On the emergence of natural singularities and state transitions in living patterns
aa r X i v : . [ phy s i c s . b i o - ph ] A p r On the emergence of natural singularities and state transitions in living patterns
Akos Dobay
Institute of Evolutionary Biology and Environmental Studies, Winterthurerstrasse 190, University of Zurich, CH-8057 Zurich, Switzerland
Abstract
As far as human perceptions and rational thinking are concerned, contradictions constitute a non negligible part of our reality. Weoften refer to these phenomena, in a more informal way, as the chicken or the egg causality dilemma. However, it is not clearwhether the chicken or the egg dilemma exists only within the scope of our perceptions, or contradictions have a deeper meaningtowards our understanding of reality. Here we argue that if there is an element of reality such that can be adequately described interms of the chicken or the egg dilemma, then it might lead to a spontaneous symmetry breaking by creating an alternate entity,capable of ultimately separating the chicken from the egg. We propose a formalism to describe such mechanism and discuss how itcan be applied to phenomena to describe the natural emergence of singularities and state transitions in living systems.
Keywords: symmetry breaking, state transition, development, emergent phenomena, chicken or the egg causality dilemma
1. Introduction
What comes first: the chicken or the egg? Everyone knowsit cannot be the egg since the chicken has to lay it first. Con-versely, it cannot be the chicken since it comes out of an egg.This problem, known as the chicken or the egg dilemma, hasbeen used to describe paradoxical situations. Paradoxes arewell known and are of great interest, in the scientific commu-nity and in mathematics, because they represent a challenge toany form of rational thinking [1]. Paradoxes have fascinatedgenerations of philosophers for the same reason. The formula-tion of the chicken or the egg dilemma has also inspired severalartists. One of the most popular is certainly the Dutch artist M.C. Escher (1898 – 1972). An example is his famous lithographcalled
Drawing hands (1948), in which he depicts two hands,facing each other on a sheet of paper in the paradoxical act ofdrawing one another into existence.A paradoxical situation or a chicken or the egg dilemma ap-pears in many di ff erent fields of knowledge. Whether it isin mathematics, if we think of G¨odel’s incompleteness theo-rems [2], in quantum mechanics and the so-called EPR paradox[3, 4], or even in biology when it comes to find out how repli-cation started during the early stages of life [5], there is always,at least in appearance, a chicken or the egg dilemma. The useof the words “in appearance” in the previous sentence conveysan important aspect of the problem we would like analyze withthe present formalism. At the scientific level, the problem canbe formulated as follow: “Is there a mechanism behind the res-olution of an internal contradiction within a given physical sys-tem?” In our case the concept of a physical system correspondsto a biological organism which has to adapt and might be sub-jected to natural selection. The answer to this question willdetermine whether all the physical constituents of an internalcontradiction can be unambiguously identified and translatedinto an algorithm or the resolution happens in a dimension for which no tangible constituents exist.Among humans, paradoxical situations occurs in the courseof time when the informational content of a system is such thatit cannot be resolved rationally. Hence, when facing a para-dox, a decision-maker might be confronted with the dilemmaof not being able to make an appropriate choice within a rea-sonable amount of time. Moreover, when the decision has tobe taken under environmental pressure, it naturally creates ten-sion, whose intensity varies among individuals. This tensionwill sooner or later induce a decision, breaking the paradox intotwo distinct solutions. But either of these solutions is absolute:they rely on each other. Paradoxes also lie at the core of fun-damental theories [1]. In our attempt to find absolute criteriaupon which to build a satisfactory rational reference frame, in-dependent of any particular perceptions of the world and freefrom contradictions, the chicken or the egg dilemma stands as atrue paradigm.Can we answer our question or circumvent its limitations ifno answer can be provided? Answering the question turns outto be still out of the scope of our present formalism. Howeverwe argue that its limitations can be at least weakened using anappropriate thought pattern. To achieve our goal, we rely onthe working hypothesis that an internal contradiction requiresan additional dimension, apart from those that constitute theparadox parameter space, to be resolved. However, there mightbe no appropriate mathematics to characterize it; consequently,no calculations are possible. If the result of such process istransposed back onto its original parameter space, it might bepossible to develop a measure to quantify its e ff ects.Hence, the chicken or the egg dilemma becomes a paradigmwhere the question does not necessarily require the answer towhich one came first, and to subsequently establish a logicalorder. Rather , it is a paradigm for situations where the two ele-ments can co-exist, but accompanied by a tension that forces the1ontradiction to be resolved into measurable elements such asspace and time [6]. What appears to be a limitation in our ratio-nal thinking or our perception of the world can be instrumentalin understanding phenomena. This paradigm may explain theemergence of individuals or elements capable of mirroring theirenvironment, whenever paradoxical situations arise where theavailable information is not su ffi cient to drive a state transition,but where external constraints are necessitating the transition tooccur nonetheless: e ff ectively, a decision-making process [6].It might also serve as a hypothetical explanation for the emer-gence of particle-antiparticle pairs from vacuum breakdown [7],in case the boson is a form of tension traveling though space andtime until it materializes into a pair.The class concept is useful for describing similar entities.Formally, a class is a collection of objects that unambiguouslyshare the same properties. Similarly, in object-oriented pro-gramming, the notion of class is an abstraction that permits theencapsulation of a collection of properties common to objectsbelonging to it. By considering the chicken or the egg dilemmaas a class of related problems, we can formulate the paradigm inmore general terms and propose a specific formalism to test theunderlying mechanism behind such phenomena, and their im-plications on our understanding of nature. Moreover, any sys-tem, which has the ability to processt information from its envi-ronment, can potentially take decisions. Commonly, this abilityis attributed to higher order organisms capable of cognition, butnot to other systems such as viruses or bacteria, although someauthors are likely to investigate this possibility [8, 9]. It is evenharder to conceive that this ability can be transposed to purelyphysical systems. For this essay, we assume that reflection isa natural phenomenon that emerges when a physical system– living or not – composed of a combinatorial set of buildingblocks reaches the critical situation where environmental pres-sure forces the system to undergo a morphological or a topolog-ical transition even if temporal delays in information processinghinder a logical decision from being made. Information lags re-sult in the superposition of the past and present states, creatingan internal contradiction. From such a starting set of combina-torial building blocks, any structure from a single atom to anentire galaxy may run into a paradoxical situation.
2. Description of the formalism
As we mentioned in the introduction, the present formalismpresents the chicken or the egg dilemma as a class of relatedproblems C αβ . Therefore, any element of C αβ is characterizedby an internal contradiction that transitions from a dimension d into a measurable space d . The transition from d to d requiresthe emergence of a singularity that we formally indicate as T .Here, we show how to quantify an element of C αβ .The chicken or the egg dilemma can be represented as thesuperposition of two mutually exclusive entities α and β in thenon-spatial dimension d : αβ ! d . (1) Once the internal contradiction has been resolved in any type ofspace d with a well-defined measure, then (1) becomes (cid:16) α β (cid:17) d . (2)Hence, at the very basic level we have αβ ! d T ←−−→ (cid:16) α β (cid:17) d (3)where T results from the tension created in d by the internalcontradiction. The relation given by (3) is a representation of anobject of C αβ , basically the atomic structure of an internal con-tradiction. The bidirectional arrow between the two membersof (3) means that we make no assumptions regarding the ori-gin of internal contradictions. Now that we have given a formalrepresentation of a particular element of C αβ we need to pro-vide a mathematical toolset able to quantify it. As in Bayesianinference method [10], our approach relies on the concept ofprobability as a measure of uncertainty, as partly introduced in[6].Briefly, to describe the relation between the information con-tent of the system and a transition, we characterize the state of asystem by its confinement in a parametric space, i.e. by its con-tour c . In other words, behind each parameter of a system, weassume there is a physical agent ( a i ), such that the existence ofa contour can be represented by a set of parameters ( p i ) i = , ,..., n and their specific values x = ( x i ) i = , ,..., n , leading to a given mor-phology or topology that defines the state; these parametric val-ues belong to a statistical ensemble. Therefore, if we assumethat the specific values are variables which have a probabilisticdistribution, then the contour c ( x i ) i = , ,..., n can be defined as adistribution function of the set of parameters and their respec-tive variable x i . We further assume that f ( x ) represents the jointprobability density function associated with c . We also assumethat the ranges within which the specific values of the param-eters while variable are still representative of the state, and liein the space of real values Ω = Q ni = [ a i , b i ] which is includedin c ( x i ) i = , ,..., n . Thus, the probability of finding the parameters( p i ) i = , ,..., n