Owing to its universality, this mode of thinking demonstrates beweist a character of he human race at large and all at once, and owing to its disinterestedness, a moral moralisch character of humanity, at least in its predisposition Anlage , a character which not only permits people to hope for progress toward the better, but is already itself progress insofar as its capacity is sufficient for the present. Despite the disasters of the Terror and the Final Solution, these Events started the Modern and the Postmodern respectively.
The French Revolution gave rise, despite the bloodbaths and rolling heads in city squares, to the Modern era and both responded to and gave rise to modern philosophy, while the Holocaust brought all the hopeful optimism of modernity crashing down. In his extended discussion of Kant in this book, Lyotard wrote that. Enthusiasm is a modality of the feeling of the sublime. The imagination tries to supply a direct, sensible presentation for an Idea of reason for the whole is an object of an Idea, as for example, in the whole of practical, reasonable beings.
It does not succeed and it thereby feels its impotence, but at the same time, it discovers its destination, which is to bring itself into harmony with the Ideas of reason through an appropriate presentation. This in-between zone, between the pure and the practical, needed its own critique in which Kant sought to investigate the grounds for judgment where the elements are indeterminate. The feeling of enthusiasm was, par excellence, the experience of the sublime, sublime because the feeling could not be presented.
The inability to present is related to the fundamental incompatibilities within the sublime itself, a clash between an intensity of pleasure that becomes pain. Despite the excesses of the French Revolution, the spectacle of the Fall of the Bastille, the drama of the Oath of the Tennis Court and the promise of the Declaration of the Rights of Man excited the imagination of those level-headed disinterested enough to see to the future.
Great changes, like the French Revolution, are not, in principle, sublime, by themselves.. The Begebenhe it which ought to make a sign of history could be found only on the side of the audience watching he spectacle of the upheavals.. The Teilnehmung through desire is not a participation in the act. But it is worth more, because the feeling of the sublime, for its sake, is in fact spread out onto all national stages. If you have found this material useful, please give credit to. The work was much more than a stereotypic application of the idea of local activation and lateral inhibition. Meinhardt tried to work out the developmental logic underlying experimental findings and tested the validity of his ideas by computer simulations.
Three examples might suffice to exemplify this point. First, in , French and Bryant had formulated the polar coordinate model for appendage patterning French, et al. This model worked with circumferential positional values, a shortest intercalation and a complete circle rule. It could explain a number of complex regeneration experiments, but made assumptions which were hard to connect to known molecular mechanisms and simple morphogen models. Meinhardt realised that an organiser region defined by the crossing of an anterior—posterior and a dorsoventral compartment would provide the same explanatory power as the polar coordinate model and, in addition, could be linked to the recent discovery of compartments as clonally restricted groups of cells in Drosophila Meinhardt Such insights were not dependent on particular local activation-lateral inhibition equations, but rather were derived from the attempt to find a minimal realistic model which could explain complex phenomena.
The second example concerns stripe formation. Meinhardt and Gierer had recognised that certain types of patterns were difficult to simulate with a lateral inhibition mechanism Meinhardt and Gierer In particular, this applied to stripe formation, an important prerequisite to understand segmentation. To produce stable stripes they introduced the idea of lateral activation of mutually exclusive states. Two cell states each producing an autocatalytic substance exchange signals which mutually promote their autocatalysis.
Thus, they can only exist next to each other. However, they also stay separated since each cell state locally inhibits the other state. This model made stripe formation possible and thus, Meinhardt used it to simulate segmentation. Much later and without reference to the early work, this mechanism was rediscovered as the core kinetics of the Drosophila segment polarity network von Dassow et al.
It is of particular interest that the new simulations did not start with a specific kinetic model, but used model equations entirely based on experimental data and the assumption of robustness to identify relevant parameter sets. Thus, remarkably, an unbiased, data-driven approach discovered an intricate mechanism, which had been suggested entirely on theoretical grounds.
The third example refers to colour patterns on the surface of sea shells Meinhardt and Klinger Here, it is not so much the particular logic of a mechanism which made the simulations attractive, but the sheer similarity, indeed frequently the complete accordance, of highly complex patterns with the computer-generated images Meinhardt It seemed that an agreement between natural pattern and simulation, capturing so many details, could not be accidental.
One was forced to assume that the theory has captured at least some correct aspects of the underlying patterning mechanisms. Recently, the same shell patterns have been simulated using a neurosecretory model supported by experimental findings on the anatomy of the mollusc mantle, the shell-making machinery Boettiger et al. The neural circuitries used in this model combine local activation and lateral inhibition and thus demonstrate the equivalence between different molecular and cellular realisations of the Gierer—Meinhardt model.
