Turing Model Research Articles

Structural insights into HetR-PatS interaction involved in cyanobacterial pattern formation.

The one-dimensional pattern of heterocyst in the model cyanobacterium Anabaena sp. PCC 7120 is coordinated by the transcription factor HetR and PatS peptide. Here we report the complex structures of HetR binding to DNA, and its hood domain (HetRHood) binding to a PatS-derived hexapeptide (PatS6) at 2.80 and 2.10 Å, respectively. The intertwined HetR dimer possesses a couple of novel HTH motifs, each of which consists of two canonical α-helices in the DNA-binding domain and an auxiliary α-helix from the flap domain of the neighboring subunit. Two PatS6 peptides bind to the lateral clefts of HetRHood, and trigger significant conformational changes of the flap domain, resulting in dissociation of the auxiliary α-helix and eventually release of HetR from the DNA major grove. These findings provide the structural insights into a prokaryotic example of Turing model.

Upper Bounds for Newton’s Method on Monotone Polynomial Systems, and P-Time Model Checking of Probabilistic One-Counter Automata

A central computational problem for analyzing and model checking various classes of infinite-state recursive probabilistic systems (including quasi-birth-death processes, multitype branching processes, stochastic context-free grammars, probabilistic pushdown automata and recursive Markov chains) is the computation of termination probabilities, and computing these probabilities in turn boils down to computing the least fixed point (LFP) solution of a corresponding monotone polynomial system (MPS) of equations, denoted x = P ( x ). It was shown in Etessami and Yannakakis [2009] that a decomposed variant of Newton’s method converges monotonically to the LFP solution for any MPS that has a nonnegative solution. Subsequently, Esparza et al. [2010] obtained upper bounds on the convergence rate of Newton’s method for certain classes of MPSs. More recently, better upper bounds have been obtained for special classes of MPSs [Etessami et al. 2010, 2012]. However, prior to this article, for arbitrary (not necessarily strongly connected) MPSs, no upper bounds at all were known on the convergence rate of Newton’s method as a function of the encoding size |P| of the input MPS, x = P ( x ). In this article, we provide worst-case upper bounds, as a function of both the input encoding size |P|, and ε > 0, on the number of iterations required for decomposed Newton’s method (even with rounding) to converge to within additive error ε > 0 of q*, for an arbitrary MPS with LFP solution q*. Our upper bounds are essentially optimal in terms of several important parameters of the problem. Using our upper bounds, and building on prior work, we obtain the first P-time algorithm (in the standard Turing model of computation) for quantitative model checking, to within arbitrary desired precision, of discrete-time QBDs and (equivalently) probabilistic 1-counter automata, with respect to any (fixed) ω -regular or LTL property.

What has mathematics done for biology?

“Whathasmathematicsdoneforbiology?”isaquestionthateverymathematicalbiol-ogisthasbeenaskedatsometime,orhasaskedthemselves.Whilebioinformaticshasbeen very successful and widely accepted in biology, the acceptance of mathematicalbiology has been slower. Of course, there are notable exceptions—in ecology andepidemiology there is a long history of mathematical modelling [see, e.g., the booksby Murray (2002, 2003) and May and McLean (2007)], in physiology the Hodgkin–Huxley model is well known [see, e.g., the book by Keener and Sneyd (2009)], whilein pattern formation the Turing model (Turing 1952), while still controversial, hascertainlystimulatedanenormousamountofexperimentalactivityandledtothepara-digm patterning principle of short-range activation, long-range inhibition (Gierer andMeinhardt 1972).These successes for mathematical biology are significant, but still relatively rare,bearing in mind that the field has matured and grown substantially over the past50years. In fact, during the past 20years bioinformatics has come into being andtaken over mathematical biology in the race to the laboratory. There are a number ofreasonsforthis:(1)therehavebeenfewmajorproblemsthatmathematicalbiologyhassuccessfullyaddressed(comparedwiththegenomeproject),(2)forvalidation,modelsinmathematicalbiologytypicallyrelyonspatiotemporallyresolveddataand,todate,datatendtobestatic(electrophysiologyisanexception)andthereforemoreamenableto the tools of bioinformatics. However, mathematical biology has made numerousadvancesinbiology,andadvancesintechnology(particularlyimaging)areleadingto

Forging patterns and making waves from biology to geology: a commentary on Turing (1952) 'The chemical basis of morphogenesis'.