inside a contour can be calculated by integrating thejoint probability density function over all these intervalsPr (cid:16) c ( x i ) i = , ,..., n (cid:17) = Z b a Z b a . . . Z b n a n f ( x ) d x . (4)In addition, the probability associated with the parameter p i alone can be deduced from equation 4 by integrating the jointprobability density function over all values of the n − f x ( x i ) = Z f ( x ) dx . . . dx i − dx i + . . . dx n . (5)From equation 5, one can finally calculate the probability asso-ciated with the parameter p i at a given point in timePr (cid:16) p c ( x i ) i = , ,..., n i , t (cid:17) = Z b i a i f x ( x i , t ) dx i . (6)2uring a transition, the system abandons its old contour andadopts a new one. An orderly transition is characterized by theabsence of an overlap between the two contours, i.e. the para-metric values of the old and new states are variable within non-overlapping intervals. Now let p i , be the i -th parametric valueat a given time point, then the di ff erence d t ( c , c ) ∈ [ − , c and c , can be defined as d t ( c , c ) : = n n X i = Pr (cid:16) p c i , t (cid:17) − Pr (cid:16) p c i , t (cid:17) (7)wherein Pr (cid:16) p c j i , t (cid:17) stands for the probability of the i -th parameterbeing in the domain of the j -th contour. The value of d t ( c , c )indicates at a given time point how many agents contributeto each contour c and c . Another way to define d t ( c , c )would have been by the Kullback-Leibler divergence [11]. TheKullback-Leibler divergence is formally given by D KL ( f c || f c ) = Z ∞−∞ ln (cid:16) f c ( x ) f c ( x ) (cid:17) f c ( x ) dx (8)and expresses the di ff erence between two probability distribu-tions f c and f c when the distribution f c represents the refer-ence model against which one quantifies the loss of informationwhen using the model represented by the distribution f c . How-ever, the contours c and c are not necessarily comparable –and most of the time they are not. This brings us to introduce asimple distance function.We can now define the strain S t ( c , c ) ∈ [0 ,
1] associatedwith the internal contradiction as the inverse of the absolutevalue of d t ( c , c ) S t ( c , c ) : = | d t ( c , c ) | . (9)Here we assume S t ( c , c ) to be strictly positive. To the notationdefined by (2) can now correspond a probabilistic distributionmeasure such that (cid:16) c c (cid:17) d : = d t ( c , c ) . (10)Similarly the internal contradiction represented by (1) can bewritten as c c ! d : = S t ( c , c ) . (11)What we described as an emerging singularity T can now bequantified by its dual manifestation in d and d .
3. Discusion
The physicist Erwin Schr¨odinger wrote that physical mod-els do not represent reality. They are simply useful when theyare adequate for describing we observe [12]. The present for-malism has multiple motivations. First, it reconnects with thevery basic desire to find general design principles that could ex-plain the features of underlying phenomena. Second, it repre-sents an attempt to rationally model what we experience. Using Schr¨odinger’s terms, the formalism we propose aims to ade-quately describe what we observe and connect di ff erent phe-nomena.Many authors, whether they were scientists, philosophers,poets or writers came to a similar conclusion: a fundamentaluncertainty, or ambiguity, prevents us from accessing a com-plete picture of the reality within a single rational framework[2, 3, 13]. It seems that at the core of every natural systemlies a fundamental contradiction, and this observation encom-passes rational thinking. Like in philosophy, one can pose thenon-existence of God as a postulate and draw the ethical conse-quences [14], one can also assume the existence of an internalcontradiction in every natural system and explore its ramifica-tions.In this respect, the morphological distance d t ( c , c ) and thetension S t ( c , c ) associated with it constitute a very general ap-proach for quantifying multifactorial situations whose outcomeis in an internal contradiction. A good example of such a situa-tion are transitions in single cell organism.Cellular mechanisms are often subjected to transitions; in eu-caryotes, transitions during cell division is orchestrated into or-dered states known as the cell cycle. The cycle is flanked bytwo main checkpoints between the passage from G to S andthe passage from G to M . G is the state prior to genome du-plication or DNA synthesis ( S ). G is the state prior to mitosis( M ) or cell division. The transition from one state to anotheris most likely based on feedback loops and molecular signal-ing, where cycle-specific molecules and proteins are synthe-sized and degraded based on extracellular inputs. In the caseof the cell cycle, cyclin-dependent kinases (Cdks) are knownto play an important role [15]. Conflicting or incomplete in-formation is not unusual in this process, leading to unresolvedsituations [16]. Hence, being able to measure the level of eachcontributing signals permits the computation of d t ( G , S ) and S t ( G , S ), or d t ( G , M ) and S t ( G , M ). The problem of havingconflicting or incomplete information can also result from anincomplete screening of the environment. In this case, a con-tradiction can be minimized by a more appropriate