However, they also provide a clue about how difficult it is to predict the actual nature of the mechanisms underlying a patterning process even if the simulations recover the output pattern with astonishing accuracy. The members of the research group felt that it was only a matter of time until they had isolated the autocatalytic activator s and the broadly diffusible long-range inhibitor s. The first candidates had been already purified Schaller and Gierer ; Schaller A splendid validation of the theoretical work seemed imminent.
However, this turned out to be much more difficult than expected. Even today, the regulatory behaviour of hydra is not fully understood at the molecular level, and the postulated diffusible activators and inhibitors have not been identified Bode It transpired that biochemical methods were much less powerful for the dissection of developmental pathways than expected. In the long run, this work would completely change the basis for any theory of pattern formation.
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For the first time, complete, or almost complete genetic networks for particular patterning problems such as segmentation or neuroblast selection would become available and would provide test cases for modelling approaches. The experience gained with the zygotic whole genome screens made it possible to conduct large-scale screens for maternal-effect mutations.
The analysis of mutant phenotypes together with transplantation experiments soon revealed that the AP morphogen gradients in the Drosophila egg emanate from localised determinants at the anterior and posterior egg poles. Later, these were shown to be localised mRNAs. At the level of the early embryo, gradient formation did not require self-organised patterning on which the egg asymmetries would only have a weakly orienting influence.
For a while, the dorsoventral DV axis seemed to be different. Apparently, DV axis formation did not require localised components within the egg, and thus appeared to be an example for a self-organised patterning system. However, the exciting observations of ectopic axis formation and axis inversions along the DV axis finally could be explained by local signals emanating from cues in the eggshell Anderson et al. The transition from eggshell cues, though, to the embryonic DV pattern occurs in a dynamic way which apparently has some self-organising properties Meinhardt ; Moussian and Roth ; Roth and Schupbach However, the eggshell cues themselves provide fairly accurate information which can be traced back to a mRNA localised within the developing oocyte Neuman-Silberberg and Schupbach Thus, for those instances where we had expected that the theory would be most helpful, a completely different picture had emerged.
It became necessary to study how the egg was constructed during oogenesis in order to understand axis formation. One might object that Drosophila and insects in general are unusual because they produce eggs that display bilateral symmetry presaging the two body axes before fertilisation. However, it seems that the lesson learned from Drosophila reflects a more general problem of the early theories of pattern formation.
These theories systematically underestimated the potential and importance of intracellular patterning to provide spatial information for the multicellular level or to influence and constrain mechanisms of multicellular patterning. The description of intracellular patterning requires, however, at least in part, approaches different from the mechanisms employed in the original Gierer—Meinhardt model.
Since every eukaryotic cell already represents a highly structured three-dimensional object, the spatial information contained in the cell can be used to initiate symmetry-breaking events. Thus, in Drosophila , DV asymmetry arises through asymmetric movement of the oocyte nucleus Roth and Lynch or in amphibians, bilateral symmetry depends on rotation of the oocyte cell cortex with regard to the rest of the cell Larabell et al.
These phenomena and other aspects of cell polarisation require dynamic changes of cytoskeletal elements. At this level, theory again becomes relevant. A deeper understanding of cytoskeletal dynamics requires biophysical modelling approaches Howard The same applies for other aspects of intracellular patterning which increasingly become subject of mathematical modelling Karsenti Meinhardt himself has contributed to some of this work addressing dynamic aspects of cell division in bacteria Hale et al.
While the inadequate representation of intracellular patterning addresses the reference level of the theory, the more general problem of early pattern formation theories was the lack of molecular detail. Since none of the developmental processes which the theories described was understood at the molecular level, the theories could not work with realistic molecular assumptions. Activators and inhibitors remained abstract entities. As pointed out, this did not prevent the early theoreticians from gaining major insights which were validated by later research.
However, these insights addressed a more generic level representing both their strength—they could be applied to many cases, and their weakness—they could not capture any specific molecular process. With detailed molecular data, in particular from Drosophila , becoming available in the mids, a new generation of theoretical approaches became possible. Modelling was no longer restricted to general mechanism, such as local activation and lateral inhibition, but could incorporate genetic interaction data or even real measurements of concentrations of respective components.
Now finally, one could hope, the real molecular processes could be simulated in the computer. This should lead to realistic models providing a deep understanding of multi-component systems. The step from the linear stability analysis to which Turing was largely restricted to the simulation of non-linear systems by Gierer and Meinhardt was linked to increased computer power.
Likewise, the new simulations again were only possible by another large leap in simulation capacity. In particular, now it was possible to perform systematic explorations of parameter spaces or to apply complex algorithms of non-linear fitting simulated annealing. The idea of robustness started to play an important role.
