Self-Organization of Molecular Systems

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The part must be an organ producing the other parts—each, consequently, reciprocally producing the others Only under these conditions and upon these terms can such a product be an organized and self-organized being, and, as such, be called a physical end. Sadi Carnot — and Rudolf Clausius — discovered the second law of thermodynamics in the 19th century. It states that total entropy , sometimes understood as disorder, will always increase over time in an isolated system.

This means that a system cannot spontaneously increase its order without an external relationship that decreases order elsewhere in the system e. This idea became associated with Lamarckism and fell into disrepute until the early 20th century, when D'Arcy Wentworth Thompson — attempted to revive it. The psychiatrist and engineer W. Ross Ashby introduced the term "self-organizing" to contemporary science in Self-organization was associated [ by whom? Around , a concept of guided self-organization started to take shape.

This approach aims to regulate self-organization for specific purposes, so that a dynamical system may reach specific attractors or outcomes. The regulation constrains a self-organizing process within a complex system by restricting local interactions between the system components, rather than following an explicit control mechanism or a global design blueprint.

The many self-organizing phenomena in physics include phase transitions and spontaneous symmetry breaking such as spontaneous magnetization and crystal growth in classical physics , and the laser , [28] superconductivity and Bose—Einstein condensation in quantum physics. It is found in self-organized criticality in dynamical systems , in tribology , in spin foam systems, and in loop quantum gravity , [29] river basins and deltas, in dendritic solidification snow flakes , and in turbulent structure. Self-organization in chemistry includes molecular self-assembly , [31] reaction-diffusion systems and oscillating reactions , [32] autocatalytic networks, liquid crystals , [33] grid complexes , colloidal crystals , self-assembled monolayers , [34] [35] micelles , microphase separation of block copolymers , and Langmuir-Blodgett films.

Self-organization in biology [3] [37] can be observed in spontaneous folding of proteins and other biomacromolecules, formation of lipid bilayer membranes, pattern formation and morphogenesis in developmental biology , the coordination of human movement, social behaviour in insects bees , ants , termites , [38] and mammals , flocking behaviour in birds and fish.

The mathematical biologist Stuart Kauffman and other structuralists have suggested that self-organization may play roles alongside natural selection in three areas of evolutionary biology , namely population dynamics , molecular evolution , and morphogenesis. However, this does not take into account the essential role of energy in driving biochemical reactions in cells. The systems of reactions in any cell are self-catalyzing but not simply self-organizing as they are thermodynamically open systems relying on a continuous input of energy.

Phenomena from mathematics and computer science such as cellular automata , random graphs , and some instances of evolutionary computation and artificial life exhibit features of self-organization. In swarm robotics , self-organization is used to produce emergent behavior. In particular the theory of random graphs has been used as a justification for self-organization as a general principle of complex systems. In the field of multi-agent systems , understanding how to engineer systems that are capable of presenting self-organized behavior is an active research area.

If the solution is considered as a state of the iterative system, the optimal solution is the selected, converged structure of the system. These emerge from bottom-up interactions, unlike top-down hierarchical networks within organizations, which are not self-organizing. Norbert Wiener regarded the automatic serial identification of a black box and its subsequent reproduction as self-organization in cybernetics. Eric Drexler sees self-replication as a key step in nano and universal assembly.

By contrast, the four concurrently connected galvanometers of W. Ross Ashby 's Homeostat hunt , when perturbed, to converge on one of many possible stable states. Nyquist stability criterion. Warren McCulloch proposed "Redundancy of Potential Command" [56] as characteristic of the organization of the brain and human nervous system and the necessary condition for self-organization. In the s Stafford Beer considered self-organization necessary for autonomy in persisting and living systems.

He applied his viable system model to management. It consists of five parts: the monitoring of performance of the survival processes 1 , their management by recursive application of regulation 2 , homeostatic operational control 3 and development 4 which produce maintenance of identity 5 under environmental perturbation.

Focus is prioritized by an alerting "algedonic loop" feedback: a sensitivity to both pain and pleasure produced from under-performance or over-performance relative to a standard capability. In the s Gordon Pask argued that von Foerster's H and Hmax were not independent, but interacted via countably infinite recursive concurrent spin processes [60] which he called concepts. His strict definition of concept "a procedure to bring about a relation" [61] permitted his theorem "Like concepts repel, unlike concepts attract" [62] to state a general spin-based principle of self-organization.