distributionof cellular receptors or with a di ff erent set of pathways bettersuited to analyze the situation. Whether the cell can fully re-solved the space- and time-dependent orchestration of its owndivision is still, in our opinion, an open question.Another concrete example can be drawn from the field ofepigenetics. Bivalent lineage-specific genes constitute a largefamily composed of more than 2,000 members [17, 18, 19, 20].They are characterized by the presence of both activating andrepressive chromatin marks, H3K4me3 and H3K27me3. Mostbivalent domains are often associated with the loci of key de-velopmental transcription factors, morphogens, as well as cellsurface molecules [17, 21]. A prevailing hypothesis about bi-valent genes is that H3K27me3 suppresses the expression oflineage control genes, but the H3K4me3 keeps it in a poisedstate for activation [17], although the idea of epigenetic pre-determination of such poised states was recently challengedin [20]. In thist study, the authors emphasize that the lev-els of H3K4me3 and H3K27me3 associated with bivalent do-mains are widely variable; together with RNA polymerase oc-3upancy data and transcript information, they hypothesize thatH3K4me3 is a marker for variable degrees of stochastic tran-scription, while H3K27me3 serves to regulate the expressionof bivalent genes. Nonetheless, the incidences of stochas-tic transcription reported were at basal level, and the bivalentgenes could still be considered poised [20]. On di ff erentiation,most of the bivalent modifications are resolved into either ac-tivated or repressed states [17]. It is consequently plausible tohypothesize that ambivalence at the level of gene expressioncorresponds to cases where two, or possibly, more gene reg-ulation outcomes are linked in a shared state which is unde-fined until commitment. The final outcome can be modeled interms of the distance d t ( H K me , H K me
3) and the tension S t ( H K me , H K me
3) depending on the level of histonemethylation and how many transcription factors are e ff ectivelypresent to trigger the state transition.In a similar way, with the rise of virtual social networks, itis possible to have access to a very large set of indexes aboutwhat individuals in a society feel at a given time point. Trans-lating opinions into variables, one could technically calculatetheir probability density function to estimate the social tensionin a society.To further strengthen the essence of our formalism, the con-cept of cooperation among individuals provides a straightfor-ward example. Individuals like humans or mammals can eithercooperate or compete. If we assume that competition is the dualaspect of cooperation (i.e. that cooperation cannot exist withoutcompetition), what we consider as competition or cooperationis only a matter of where we set the limit. One can increasecompetition and lower cooperation or the other way around.What makes competition really e ff ective is the ability of someentities to find a way to take advantage of other entities. In thismanner, the tension of the internal contradiction is broken, re-sulting in an asymmetry in space and time. This asymmetry canbe amplified to the point that the duality becomes residual foran external observer.We also want to emphasize that in our opinion, neither d t ( c , c ) nor S t ( c , c ) reach extrema, which are {− , } and { , } respectively. Indeed, a well-defined morphology alwayscontains a fraction of its parameter values outside what intrin-sically constitutes its manifestation. Consequently, the tensioncan never be totally absent from the system.Our formalism also constitutes an interesting framework todiscuss randomness. Randomness is assumed to be consub-stantial with several natural phenomena. The disintegration ofparticles is a typical example of process in which events occurin a random manner. Brownian motion of protein complexeswithin cell nucleus is another example [22]. Predictions canbe made only based on probabilities. Similarly, the mathemat-ical development realized for building random number gener-ators turns out to be a non-trivial task, and only finite seriescan be achieved. Such series are usually called pseudo-randomnumber generators (PRNGs). How random are transcendentalnumbers such as π or e is still an open question [23]. Here weargue that a true paradoxical situation can only generate a ran-dom outcome and constitutes a natural source of randomnessin living processes, which in turn may act as one of the evo- lutionary forces in nature. In that case, a clear venue whereour formalism may have interesting applications is in artificialintelligence. Being able to realize a device based on an intrin-sic tension, and capable of producing its own decisions, via asingularity T , can lead to a new generation of self-evolving en-tities.Finally, one can point out that we have not investigated mul-tiple morphologies, where the outcome relies on multiple con-tours. Having to choose between multiple possibilities clearlyconstitutes the most common situations in nature. Howeveronce the possibilities have been weighted for a subsequent ac-tion, all these possibilities are narrowed down to at most twopossibilities; and this is clearly what constitutes the di ffi cultyof making a choice.