By systematic variation of the parameters of given model equations, one could identify those conditions which provided the most robust mechanism. In the following, three examples from recent work on Drosophila pattern formation will be discussed which reveal the power and limitations of the new approaches. Each of them demonstrates the need for mathematical modelling. They will be presented in a sequence of increasing closeness to quantitative experimental data: the first example deals with models for the formation of a BMP morphogen gradient in the embryo, the second with the gene circuit approach to explain the segmentation process, and the third with models for the formation of the Bicoid morphogen gradient.
Dorsal forms a nuclear concentration gradient with peak levels along the ventral midline of the early embryo. Dorsal activates and represses target genes in a concentration-dependent manner and thereby specifies different gene expression domains along the DV axis. Interestingly, the Dorsal gradient is confined to the ventral half of the embryonic circumference, but nevertheless is also required for patterning the dorsal half. How is such a precisely controlled long-range influence possible? Mutant analysis showed that, paradoxically, a BMP inhibitor expressed ventrally under the control of Dorsal is required for establishing BMP peak levels at the dorsal midline.
This observation remained a conundrum until the suggestion was made that BMP—inhibitor complexes forming in ventral regions diffuse dorsally and thereby transport BMP to the dorsal side Ashe and Levine ; Ferguson and Anderson , Here, the inhibitor is cleaved and BMP is released. Since the protease cleaving the inhibitor is not confined to the dorsal midline, it was still hard to understand how this mechanism accounts for spatial precision.
A general mathematical treatment of the reaction-diffusion system encompassing BMP, the inhibitor and the protease revealed that rather simple interactions are sufficient to reproduce the experimentally observed patterning capability Eldar et al. The authors started with general model equations for the three-component reaction-diffusion system and calculated the steady-state BMP activation profile for thousands of randomly chosen parameter sets. The parameters included rate constants for production, decay and diffusion of the individual proteins and complexes.
Parameter sets, which produced steady-state solutions mirroring the experimentally observed BMP activation pattern, were tested for robustness using the following rationale: Genetic experiments show that reducing the gene dosage of some network components by a factor of two does not significantly change the final BMP activation pattern. Thus, realistic parameter sets should also lead to solutions which are robust, given twofold changes in network components. Parameter sets fulfilling this condition were identified and suggested that free BMP does not significantly diffuse and the free inhibitor is not a good target for proteolysis.
Thus, BMP should be predominantly transported in a complex with the inhibitor, and this complex should be the main target of proteolysis. This assumption led to simplified equations which could be treated analytically, providing precise mathematical arguments for the proposed mechanism. The entire approach seemed to be highly convincing Meinhardt and Roth The derivation of an analytical expression capturing the mechanism of robustness could be almost regarded as a proof of the correctness of the proposed interactions.
However, several complications arose that revealed the intricateness and unpredictability of biological systems. First, one component of the network, the BMP ligand Dpp was known for a long time to be dosage-sensitive. Indeed, dpp is the only developmental gene in Drosophila which is haplolethal, meaning that a twofold reduction of its dose leads to lethality Irish and Gelbart Irrespective of the particular implications for the proposed mechanism, this observation raises the general question of how valid robustness is as a universal criterion for identifying appropriate modelling parameters.
In the case of dpp , one could argue that its employment in different patterning contexts leads to an evolutionary optimisation problem which has no ideal robust solution. However, the mere fact that such exceptions exist brings into question the blind reliance on robustness for selecting appropriate parameter sets. It seems that assumptions about robustness have to be analysed in each particular case. Second, biochemical studies and additional modelling approaches revealed different interaction networks, which could also produce sharp BMP signalling peaks.
An interesting assumption was that Sog acts as a competitive inhibitor of BMP binding to its receptor and that receptor-bound BMP is degraded Mizutani et al. Thus, Sog protects BMP from receptor-mediated degradation and thereby enhances its range of action. Another paper showed that the main signalling molecule is a heterodimer of two BMPs and that it is the heterodimer which is bound by the inhibitor, while mathematical modelling had suggested alternative interactions Shimmi et al. While the former two papers demonstrated that the interactions of even a small number of secreted proteins and their receptors can lead to several alternative models which can account for observed phenotypes, another publication opened up completely new aspects of the system.
Eldar et al.
A transcription factor was shown both to be a target of, and to promote, dorsal BMP signalling. Apparently, this feedback loop promotes future BMP binding to its receptor as a function of previous signalling strength. This finding led to a completely new series of modelling approaches Umulis et al.