Alan Turing was neither a biologist nor a chemist, and yet the paper he published in 1952, ‘The chemical basis of morphogenesis’, on the spontaneous formation of patterns in systems undergoing reaction and diffusion of their ingredients has had a substantial impact on both fields, as well as in other areas as disparate as geomorphology and criminology. Motivated by the question of how a spherical embryo becomes a decidedly non-spherical organism such as a human being, Turing devised a mathematical model that explained how random fluctuations can drive the emergence of pattern and structure from initial uniformity. The spontaneous appearance of pattern and form in a system far away from its equilibrium state occurs in many types of natural process, and in some artificial ones too. It is often driven by very general mechanisms, of which Turing's model supplies one of the most versatile. For that reason, these patterns show striking similarities in systems that seem superficially to share nothing in common, such as the stripes of sand ripples and of pigmentation on a zebra skin. New examples of ‘Turing patterns' in biology and beyond are still being discovered today. This commentary was written to celebrate the 350th anniversary of the journal Philosophical Transactions of the Royal Society.

A computational analysis of bone formation in the cranial vault in the mouse.

Bones of the cranial vault are formed by the differentiation of mesenchymal cells into osteoblasts on a surface that surrounds the brain, eventually forming mineralized bone. Signaling pathways causative for cell differentiation include the actions of extracellular proteins driven by information from genes. We assume that the interaction of cells and extracellular molecules, which are associated with cell differentiation, can be modeled using Turing’s reaction–diffusion model, a mathematical model for pattern formation controlled by two interacting molecules (activator and inhibitor). In this study, we hypothesize that regions of high concentration of an activator develop into primary centers of ossification, the earliest sites of cranial vault bone. In addition to the Turing model, we use another diffusion equation to model a morphogen (potentially the same as the morphogen associated with formation of ossification centers) associated with bone growth. These mathematical models were solved using the finite volume method. The computational domain and model parameters are determined using a large collection of experimental data showing skull bone formation in mouse at different embryonic days in mice carrying disease causing mutations and their unaffected littermates. The results show that the relative locations of the five ossification centers that form in our model occur at the same position as those identified in experimental data. As bone grows from these ossification centers, sutures form between the bones.

Mathematically guided approaches to distinguish models of periodic patterning.

How periodic patterns are generated is an open question. A number of mechanisms have been proposed--most famously, Turing's reaction-diffusion model. However, many theoretical and experimental studies focus on the Turing mechanism while ignoring other possible mechanisms. Here, we use a general model of periodic patterning to show that different types of mechanism (molecular, cellular, mechanical) can generate qualitatively similar final patterns. Observation of final patterns is therefore not sufficient to favour one mechanism over others. However, we propose that a mathematical approach can help to guide the design of experiments that can distinguish between different mechanisms, and illustrate the potential value of this approach with specific biological examples.

An interplay of geometry and signaling enables robust lung branching morphogenesis.

Early branching events during lung development are stereotyped. Although key regulatory components have been defined, the branching mechanism remains elusive. We have now used a developmental series of 3D geometric datasets of mouse embryonic lungs as well as time-lapse movies of cultured lungs to obtain physiological geometries and displacement fields. We find that only a ligand-receptor-based Turing model in combination with a particular geometry effect that arises from the distinct expression domains of ligands and receptors successfully predicts the embryonic areas of outgrowth and supports robust branch outgrowth. The geometry effect alone does not support bifurcating outgrowth, while the Turing mechanism alone is not robust to noisy initial conditions. The negative feedback between the individual Turing modules formed by fibroblast growth factor 10 (FGF10) and sonic hedgehog (SHH) enlarges the parameter space for which the embryonic growth field is reproduced. We therefore propose that a signaling mechanism based on FGF10 and SHH directs outgrowth of the lung bud via a ligand-receptor-based Turing mechanism and a geometry effect.