4. Conclusion
When the Copenhagen interpretation of quantum mechan-ics was formulated by physicists such as Niels Bohr, WernerHeisenberg and Erwin Schr¨odinger among others [24], con-cepts like entanglement, wave function and uncertainty under-mined the idea of a complete picture of a physical reality. Evenif these concepts sounded new to scientists used to work withclassical mechanics, it is clear that any attempt to fully describereality in a deterministic way would have sooner or later failed.Who can possibly predict what will happen tomorrow? Possi-bilities in the future are entangled and even after they happened,there is not always an objective measure to decide which ofthese possibilities really occurred.The present formalism suggests the central perspective of in-ternal contradiction to examine phenomena; this might resem-ble to some extent the concept of entanglement in quantummechanics, since the intrinsic ambiguity contained in the wavefunction of a multiple states system is resolved upon measure-ment. In contrast to the formalism developed in quantum me-chanics, the physical manifestation of the internal contradictiondescribed by the the chicken or the egg dilemma does not leadto a complete loss of correlation between the contours. As wealready mentioned, an internal contradiction is always charac-terized by a tension and a di ff erence. Hence, when a patternemerges from an internal contradiction, it has to minimized itstension into one of the contours. In other words, the patterndoes not have an independent reality. A pattern emerges whenthe circumstances allow the reduction of tension towards oneof the contours, but the ambiguity remains as manifestation oftension.The formalism we proposed to conceptualize an observationis believed to be essential in many di ff erent phenomena – thatis, a scale-free design principle, whose applications are not nec-essarily limited to biological systems.
5. Acknowledgment
The author thanks Homayoun C. Bagheri for the fruitful dis-cussions and his support during the writing of the manuscriptand Maria Pamela Dobay for her valuable comments and proof-reading.4 eferences [1] D. R. Hofstadter, G¨odel, Escher, Bach : an eternal golden braid, BasicBooks, New York, 1979.[2] K. G¨odel, ¨Uber formal unentscheidbare s¨atze der principia mathematicaund verwandter systeme, I. Monatshefte f¨ur Mathematik und Physik 38(1931) 173–198.[3] A. Einstein, B. Podolsky, N. Rosen, Can quantum-mechanical descriptionof physical reality be considered complete?, Phys. Rev. 47 (1935) 777–780.[4] H. Everett, “relative state” formulation of quantum mechanics, Rev. Mod.Phys. 29 (1957) 454–462.[5] M. Eigen, Selforganization of matter and the evolution of biologicalmacromolecules, Naturwissenschaften 58 (10) (1971) 465–523.[6] A. Dobay, W. van Toledo, Informational processing of morphological andtopological transitions in biology, Arkhai 13 (2008) 5–16.[7] J. Schwinger, On gauge invariance and vacuum polarization, Phys. Rev.82 (1951) 664–679.[8] Z. Xie, L. E. Ulrich, I. B. Zhulin, G. Alexandre, Pas domain containingchemoreceptor couples dynamic changes in metabolism with chemotaxis,Proceedings of the National Academy of Sciences 107 (5) (2010) 2235–2240.[9] J. S. Weitz, Y. Mileyko, R. I. Joh, E. O. Voit, Collective decision makingin bacterial viruses, Biophysical Journal 95 (6) (2008) 2673–2680.