In addition, new precise measurements of BMP signalling in whole embryos provided the basis for organism-scale modelling using realistic geometries Umulis et al. Unfortunately, until now the molecular basis of the positive feedback has remained elusive. The most recent publication introduces eight different models for possible positive feedback mechanisms encompassing a variety of cell biological and biochemical details as well as newly observed interactions with extracellular matrix molecules.
The most important outcome of the paper is that positive feedback mechanisms are indeed producing the best fit to the data and that, among these mechanisms, a positive feedback involving a surface-bound BMP binding protein is slightly superior to other mechanisms. However, the modelling approaches veered away from elegant analytical results to numerical solutions of high dimensional systems of more than ten coupled partial differential equations and 17 parameters.
Despite whole embryo 3D representations of BMP signalling in wild-type and mutant embryos at different developmental stages, and even comparisons of different Drosophila species, the discriminatory power of modelling results appears to be weak with regard to the mechanistic alternatives. Even the most ardent aficionado of pattern formation theory might come to the conclusion that we have moved further away from understanding the system.
The complexity of the embryo appears to evade full mathematical treatment. First, the gap genes are activated in broad domains and in turn control the pair-rule genes which represent the first level of periodic gene expression in the embryo, albeit with double-segment periodicity. Segmental periodicity is only reached at the next tier, the expression of the segment polarity genes, which receive their regulatory input largely from the pair-rule genes.
At the end of the blastoderm stage, the patterning process has reached its highest possible resolution: single-cell wide stripes of segmental polarity gene expression defining the position of segment and compartment boundaries.
The gap and pair-rule genes code for transcription factors which can diffuse between nuclei since patterning occurs before the cell membranes are formed. Thus, the system is basically an interaction network of factors which mutually regulate their expression. Besides nuclear divisions, no morphogenetic events take place, allowing a description of the process with one spatial coordinate representing the AP axis or rather the position of the nuclei along the AP axis.
Inspired by models for neural networks Hopfield , a data-driven modelling approach for gene regulatory networks was devised called the connectionist gene circuit method and was specifically applied to the Drosophila segmentation cascade Mjolsness et al. Usually, theoretical modelling is based on proposed interactions of the relevant components inferred from genetic and molecular experiments.
Only minimal a priori assumptions are made about potential regulatory interactions. A system of general model equations is formulated which describes the concentration change of each transcription factor in a particular nucleus as a function of gene expression, diffusion and decay. The expression of a particular transcription factor depends on regulatory inputs from other transcription factors. To capture these inputs, an interconnectivity matrix T is introduced, in which the regulatory effect of gene a on gene b is represented by the matrix element T ab.
Depending on whether this element is positive or negative, gene a activates or represses gene b , respectively. The quantity of T ab defines the strength of the interaction; if it is zero, the two genes do not interact. The main point of the approach is that no a priori assumptions are made about the numerical values for the elements of the matrix or the diffusion and decay rate of the transcription factors.
Instead, prior to modelling, a set of quantitative expression data is collected. In this particular case, antibodies were produced against most segmentation genes, allowing the measurement of the protein distribution in the entire embryo at different time points. The resulting highly accurate quantitative descriptions of the spatiotemporal expression changes were used to determine the numerical values of the matrix elements and the diffusion and decay rates of the transcription factors by procedures of non-linear fitting. Thus, the network topology was recovered from experimental data, rather than being implemented prior to the actual modelling work.
Initially, this approach encountered doubts and scepticism from molecular biologists studying the transcriptional regulation of individual segmentation genes. The analysis of cis-regulatory elements should uncover those interactions which actually occur in the embryo. Why would one need a more global indirect approach? However, even early gene circuit modelling using still limited experimental data sets for fitting led to some highly non-trivial predictions. For example, a paper on the formation of pair-rule gene expression stripes predicted that stripe formation requires very low diffusion rates of the pair-rule gene products Reinitz and Sharp This prediction later gained experimental support by the demonstration that the mRNA of pair-rule genes is tightly localised to the cell cortex above the nuclei.
Translation of the localised mRNA at the cortical positions is likely to hinder the spreading of the protein to neighbouring nuclei, accounting for low diffusion rates. More importantly, later modelling of the gap gene network using improved data sets for fitting and more sophisticated optimisation programmes led to results not predicted by any experimental work and in addition, provided a deeper understanding of one of the central questions in developmental biology: how spatial precision emerges despite noisy starting conditions.
A careful analysis of the gap gene expression pattern showed that the gap gene domains undergo a coordinated anterior shift after their initial establishment under the control of maternal gradients Jaeger et al. This had not been noticed by experimentalists and implies that the readout of morphogen gradients, at least in this particular case, is not a static but rather a dynamic process.