His edict, an exclusion principle, "There are No Doppelgangers " means no two concepts can be the same. After sufficient time, all concepts attract and coalesce as pink noise. The theory applies to all organizationally closed or homeostatic processes that produce enduring and coherent products which evolve, learn and adapt. The self-organizing behaviour of social animals and the self-organization of simple mathematical structures both suggest that self-organization should be expected in human society.

Tell-tale signs of self-organization are usually statistical properties shared with self-organizing physical systems. Examples such as critical mass , herd behaviour , groupthink and others, abound in sociology , economics , behavioral finance and anthropology. In social theory, the concept of self-referentiality has been introduced as a sociological application of self-organization theory by Niklas Luhmann For Luhmann the elements of a social system are self-producing communications, i.

For Luhmann human beings are sensors in the environment of the system. Luhmann developed an evolutionary theory of Society and its subsystems, using functional analyses and systems theory. In economics, a market economy is sometimes said to be self-organizing. Paul Krugman has written on the role that market self-organization plays in the business cycle in his book "The Self Organizing Economy".

Neo-classical economists hold that imposing central planning usually makes the self-organized economic system less efficient. On the other end of the spectrum, economists consider that market failures are so significant that self-organization produces bad results and that the state should direct production and pricing.

Most economists adopt an intermediate position and recommend a mixture of market economy and command economy characteristics sometimes called a mixed economy. When applied to economics, the concept of self-organization can quickly become ideologically imbued.

Enabling others to "learn how to learn" [70] is often taken to mean instructing them [71] how to submit to being taught. Self-organised learning S. The self-organizing behavior of drivers in traffic flow determines almost all the spatiotemporal behavior of traffic, such as traffic breakdown at a highway bottleneck, highway capacity, and the emergence of moving traffic jams. In — these complex self-organizing effects were explained by Boris Kerner 's three-phase traffic theory.

Order appears spontaneously in the evolution of language as individual and population behaviour interacts with biological evolution. Self-organized funding allocation SOFA is a method of distributing funding for scientific research. In this system, each researcher is allocated an equal amount of funding, and is required to anonymously allocate a fraction of their funds to the research of others.

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Proponents of SOFA argue that it would result in similar distribution of funding as the present grant system, but with less overhead. Most scientists would agree with the critical view expressed in Problems of Biological Physics Springer Verlag, by the biophysicist L. Blumenfeld, when he wrote: "The meaningful macroscopic ordering of biological structure does not arise due to the increase of certain parameters or a system above their critical values.

These structures are built according to program-like complicated architectural structures, the meaningful information created during many billions of years of chemical and biological evolution being used. Of course, Blumenfeld does not answer the further question of how those program-like structures emerge in the first place.

Chemical Complexity: Self-Organization Processes in Molecular Systems (A.S. Mikhailov & G. Ertl)

His explanation leads directly to infinite regress. In short, they [Prigogine and Stengers] maintain that time irreversibility is not derived from a time-independent microworld, but is itself fundamental. The virtue of their idea is that it resolves what they perceive as a "clash of doctrines" about the nature of time in physics. Most physicists would agree that there is neither empirical evidence to support their view, nor is there a mathematical necessity for it.

There is no "clash of doctrines. In theology , Thomas Aquinas — in his Summa Theologica assumes a teleological created universe in rejecting the idea that something can be a self-sufficient cause of its own organization: [94]. Since nature works for a determinate end under the direction of a higher agent, whatever is done by nature must needs be traced back to God, as to its first cause. So also whatever is done voluntarily must also be traced back to some higher cause other than human reason or will, since these can change or fail; for all things that are changeable and capable of defect must be traced back to an immovable and self-necessary first principle, as was shown in the body of the Article.

From Wikipedia, the free encyclopedia. It has been suggested that Spontaneous order be merged into this article. Discuss Proposed since May Further information: Spontaneous order. See also: Self-assembly and Self-assembly of nanoparticles. Further information: Biological organisation. Main article: Self-organization in cybernetics. Main article: Spontaneous order. Main article: Three-phase traffic theory.

Autopoiesis Autowave Self-organized criticality control Free energy principle Information theory Constructal law Emergence. Journal of Materials Chemistry A.