A Note on the Complexity of Comparing Succinctly Represented Integers, with an Application to Maximum Probability Parsing

The following two decision problems capture the complexity of comparing integers or rationals that are succinctly represented in product-of-exponentials notation, or equivalently, via arithmetic circuits using only multiplication and division gates, and integer inputs. Input instance: Four lists of positive integers: a 1,..., an ∈N+ n ; b 1,..., bn ∈N+ n ; c 1,..., cm ∈N+ m ; d 1, ..., dm ∈N+ m ; where each of the integers is represented in binary. Problem 1 (equality testing): Decide whether a 1 b 1 a 2 b 2⋯ anbn = c 1 d 1 c 2 d 2⋯ cmdm . Problem 2 (inequality testing): Decide whether a 1 b 1 a 2 b 2⋯ anbn ≥ c 1 d 1 c 2 d 2⋯ cmdm . Problem 1 is easily decidable in polynomial time using a simple iterative algorithm. Problem 2 is much harder. We observe that the complexity of Problem 2 is intimately connected to deep conjectures and results in number theory. In particular, if a refined form of the ABC conjecture formulated by Baker in 1998 holds, or if the older Lang-Waldschmidt conjecture (formulated in 1978) on linear forms in logarithms holds, then Problem 2 is decidable in P-time (in the standard Turing model of computation). Moreover, it follows from the best available quantitative bounds on linear forms in logarithms, namely, by Baker and Wüstholz [1993] or Matveev [2000], that if m and n are fixed universal constants then Problem 2 is decidable in P-time (without relying on any conjectures). This latter fact was observed earlier by Shub [1993]. We describe one application: P-time maximum probability parsing for arbitrary stochastic context-free grammars (where ε -rules are allowed).

The PCP Theorem for NP Over the Reals

In this paper we show that the PCP theorem holds as well in the real number computational model introduced by Blum, Shub, and Smale. More precisely, the real number counterpart $$\mathrm{NP}_{{\mathbb {R}}}$$NPR of the classical Turing model class NP can be characterized as $$\mathrm{NP}_{{\mathbb {R}}}= \mathrm{PCP}_{{\mathbb {R}}}(O(\log {n}), O(1))$$NPR=PCPR(O(logn),O(1)). Our proof structurally follows the one by Dinur for classical NP. However, a lot of minor and major changes are necessary due to the real numbers as underlying computational structure. The analogue result holds for the complex numbers and $$\mathrm{NP}_{{\mathbb {C}}}$$NPC.

Testing Turing's theory of morphogenesis in chemical cells.

Alan Turing, in "The Chemical Basis of Morphogenesis" [Turing AM (1952) Philos Trans R Soc Lond 237(641):37-72], described how, in circular arrays of identical biological cells, diffusion can interact with chemical reactions to generate up to six periodic spatiotemporal chemical structures. Turing proposed that one of these structures, a stationary pattern with a chemically determined wavelength, is responsible for differentiation. We quantitatively test Turing's ideas in a cellular chemical system consisting of an emulsion of aqueous droplets containing the Belousov-Zhabotinsky oscillatory chemical reactants, dispersed in oil, and demonstrate that reaction-diffusion processes lead to chemical differentiation, which drives physical morphogenesis in chemical cells. We observe five of the six structures predicted by Turing. In 2D hexagonal arrays, a seventh structure emerges, incompatible with Turing's original model, which we explain by modifying the theory to include heterogeneity.

Turing Pattern Formation from the Cooperation of Competition and Cross-Diffusion

This paper investigates the pattern formation in a reaction–diffusion (R-D) system where two interacting species form coupled positive and negative feedback loops. It is found that the cooperation of competition and cross-diffusion can lead to the Turing pattern formation for which an adequate set of conditions are analytically derived. Such a mechanism of generating Turing patterns is different from the case that self-diffusion is sufficient to generate Turing patterns in a paradigm model (proverbially called as the Turing model) where two interacting species constitute a single negative feedback loop. Therefore, this work not only provides another model paradigm for studying the pattern formation but also would be helpful for understanding the formation of, for example, diversiform skin patterns in the mammalian world where coupled positive and negative feedback loops are ubiquitous.