The positions of target gene domains are not ultimately fixed by particular morphogen concentrations. Rather, interactions among the target genes define the final coordinates of target gene expression. The data-fitting algorithms revealed the regulatory parameters responsible for the domain shifts and allowed an interpretation of the underlying mechanism Jaeger et al. The elaborate picture of the gap gene network which emerged from these studies was later expanded and used to address the problem of canalisation.
The segmentation cascade of Drosophila allows one to detect canalisation at the molecular level and to analyse its mechanism. The crucial maternal morphogen responsible for activating the gap genes is the gradient of the transcription factor Bicoid. Careful measurements were used to detect the embryo-to-embryo variability of the Bicoid gradient and of the early and the late gap domains. A comparison of the results revealed a progressive reduction of the variation with developmental time. The shape of the Bicoid gradient as well as the early expression domains of the gap genes were noisier than the late gap domains.
By applying the gene circuit modelling approach, the specific regulatory interactions could be identified which are responsible for the noise reduction Manu et al. A combination of strong and weak mutual inhibition was shown to be crucial. The mathematical analysis of a precise dynamical model which contained the experimentally derived parameters revealed that the gap gene network possesses certain attractors to which the system trajectories converge, irrespective of small variations in the starting conditions Manu et al.
Thus, in this particular case, a mathematical explanation of the stability, i. The fact that modelling in this case was an a posteriori analysis of interactions occurring in the embryo is a distinctive feature of this work. The stability of steady state solutions in the face of variable starting conditions had always been a strong motive in pattern formation theory.
Gierer and Meinhardt also considered the stability of the outcome of pattern formation processes in spite of system perturbations as one of the most essential aspects of their theory Gierer ; Meinhardt However, the papers on the gap gene network represent probably the first case in which quantitative data, including an assessment of the actual noisiness of the system have been used as the basis for modelling.
Despite these seminal achievements and the obvious closeness to biological reality, the gene circuit approach remains at the phenomenological level in one basic respect. The elements of the interconnectivity matrix are open to a number of molecular interpretations and they are also not free from a priori assumptions which might not fully reflect the complexity of the transcriptional process. For example, the model does not allow that a transcription factor activates and represses at low or high concentrations, respectively.
The fact that the authors can produce a self-consistent model for the gap gene network without assuming more complex interactions may have three explanations. A crucial question will be how the model fares when applied to the next level of the segmentation process: the emergence of the pair-rule gene expression pattern. At this level, several fundamental questions still need to be addressed, in particular the precise phase shift between partially overlapping pair-rule stripe patterns.
This phase shift is crucial for initiating segment formation Klingler and Gergen ; Warrior and Levine , but we are almost completely ignorant with regard to the underlying molecular mechanisms. Here, the gene circuit model might indicate where to look at the molecular level, and thus could have a crucial heuristic role for the working molecular biologist.
On the other hand, there is some indication that besides protein—DNA interactions, protein—protein interactions between the pair-rule gene transcription factors also contribute to the patterning process Fitzpatrick et al. Such interactions have not been included in the gene circuit model equation to date, and it is doubtful whether they can be fully represented. Any mechanism which necessitates the elements of the interconnectivity matrix to become functions of space and time will probably render the optimisation problem unsolvable. Thus, it is possible that the gene circuit model in its current form has a very limited applicability.
There might be few other instances in all developmental biology, which suit this approach as much as the first step in the Drosophila segmentation cascade, the gap gene network. Crick had already provided a special solution to this problem. The particular case in question is the aforementioned gradient of the transcription factor Bicoid, which provides the input for the gap gene pattern Porcher and Dostatni ; Grimm et al. Even intuitively, one can imagine how local production and diffusion are able to produce a long-range concentration gradient.
This intuitive concept can be captured in a simple equation describing the spatiotemporal change of Bicoid concentration as a function of s ynthesis, d ecay and d iffusion SDD model. For SDD models, steady-state distributions can be calculated in which synthesis, diffusion and decay are balanced such that the resulting concentration profiles do not change in time Gregor et al. A localised source and spatially uniform decay give rise to an exponential steady-state profile characterised by a length scale which only depends on the diffusion and decay rates.
Furthermore, measurements of the diffusion of fluorescent dextran molecules which had a molecular mass comparable to that of Bicoid were in agreement with the SDD model for gradient formation Gregor et al. However, one phenomenon suggested a more complex scenario. Closely related fly species show almost identical early segmentation gene expression.
However, their egg size may vary over more than a factor of five in length Gregor, et al. The scaling of the segmentation gene expression pattern can be traced back to the scaling of the Bicoid gradient, i. Thus, in a large egg, the gradient has a proportionally larger length scale which might result either from an increased diffusion or a decreased decay rate of Bicoid protein.