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International Journal of Signs and Semiotic Systems. Self-organization in Biological Systems. Princeton studies in complexity reprint ed. Princeton University Press. Retrieved Cellular Automata: A Discrete Universe. World Scientific. Self-organization and Emergence in Life Sciences. Swarm intelligence: from natural to artificial systems. The Journal of General Psychology. Heinz von Foerster and George W. Zopf, Jr. Office of Naval Research. Yovits and S. Cameron eds. International Encyclopedia of Systems and Cybernetics 2nd ed.

Berlin : Walter de Gruyter. Self-organization in nonequilibrium systems: From dissipative structures to order through fluctuations. Wiley, New York. Order out of chaos: Man's new dialogue with nature. Bantam Books. Applied Soft Computing. Reading Lucretius in the Renaissance. Harvard University Press. Ada Palmer explores how Renaissance readers, such as Machiavelli, Pomponio Leto, and Montaigne, actually ingested and disseminated Lucretius, CUP Archive. Active sets of molecules are transient and what is observed in experiments is a superposition of subsequent active sets, which we call the expressed set of agents.

The regulatory interactions between the expressed agents we call the expressed regulatory network. To find the property of differentiability in a regulatory network model therefore requires that one network is capable of producing different expression modes while perturbations external signal only modify active sets locally and the particular expression mode can be restored. The six dynamical properties we have listed above have been addressed with a variety of conceptually different models.

The essence of all these models is that they try to capture the dynamics induced by positive and negative feed back loops within the GRN. The choice of model depends largely on the type and resolution coarse graining of experimental data. At the single cell level cellular activity can be modeled by nonlinear stochastic differential equations [34] , [35] which can explain homeostasis, periodic and multi-stable behavior. To do so one considers a dynamic variable associated with each agent. If index represents a type of effector molecule, then denotes the concentration abundance of those molecules.

If index for instance marks the protein is the -protein concentration abundance in the cell. If represents a gene, then represents the frequency of being expressed. The dynamics governed by a GRN is given by a set of coupled nonlinear differential equations 1 where is a nonlinear function capturing the GRN. Note that can have stochastic components. Analysis of such systems is often complicated by the interplay between fluctuations and nonlinearities [36]. Differential equation models can be approximated by cellular automata, Boolean or piecewise-linear models. SOC dynamics was also discussed in continuous differential equation based models [43] , [44].

Boolean and piecewise-linear models share common origins in the work of Glass and Kauffman, [45] , and have extensively been used for modeling and analyzing GRN [46] — [49]. For their superior properties in approximating nonlinear systems in principle to any suitable precision piecewise-linear models also are applied in different disciplines, for instance for modeling highly nonlinear electronic circuits [50]. In the context of GRN both boolean and piecewise-linear models usually are used for describing nonlinear dynamics with switch-like regulatory elements frequently observed in biological regulatory processes [51] , [52].

Such switches react if the concentration of an agent the signal crosses a specific threshold level. To model such switches in regulation networks of molecular agents with concentrations the range space of concentrations is cut into segments defined by the threshold values where the concentration can trigger a regulatory switch.

These segments are called regulatory domains e. In each such domain Eq. If in a regulatory domain , then promotes the production of. If , then suppresses. If has no influence on. On the full range of concentration values both and are step-functions depending on all concentrations in principle, but being constant in each regulatory domain. In other words, since depends only on the regulatory domain it can be decomposed into boolean functions on the regulatory domains.

Let index regulatory domains of the system. The function if is contained in the 'th regulatory domain and zero otherwise. Then can be written as with being the value of the production term in the 'th regulatory domain. Between regulatory domains the system switches from one linear behavior to another. As an example for interpreting Eq. In the model this corresponds to interaction weights and. Further promotes the transcription of , so that , and can block the N-terminal transcription-activation domain TAD of so that for transcription factors which are activated by -TAD implying.

Assuming that -protein does not degrade on its own, i. Given that the interaction matrix of the regulatory network is invertible which is almost certainly true for the biologically relevant range of connectivities of GRN Eq. The fixed-point is stable unstable and will be attracted repelled by. If is stable and for all then is one of possibly many stationary solution of Eq. Not all models approximating nonlinear differential equation descriptions of GRN are equally suited to capture all GRN properties discussed above simultaneously depending on whether discrete Boolean, cellular automata or smooth differential equation features dominate the model.