[10] M. Bayes, M. Price, An essay towards solving a problem in the doctrineof chances. by the late rev. mr. bayes, f. r. s. communicated by mr. price, ina letter to john canton, a. m. f. r. s., Philosophical Transactions 53 (1763)370–418.[11] S. Kullback, R. A. Leibler, On information and su ffi ciency, Annals ofMathematical Statistics 22 (1) (1951) 79–86.[12] E. Schr¨odinger, Science and Humanism: Physics in Our Time, Cambridgeuniv. Press, New York, 1952.[13] W. Heisenberg, ¨Uber den anschaulichen inhalt der quantentheoretischenkinematik und mechanik, Zeitschrift f¨ur Physik 43 (3-4) (1927) 172–198.[14] J.-P. Sartre, L’ ˆEtre et le N´eant, ´Editions Gallimard, Paris, 1943.[15] W. A. Murray, Recycling the cell cycle: Cyclins revisited, Cell 116 (2)(2004) 221–234.[16] H. W. Mankouri, D. Huttner, I. D. Hickson, How unfinished business froms-phase a ff ects mitosis and beyond, The EMBO Journal 32 (20) (2013)2661–2671.[17] B. E. Bernstein, T. S. Mikkelsen, X. Xie, M. Kamal, D. J. Huebert, J. Cu ff ,B. Fry, A. Meissner, M. Wernig, K. Plath, R. Jaenisch, A. Wagschal,R. Feil, S. L. Schreiber, E. S. Lander, A bivalent chromatin structuremarks key developmental genes in embryonic stem cells, Cell 125 (2)(2006) 315–26.[18] A. Barski, S. Cuddapah, K. Cui, T. Roh, D. Schones, High-resolutionprofiling of histone methylations in the human genome, Cell 129 (2007)823–37.[19] K. Cui, C. Zang, T. Roh, D. Schones, R. Childs, W. Peng, K. Zhao, Chro-matin signatures in multipotent human hematopoietic stem cells indicatethe fate of bivalent genes during di ff erentiation, Cell Stem Cell 4 (1)(2009) 80–93.[20] M. D. Gobbi, D. Garrick, M. Lynch, D. Vernimmen, J. R. Hughes,N. Goardon, S. Luc, K. M. Lower, J. A. Sloane-Stanley, C. Pina, S. Soneji,R. Renella, T. Enver, S. Taylor, S. E. W. Jacobsen, P. Vyas, R. J. Gibbons,D. R. Higgs, Generation of bivalent chromatin domains during cell fatedecisions, Epigenetics & Chromatin 2010 3:2 4 (1) (2011) 9.[21] T. Mikkelsen, M. Ku, D. Ja ff e, B. Issac, E. Lieberman, G. Giannoukos,P. Alvarez, W. Brockman, T. Kim, R. Koche, Genome-wide maps of chro-matin state in pluripotent and lineage-committed cells, Nature 448 (7153)(2007) 553–560.[22] J. P. Siebrasse, R. Veith, A. Dobay, H. Leonhardt, B. Daneholt, U. Ku-bitscheck, Discontinuous movement of mrnp particles in nucleoplasmicregions devoid of chromatin, Proceedings of the National Academy ofSciences 105 (51) (2008) 20291–20296.[23] S. Pincus, R. E. Kalman, Not all (possibly) “random” sequences are cre-ated equal, Proceedings of the National Academy of Sciences 94 (8)(1997) 3513–3518.[24] J. Mehra, H. Rechenberg, The Historical Development of Quantum The-ory, Springer-Verlag, New York, 1982.e, B. Issac, E. Lieberman, G. Giannoukos,P. Alvarez, W. Brockman, T. Kim, R. Koche, Genome-wide maps of chro-matin state in pluripotent and lineage-committed cells, Nature 448 (7153)(2007) 553–560.[22] J. P. Siebrasse, R. Veith, A. Dobay, H. Leonhardt, B. Daneholt, U. Ku-bitscheck, Discontinuous movement of mrnp particles in nucleoplasmicregions devoid of chromatin, Proceedings of the National Academy ofSciences 105 (51) (2008) 20291–20296.[23] S. Pincus, R. E. Kalman, Not all (possibly) “random” sequences are cre-ated equal, Proceedings of the National Academy of Sciences 94 (8)(1997) 3513–3518.[24] J. Mehra, H. Rechenberg, The Historical Development of Quantum The-ory, Springer-Verlag, New York, 1982.