However, the Bicoid proteins of different fly species have a very similar structure and thus should have similar diffusion and decay rates implying similar length scales. This assumption was strongly corroborated by generating Drosophila melanogaster embryos expressing a Bicoid protein derived from a fly species with large eggs Gregor et al. The heterologous Bicoid protein formed a gradient indistinguishable from the endogenous one. Thus, scaling seems to be a feature of the embryonic system rather than of the respective Bicoid protein. To explain this fact, a cellular property had to be found which also showed scaling behaviour.
An obvious candidate was the number of nuclei, which is conserved in different fly species. Flies with large eggs have a lower nuclear density, and those with small eggs a higher one. As a transcription factor, Bicoid acts within the nuclei. Thus, for its biological function the nuclear, as opposed to the cytoplasmic, concentrations are of crucial importance. If the degradation of Bicoid mainly occurs in the nuclei, then the scaling of the gradient could be explained because the length scale of the Bicoid gradient would depend on the nuclear density.
The production of transgenic flies expressing fully functional GFP-tagged Bicoid protein partially allowed such measurements and produced two surprising results Gregor et al. This observation is in direct conflict with the explanation for scaling of the gradient in different flies because scaling requires that the gradient changes concomitantly with changing nuclear densities. In addition, nuclear stability by itself is not easy to explain given a steady state model of the gradient. Gregor et al. Several models were developed to address this question.
One of them assumes that the gradient is not in a steady state, but that the total amount of Bicoid protein increases, and this increase balances the increase in nuclear volume so that the local nuclear concentrations remain constant Coppey et al. The second surprising finding concerned the measured diffusion constant of Bicoid, which was an order of magnitude too low to account for the length scale of the gradient Gregor et al.
Even assuming non-stationary models the measured rate of Bicoid diffusion cannot explain the shape of the gradient Grimm et al. To tackle this problem, mathematical models were developed that paid more attention to the actual cellular complexity of the early Drosophila embryo.
A coarse-grained model for the syncytial blastoderm was derived which combined cytoplasmic diffusion and nucleocytoplasmic shuttling of Bicoid Sample and Shvartsman ; Kavousanakis et al. The model uses a homogenisation approach that is applied in physics and engineering for the description of heterogeneous materials. If materials are structured at two clearly distinct length scales, the structure at the larger length scale might result from the repetition of a small-scale unit with internal structural complexity the reference cell. In the case of the syncytial blastoderm, the reference cell was assumed to be a nucleus with its surrounding cytoplasm.
The homogenisation approach allowed the definition of an effective large scale diffusivity for Bicoid as a function of the geometry of the nucleus together with the surrounding cytoplasmic island, the diffusivity in the cytoplasm and nucleocytoplasmic shuttling Kavousanakis et al. In particular, the last point remained disconcerting since the explanations invoked for each phenomenon required conflicting assumptions.
Additional experimentation was required. In a recent paper, altered Bicoid proteins were analysed that had an impaired ability for nuclear transport or were lacking nuclear transport altogether Grimm and Wieschaus Surprisingly, these Bicoid versions produced gradients indistinguishable from that of the wild-type protein.
This is a striking observation since wild-type Bicoid concentrations show huge local fluctuations depending on the nuclear cycle. Thus, one would assume intuitively that these local changes have an effect on the overall gradient. These new findings also rule out explanations for scaling that invoke nuclear density. According to a recent suggestion, scaling might result from the positive correlation between the amount of bicoid mRNA and egg size Cheung et al. Finally, the new findings re-open the whole discussion on the mechanisms of gradient formation.
No doubt, we are currently unable to provide a mechanistic explanation of one of the simplest patterning problems in biology, the formation of the exponentially decaying gradient of a single protein species emanating from a local source. This problem seems to be much simpler than that of self-organised patterning which motivated Turing or Gierer and Meinhardt; it is also much simpler than BMP gradient formation or the gap gene network, the two situations exemplified above.
However, it touches on one key feature of development: the capability to produce the same pattern at different length scales. It was precisely this feature, in some of its extreme realisations, that Driesch believed could not be explained by physics and chemistry and for which he developed his harmonious equipotential system reflecting a type of lawfulness only found in organisms.
As we do not want to take refuge in vitalistic tenets, the question arises of where we could obtain additional physicochemical explanations to explain Bicoid gradient formation. The problem here is not the lack of potential candidates, but rather the multitude of options. Indeed, bicoid mRNA is not tightly localised to the anterior cortex, but forms a steep gradient Spirov et al. Thus, Bicoid might emerge from a graded source. This might certainly help to explain the discrepancy between low diffusivity and large length scale.