However there exists a surprisingly simple class of models which exhibits all desired GRN properties. Here we present such a simple model that captures all of the above dynamical properties. We find that the alternating dynamics plays a key role for the stability of regulatory systems and for the formation of SOC dynamics in particular [43] , [44]. We show that experimental facts, linking variations of decay rates observed between different cell-types of an organism to variations of the abundance of intra-cellular biochemical agents in these cell-types, correspond to a differences in the expressed genetic regulatory network, and b these differences can be controlled via decay rates of intracellular agents.

In other words typical expression modes cyclical sequences of successive active sub-networks of the GRN can be altered and switched by controlling decay rates. Setting in Eq. Since and may depend on the regulatory domain this corresponds exactly to the class of Glass-Kauffman piece-wise linear models, [45] , [53]. In Glass-Kauffman systems, [45] , concentrations usually remain positive for all times , given positive initial conditions and for all since concentrations can at best decay exponentially with time.

This makes it impossible to produce alternating activity of agents. For , in a Glass-Kauffman system, to become zero within a finite time, production rates - which are non-negative by definition - would have to become negative. Equation 2 generalizes this class of models to systems allowing to explicitly model linear regulatory interactions between agents within each regulatory domain. Suppose suppresses then can in principle down-regulate in a finite time and positivity of solutions of Eq.

Positivity non-negativity of solutions needs to be ensured as a constraint on the piece-wise linear dynamics 4 This constraint alters the linear dynamics of Eq. Whenever a concentration becomes zero at time then remains zero for for as long as , according to Eq. If for then is no longer subject to the positivity constraint and continues to evolve according to Eq. Agent is said to be active at time , if and inactive , if. To simplify the discussion in the following we only consider systems with a single regulatory domain - such that all nonlinear behavior of the dynamics is solely due to the positivity-constraint.

The positivity constraint Eq. At any point in time there will be a sub-set of agents with non-vanishing concentrations which we call the active set of agents. The remaining agents have zero concentration, and therefore do not actively influence the concentrations of any of the non-vanishing agents.

There exist different active sets, i. Each active set can be uniquely identified by an index e. In the course of time some agents will vanish while others re-appear, so that one effectively observes a sequence of sets of active agents 5 being the initial active set. The active set switches to active set at time. In each time interval of duration it is thus possible to only consider the regulatory sub-network acting on the set of active agents.

This sub-network is described by the part of the full interaction matrix , where and are restricted to the set of active agents. These sub-matrices we call active networks and denote them by. The concentration vector of active agents we call. The attractiveness of this description arises through the fact that it becomes possible to understand the dynamics by considering the sequences of active networks 7 which allows to analyze dynamical properties in terms of eigenvalues and eigenvectors of the active sub-matrices see materials and methods.

This model can be shown to be mathematically equivalent to [43] , [44]. The dynamics of nonlinear systems in general and sequentially linear system in particular converges to different attractors of the dynamics fixed points, limit cycles. Sequentially linear systems can possess multiple distinct limit cycles and fixed points.

Perturbations or different initial conditions may push a system from one to another attractor. The question of how many different attractors a sequentially linear system possesses goes beyond the scope of this paper and will be discussed elsewhere. In the picture of sequentially linear dynamics it becomes possible to identify operational modes of a cell as a particular sequence of active networks. Cell types in ordinary operational modes may be classified by specific sequences of active networks. Two distinct possibilities for such sequences exist.

One possibility is that, after some initial switching events, a system ends up in a stationary state associated with a particular active network of the system see materials and methods. The other possibility is that a system converges to a periodic dynamics with an associated periodic sequence of active networks.

As a hypothetical example a liver cell under typical conditions might be characterized by a periodic sequence , whereas an endothelial cell is given by. Note that all types share the same full regulatory network. This separates timescales of the dynamics: on the fast timescale the dynamics is continuous and characterized by linear changes of the concentrations. On the slower time-scale the dynamics is characterized by discrete changes of active sets.

The change from one sequence of active sets to another can be interpreted as the expression modes of different cell-types cell differentiation and we show that changes in decay rates of molecular species trigger switches between expression modes.

When molecules leave tire tracks: A new approach to optimizing molecular self-organization

As an example for sequentially linear dynamics we consider a system with molecular agents, , for all agents , and a regulatory network given by 8 The dynamics of this system over one period is shown in Fig. The property describing the stability of an active set is the maximal real part of the eigenvalues of the active matrix denoted.