Studies on the Existence of Unstable Oscillatory Patterns Bifurcating from Hopf Bifurcations in a Turing Model

We revisit a homogeneous reaction-diffusion Turing model subject to the Neumann boundary conditions in the one-dimensional spatial domain. With the help of the Hopf bifurcation theory applicable to the reaction-diffusion equations, we are capable of proving the existence of Hopf bifurcations, which suggests the existence of spatially homogeneous and nonhomogeneous periodic solutions of this particular system. In particular, we also prove that the spatial homogeneous periodic solutions bifurcating from the smallest Hopf bifurcation point of the system are always unstable. This together with the instability results of the spatially nonhomogeneous periodic solutions by Yi et al., 2009, indicates that, in this model, all the oscillatory patterns from Hopf bifurcations are unstable.

Involvement of Delta/Notch signaling in zebrafish adult pigment stripe patterning

The skin pigment pattern of zebrafish is a good model system in which to study the mechanism of biological pattern formation. Although it is known that interactions between melanophores and xanthophores play a key role in the formation of adult pigment stripes, molecular mechanisms for these interactions remain largely unknown. Here, we show that Delta/Notch signaling contributes to these interactions. Ablation of xanthophores in yellow stripes induced the death of melanophores in black stripes, suggesting that melanophores require a survival signal from distant xanthophores. We found that deltaC and notch1a were expressed by xanthophores and melanophores, respectively. Moreover, inhibition of Delta/Notch signaling killed melanophores, whereas activation of Delta/Notch signaling ectopically in melanophores rescued the survival of these cells, both in the context of pharmacological inhibition of Delta/Notch signaling and after ablation of xanthophores. Finally, we showed by in vivo imaging of cell membranes that melanophores extend long projections towards xanthophores in the yellow stripes. These data suggest that Delta/Notch signaling is responsible for a survival signal provided by xanthophores to melanophores. As cellular projections can enable long-range interaction between membrane-bound ligands and their receptors, we propose that such projections, combined with direct cell-cell contacts, can substitute for the effect of a diffusible factor that would be expected by the conventional reaction-diffusion (Turing) model.

The sensitivity of Turing self-organization to biological feedback delays: 2D models of fish pigmentation

Turing morphogen models have been extensively explored in the context of large-scale self-organization in multicellular biological systems. However, reconciling the detailed biology of morphogen dynamics, while accounting for time delays associated with gene expression, reveals aberrant behaviours that are not consistent with early developmental self-organization, especially the requirement for exquisite temporal control. Attempts to reconcile the interpretation of Turing's ideas with an increasing understanding of the mechanisms driving zebrafish pigmentation suggests that one should reconsider Turing's model in terms of pigment cells rather than morphogens (Nakamasu et al., 2009, PNAS, 106: , 8429-8434; Yamaguchi et al., 2007, PNAS, 104: , 4790-4793). Here the dynamics of pigment cells is subject to response delays implicit in the cell cycle and apoptosis. Hence we explore simulations of fish skin patterning, focussing on the dynamical influence of gene expression delays in morphogen-based Turing models and response delays for cell-based Turing models. We find that reconciling the mechanisms driving the behaviour of Turing systems with observations of fish skin patterning remains a fundamental challenge.

On the information-based complexity of stochastic programming

Existing complexity results in stochastic linear programming using the Turing model depend only on problem dimensionality. We apply techniques from the information-based complexity literature to show that the smoothness of the recourse function is just as important. We derive approximation error bounds for the recourse function of two-stage stochastic linear programs and show that their worst case is exponential and depends on the solution tolerance, the dimensionality of the uncertain parameters and the smoothness of the recourse function.

