this problem. Kank, Zheng and
Othmer [7] investigate the robustness of the location
of threshold boundaries to perturbations in parameters
and boundary inputs for a number of chemical pre-
pattern models in both deterministic and stochastic
settings. In particular, it investigates the ability of
different types of response functionals to dampen the
effects of noise. The paper by Maini and his co-authors
[8] on ‘Turing’s model for biological pattern formation
and the robustness problem’ investigates robustness of
patterns produced on growing domains in the face
of stochastic noise and expression delay. While
domain growth can greatly enhance the robustness of
Turing patterns in the deterministic system (without
delay), it is shown that the situation becomes less
straightforward when stochasticity is introduced.
The subsequent articles in this tribute to Alan
Turing’s inﬂuence, and the continued fruitful inter-
action between theory and nature, trace a less direct,
but still fundamental impact on the science.
Ehud Shapiro’s [9] ‘A Mechanical Turing Machine:
Blueprint for a Biomolecular Computer’ is already
something of an unpublished classic. It describes a
mechanical device based on the ideas of Turing’s 1936
paper, an embodied biomolecular Turing machine.
As one of the referees commented: ‘This paper is a
coherent articulation of a theoretical vision for molecu-
lar computing machines that operate within cells,
written years prior to the author’s seminal experimental
work (with Kobi Benenson et al.) that helped launch a
wave of research into disease-diagnosing-and-curing
biochemical circuits. . . . It would be good to publish
it, at long last—and it would be especially nice for the
paper to appear in a Turing Centenary issue’. Among
a number of improvements in presentation, the author
has updated the original manuscript with a Post-
script telling how the ﬁeld has progressed since it was
originally written. We follow the referee in hoping
that the appearance of this paper will reinforce the per-
ception of the author ‘as a visionary and luminary in
this burgeoning ﬁeld’.
Natasha Jonoska and Nadrian Seeman [10] also
take the Turing research programme to the molecular
level, reporting on recent work in relation to two
computing models by DNA self-assembly, while pro-
viding a nice introduction to current directions in
the area. According to the authors: ‘In the last few
decades, research done in biology, chemistry and
physics has resulted in an explosion of new ﬁndings
about molecular interactions. These ﬁndings often
reveal transfer of information at a molecular level result-
ing in proliferation of the science of computing within
established, on a ﬁrst glance unrelated, scientiﬁc
ﬁelds. The notion of ‘computing’, up till recently a
theoretical concept, acquires a new meaning within
these intrinsically experimental disciplines’. Turing wouldInterface Focus (2012)surely have found this incarnation of self-assembly
interesting, and would have appreciated the role
of experiment in the authors’ approach to the
challenges arising.
The paper of Anne Condon and her co-authors [11]
‘Less haste, less waste: On recycling and its limits in
strand displacement systems’, is more concerned with
the operative efﬁciency of recent incarnations of DNA
computation. The technical content, focused on DNA
strand displacement, is inevitably beyond anything,
Turing could have envisaged, with the work of
Watson, Crick, Wilkins and Franklin having only
appeared a year before Turing died. In fact, the paper
hardly mentions Turing. But his spirit can still be
detected in this visceral take on the resource-related
capabilities of DNA computing.
Dorner, Goold and Vedral’s [12] ‘Toward quantum
simulations of biological information ﬂow’ inhabits Tur-
ing’s multi-disciplinary terrain in a very contemporary
and ﬁttingly adventurous manner. It investigates trans-
port in biological molecules based on the hypothesis
that it may be possible that biological transport pro-
cesses operate between purely classical diffusion and
ballistic motion, exploiting quantum coherence. The
authors propose an analogue quantum simulator to
study electron transport in biology. The paper can be
seen as a signiﬁcant step towards the realization of ana-
logue quantum simulators using existing technology,
while providing new insights into the nature of quan-
tum effects in biology.
Alan Turing had lifelong interest in quantum mech-
anics and the computational content of biology: which
makes this cutting-edge research an appropriate
conclusion to an interesting centenary tribute.REFERENCES
1 Murray, J. D. 2012 Vignettes from the ﬁeld of mathe-
matical biology: the application of mathematics to
biology and medicine. Interface Focus 2, 397–406.
(doi:10.1098/rsfs.2011.0102)
2 Meinhardt, H. 2012 Turings’ theory of morphogenesis of
1952 and the subsequent discovery of the crucial role of
local self-enhancement and long-range inhibition. Inter-
face Focus 2, 407–416. (doi:10.1098/rsfs.2011.0097)
3 Szalai, I., Cuin˜as, D., Taka´cs, N., Horva´th, J. &
De Kepper, P. 2012 Chemical morphogenesis: recent
experimental advances in reaction–diffusion system
design and control. Interface Focus 2, 417–432. (doi:10.
1098/rsfs.2012.0010)
4 Painter, K. J., Hunt, G. S., Wells, K. L., Johansson, J. A. &
Headon, D. J. 2012 Towards an integrated experimental–
theoretical approach for assessing the mechanistic basis of
hair and feather morphogenesis. Interface Focus 2,
433–450. (doi:10.1098/rsfs.2011.0122)
5 Yoshimoto, E. & Kondo, S. 2012 Wing vein patterns
of the Hemiptera insect Orosanga japonicus differ among
individuals. Interface Focus 2, 451–456. (doi:10.1098/
rsfs.2011.0112)
6 Chen, T.-H., Guo, C., Zhao, X., Yao, Y., Bostro¨m, K. I.,
Wong, M. N., Tintut, Y., Demer, L. L., Ho, C.-M. &
Garﬁnkel, A. 2012 Patterns of periodic holes created by
increased cell motility. Interface Focus 2, 457–464.
(doi:10.1098/rsfs.2012.0001)
396 Introduction. Alan turing centenary S. B. Cooper and P. K. Maini
 on June 18, 2012rsfs.royalsocietypublishing.orgDownloaded from 7 Kang,H.-W.,Zheng,L.&Othmer,H.G.2012Theeffectof the
signalling scheme on the robustness of pattern formation in
development. Interface Focus 2, 465–486. (doi:10.1098/rsfs.
2011.0116)
8 Maini, P. K., Woolley, T. E., Baker, R. E., Gaffney, E. A.
& Lee, S. S. 2012 Turing’s model for biological pattern for-
mation and the robustness problem. Interface Focus 2,
487–496. (doi:10.1098/rsfs.2011.0113)
9 Shapiro, E. 2012 A mechanical Turing machine: blueprint
for a biomolecular computer. Interface Focus 2, 497–503.
(doi:10.1098/rsfs.2011.0118)Interface Focus (2012)10 Jonoska, N. & Seeman, N. C. 2012 Computing by
molecular self-assembly. Interface Focus 2, 504–511.
(doi:10.1098/rsfs.2011.0117)
11 Condon, A., Hu, A. J., Manˇuch, J. & Thachuk, C. 2012
Less haste, less waste: on recycling and its limits in
strand displacement systems. Interface Focus 2,
512–521. (doi:10.1098/rsfs.2011.0106)
12 Dorner, R., Goold, J. & Vedral, V. 2012 Towards
quantum simulations of biological information ﬂow.
Interface Focus 2, 522–528. (doi:10.1098/rsfs.2011.
0109)


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