Diffusivity of Bicoid has so far been measured only at the surface of the syncytial blastoderm in embryos. It could be larger in earlier stages or in the inner part of the embryo as opposed to the cortex. Recent measurements of Bicoid diffusion using different biophysical methods have questioned previous data Abu-Arish et al. In addition, a certain amount of cytoplasmic streaming occurs in the embryo which has been used to explain the length scale of the gradient Hecht et al. For example, the phosphorylation of Bicoid by the receptor tyrosine kinase Torso occurs only in a restricted anterior zone Janody et al.
Recently, a ubiquitinylation of Bicoid was demonstrated, which targets the protein for degradation Liu and Ma Thus, many different protein forms of Bicoid might exist in the embryo, each with particular diffusion and decay rates or even with particular spatial distributions.
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Taken together, organismic complexity impinges on multiple levels even for the simple process of forming a protein gradient. The most difficult problem for theory lies in the fact that no a priori physicochemical argument exists that allows us to decide which aspects are relevant and which can be neglected. Thus, for some time, bicoid mRNA distribution was not considered to be important for gradient formation since the mRNA is clearly more locally restricted than the protein.
Given the suggested low diffusivity of Bicoid protein, the potential contribution of mRNA spreading becomes more important. Conversely, nucleocytoplasmic shuttling of Bicoid appeared to be of pivotal importance for understanding gradient stability and scaling, leading to very impressive theoretical approaches Gregor et al. However, recent experimental evidence undermines the importance of this process Grimm and Wieschaus Evolutionary processes can affect and fine-tune each level of a developmental process.
From the physical point of view, no simple process exists in an organism since not a single protein or RNA molecule can be treated separately from the organismic context. This is impressively shown by the example of Bicoid proteins from large eggs that form normal gradients within smaller eggs Gregor et al. Since evolution can operate at each level e. Bicoid shows us even more impressively than the other examples, how the organism evades the attempts of the theoretician. The history of pattern formation theory presented here reveals the necessity of mathematical modelling already for simple chemical and even more so for biological systems.
The oscillatory behaviour Lotka found by solving the differential equations is not evident if one just looks at the reaction scheme. However, understanding the important difference between conservative and dissipative systems, reflected in the difference between structurally unstable and structurally stable limit cycle oscillations, requires relatively abstract mathematical arguments.
Nevertheless, this point is absolutely crucial; all interesting chemical and biological pattern-forming systems have the property of structural stability. The demonstration by Turing at the beginning of his paper that differences in diffusion rates can lead to unequal distributions of components starting from homogeneity is rather counter-intuitive.
Is not diffusion a process that diminishes concentration differences? Finally, the rich pattern-forming capabilities of systems combining non-linear autocatalysis, inhibition and differential diffusion rates Gierer—Meinhardt model could have never been demonstrated without computer simulations. The need for mathematical modelling is even more apparent in modern data-driven approaches. Already data acquisition requires sophisticated algorithms.
A large part of the efforts of the Reinitz group was devoted to the question of how to reliably retrieve expression data from microscopic images Janssens et al. Non-linear fitting or screening parameter spaces require elaborate programming. But even in rather simple cases such as Bicoid gradient formation, a model is needed to decide whether the measured diffusion constant is appropriate for a postulated mechanism or whether a substantial deviation from the expectation indicates the existence of unknown processes.
One can easily predict that with our increased ability to acquire highly accurate measurements of molecular properties in living systems, the need for model building will increase. However, in the face of organismic complexity, the work of the theoretician can appear to be a Sisyphean ordeal. In the examples we have described, successful and internally consistent modelling approaches were called into question by re-interpretations of underlying mechanisms or by new data sets, which contradicted previous assumptions. In the case of the gene circuit approach, a huge modelling effort was undertaken over more than a decade to produce one of the most impressive results in all theoretical biology: a precise description of how canalisation takes place.
However, the actual structure of the model might have only limited applicability beyond a particular stage of Drosophila development and despite its proclaimed closeness to experimental data cannot be easily linked to concrete transcriptional mechanisms. In the case of the Bicoid gradient, it even seems that after repeated rounds of modelling and experimentation, the theory currently has lost its object since it is not clear where the property responsible for scaling is to be found. In the light of these observations, the question arises whether the relation between a mathematical model and its object is fundamentally different in biology than in other areas of science.
An answer to this question requires careful consideration. Are not disagreements between theoretical modelling and experimental results anyway the normal situation with which theoreticians in all fields of science are confronted? However, here we know very well that some of the underlying equations inherently lead to unpredictability deterministic chaos. The impossibility of long-term predictions is mathematically understood. In other instances, this point is less clear. An impressive example is represented by the long-standing attempts to derive the properties of liquid water from computer simulations.