Kidney branching morphogenesis under the control of a ligand–receptor-based Turing mechanism

The main signalling proteins that control early kidney branching have been defined. Yet the underlying mechanism is still elusive. We have previously shown that a Schnakenberg-type Turing mechanism can recapitulate the branching and protein expression patterns in wild-type and mutant lungs, but it is unclear whether this mechanism would extend to other branched organs that are regulated by other proteins. Here, we show that the glial cell line-derived neurotrophic factor–RET regulatory interaction gives rise to a Schnakenberg-type Turing model that reproduces the observed budding of the ureteric bud from the Wolffian duct, its invasion into the mesenchyme and the observed branching pattern. The model also recapitulates all relevant protein expression patterns in wild-type and mutant mice. The lung and kidney models are both based on a particular receptor–ligand interaction and require (1) cooperative binding of ligand and receptor, (2) a lower diffusion coefficient for the receptor than for the ligand and (3) an increase in the receptor concentration in response to receptor–ligand binding (by enhanced transcription, more recycling or similar). These conditions are met also by other receptor–ligand systems. We propose that ligand–receptor-based Turing patterns represent a general mechanism to control branching morphogenesis and other developmental processes.

Growth in a Turing Model of Cortical Folding

The brain's cerebral cortex is folded into many gyri (hills) and sulci (valleys). Little is known about how the cortex folds or why the folds are located where they are. We have developed a spatio-temporal mathematical model of cortical folding to address this question. Our model utilizes a Turing reaction-diffusion system on an exponentially growing prolate spheroidal domain. This domain approximates the shape of the lateral ventricle (LV) during cortical development. The Intermediate Progenitor Model (IPM) of cortical folding states that regional patterning of self-amplication of intermediate progenitor cells (IPCs) in the subventricular zone of the LV corresponds with the formation of cortical folding. As self-amplication of IPCs is genetically controlled via chemical gradients, a Turing system is a logical choice to create a mathematical representation of the IPM. A growing domain model of cortical folding may be more realistic than previous static domain models of cortical folding since it incorporates the growth that naturally occurs as the brain develops. By comparing patterns generated by our growing prolate spheroid Turing system with those generated by a static prolate spheroid Turing system, we show that the addition of growth causes a significant change in system behavior; the system produces transient patterns instead of converging to one final pattern. Our model illustrates the importance of including growth in a model of cortical folding and can be utilized to explain certain human diseases of cortical folding.

Opening Statement: What is Computation?

Most people understand a computation as a process evoked when a computational agent acts on its inputs under the control of an algorithm. The classical Turing machine model has long served as the fundamental reference model because an appropriate Turing machine can simulate every other computational model known. The Turing model is a good abstraction for most digital computers because the number of steps to execute a Turing machine algorithm is predictive of the running time of the computation on a digital computer. However, the Turing model is not as well matched for the natural, interactive, and continuous information processes frequently encountered today. Other models whose structures more closely match the information processes involved give better predictions of running time and space. Models based on transforming representations may be useful.

Reflections on a Symposium on Computation

ACM Ubiquity hosted a symposium in 2010–2011 on Turing's question, ‘What is computation?’ The editor reflects on how the symposium was organized and what conclusions it reached. The authors showed strong consensus around the propositions that computation is a process, computational model matters, many computations are natural, many important computations are continuous, many important computations are nonterminating and computational thinking has emerged as a core practice of computing. They left open the questions of whether the Turing model is the best reference model, is computational necessarily a physical process, what is information and what is an algorithm.

Changing clothes easily: connexin41.8 regulates skin pattern variation

The skin patterns of animals are very important for their survival, yet the mechanisms involved in skin pattern formation remain unresolved. Turing's reaction-diffusion model presents a well-known mathematical explanation of how animal skin patterns are formed, and this model can predict various animal patterns that are observed in nature. In this study, we used transgenic zebrafish to generate various artificial skin patterns including a narrow stripe with a wide interstripe, a narrow stripe with a narrow interstripe, a labyrinth, and a 'leopard' pattern (or donut-like ring pattern). In this process, connexin41.8 (or its mutant form) was ectopically expressed using the mitfa promoter. Specifically, the leopard pattern was generated as predicted by Turing's model. Our results demonstrate that the pigment cells in animal skin have the potential and plasticity to establish various patterns and that the reaction-diffusion principle can predict skin patterns of animals.