The forces between two water molecules the pair potential include repulsion, electrostatic dipole interactions and hydrogen bonding. There is a surprising variety of how to formulate a pair potential based on these three contributions. Additionally, there are higher order interactions between more than two water molecules. All of them recover basic properties of water and provide accurate predictions for at least some of the measured physical constants of water. The process of approximation to the real object seems to be much smoother than in the case of biology.
Indeed, Guillot can precisely localise the weaknesses in the assumption of many models and suggest very specific improvements. Biological pattern formation theory is a relatively young field of research. Although its object of reference is infinitely more complex than liquid water, we cannot exclude that future modelling efforts will lead to a similar degree of approximative understanding. Beyond mere material complexity, however, organisms possess features not found in the inorganic world, which are likely to complicate every mathematical approach.
Proteins and mRNAs, the types of molecules which are and will be the main target of modelling are not only huge macromolecules compared to water but they also exhibit features fundamentally different from any molecule of the inorganic world. Their sequence-based structure depends on an evolutionary process which combines adaptive determinative and non-adaptive drift-like or stochastic events.
Evolution can target singular features individual positions within the sequence of these molecules. Therefore, their physical properties are to a large degree at the disposal of events buried deeply in the evolutionary past. For these events, it is not even clear whether they all have a straightforward physical grounding.
As long as we are dealing with adaptive processes, physical causes have to be postulated which guide adaptive events. For example, the particular modes of early development in insects might have served as a patterning environment to which proteins and their respective mRNAs had to adapt if they were to be useful as morphogens in providing positional information to the embryo. This process ought to be highly complex since its goal is the fit between the physical properties of a complex molecule to a highly structured cellular environment. However, in particular for multicellular organisms, drift-like processes might have a significant contribution to evolutionary change Lynch The reducing theory for such processes is probability theory McShea and Brandon Consequently, some of the molecular properties of proteins and RNAs might not have a physical cause linked to the biophysics of the particular process they are involved in, but rather result from collective statistical phenomena at the population level.
Thus, pattern formation theories working with functionally and physically motivated mechanisms are, already at the level of the single molecule, likely to face unexpected complications. However, patterning processes usually require coupled reactions of many components and frequently encompass several hierarchical levels e. Accordingly, evolution has almost infinite possibilities to complicate and fine-tune patterning mechanisms and thus can overcome or undermine almost every physicochemical constraint postulated by a particular theory.
Although Kant speculated about evolution in his late writings, he had no mechanistic explanations for how the functional adaptations observed in the organic world could arise Roth a. Nevertheless, he assumed that the apparent goal-directedness or purposiveness of organisms is not in contradiction with the general laws of nature. Organisms just represent unique constellations of matter. From the perspective of general laws, their structure is extremely improbable, yet highly reproducible. They are, according to Kant, examples of the lawfulness of the contingent Roth b.
This implies that we have no alternative for studying organisms to the mechanistic approach, despite its insufficiency for a full understanding of their contingent features. Thus, Kant clearly realised that his ideal of mechanistic understanding implying mathematics is faced with significant, indeed seemingly insurmountable problems when applied to living nature. I am grateful to Joachim Krug, Johannes Berg, Eric Davidson and two anonymous referees for insightful criticism and suggestions for improvement.
I am grateful to Ute Deichmann for giving me the opportunity to present this topic at the third annual international workshop of the Jacques Loeb Centre in Beer Sheva. Open Access This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author s and source are credited.
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Development Genes and Evolution. Dev Genes Evol. Published online Nov Siegfried Roth. Author information Article notes Copyright and License information Disclaimer. Siegfried Roth, Email: ed. Corresponding author. Received Oct 15; Accepted Oct This article has been cited by other articles in PMC. Keywords: Chemical oscillations, Spatial patterning, Dissipative structure, Turing pattern, Local activation, Lateral inhibition. The rise of physicochemical theories of self-organisation The history of biological pattern formation theory is indirectly linked to the discovery of self-organisation in chemical systems, which, in turn, is rooted in theoretical discussions and experimental findings on oscillating chemical reactions in homogeneous phase.
Modern theories of biological pattern formation While the inadequate representation of intracellular patterning addresses the reference level of the theory, the more general problem of early pattern formation theories was the lack of molecular detail. Conclusions: Kant revisited The history of pattern formation theory presented here reveals the necessity of mathematical modelling already for simple chemical and even more so for biological systems. High mobility of bicoid captured by fluorescence correlation spectroscopy: implication for the rapid establishment of its gradient.
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