The number denotes the number of time-domains in a periodic sequence of active networks, i. In this example there are four time-domains associated with three different active sets which are periodically repeated. The sequence starts in time-domain with active set with maximum real eigenvalue. Positive means that the fixed point of the active set is unstable and the associated leading eigenvalue implies that the concentration of one agent green is decaying to zero. The positivity condition deactivates this agent as its concentration becomes zero and the system enters time-domain as the active set switches to with.

Negative means that the fixed point is stable and tries to approach. This leads to the deactivated agent green becoming produced again and the system switches back to entering the third time-domain. In time-domain the initial conditions differs from the one in time-domain and a different node magenta gets deactivated. The system switches to with at the beginning of the fourth time-domain. This means is a stable fixed-point and the inactive node magenta eventually gets produced again as the system switches back to the beginning and enters the next period.

The system is thus precisely characterized by the sequence. The eigenvalue spectra of the sub-matrices associated with subsequent time-domains are shown in Fig. Sequentially linear system with decay rate and the fixed point for all agents simulated with time-increment. Periodic time-series organized into a sequence of four domains with three different active sets.

For each time-domain the associated spectrum of eigenvalues for the active sets is shown in b. In c a 3 d Poincare map of the limit cycle is plotted together with the projection of in the center. The domains are marked with bold numbers and switching events with dots. The shift of the spectrum along the real axis depending on the decay rate is indicated. Some details of the dynamics, like the existence of multiple stable fixed-points, the periodicity of bounded attractors and temporal self-organization, can be mathematically fully understood. In [43] , [44] it was already shown mathematically that sequentially linear models exhibit homeostasis , and multi-stability.

This has been demonstrated for a wide range of system size , and a number of interactions connectivity and fixed decay rates. Periodic dynamics , and self-organized critical dynamics have been noted in [43] , [44] but were not clarified and require further explanation which is given in detail in materials and methods , where also a simple temporal balance condition is described and derived. The temporal balance condition states that the time-average over the real parts of the leading eigenvalues of the matrices in a sequence of active networks approximate the Lyapunov exponent.

The Lyapunov exponent measures the overall stability of a system stable, instable, critical and for sequences following a periodic attractor can be shown to be exactly zero. Inserting the values for and from table 1 into the balance condition, Eq. Although the balance equation gives only a crude approximation of the Lyapunov exponent it allows to understand why the example-system spends more time in the weakly instable time-domain and , than in the stable time-domains and which is obviously true from Fig.

Strong convergence needs less time to compensate for weak divergence. Temporal balance is a consequence of the mechanism of self-organization that fine-tunes switching times such that stable parts of the dynamics compensate instable parts of the dynamics exactly. This mechanism can be understood in the following way. Sequentially linear systems try to converge to a fixed point.

If it is reached the system becomes static. The fixed point might not be accessible however, meaning that the trajectory on the way toward the fixed point hits a boundary Fig. If the system does not converge to an accessible fixed point it is either unstable and some concentrations diverge, or the system circles through some of the possible active sets and converges onto an effective attractor - characterized in the sequence of active networks. In the later case small perturbations of on the attractor will vanish with time. This allows to show that bounded dynamics that does not converge to a fixed-point has to be periodic materials and methods.

Switching times are not static but react to perturbations of concentrations. Perturbations shift the occurrence of switching times proportional to the magnitude of the perturbation. While the perturbed dynamics returns to the attractor switching times cumulate small time-shifts resulting in a phase-shift of the periodic dynamics. A perturbation is remembered as a phase-shift of the periodic dynamics which neither grows exponentially nor dies out. We first show that the model is able to explain actual empirical data, including alternating dynamics.

These four agents are all part of the human estrogen nuclear receptor dynamics. The source of the Data is Metivier et. Data points were taken from Pigolotti et al. The TRIP1 data blue shows alternating activity which is reproduced perfectly by our sequential linear model. Time series of periodic binding of four proteins to the pS2 promoter after addition of estradiol - experimental data has been extracted from [54] , where a negative feedback-loop was proposed to explain the dynamics.

Experimental data due to [25] and [26] dotted lines is compared with a simulation of a SL system, based on the network shown in the inset, with uniform decay rates for all agents and fixed point concentrations. Correlation coefficients for simulated and measured time-series are for time larger and agents in order of the legend. The model simulation uses zero concentrations for all agents as initial condition and a time increment. For matching the simulation with experiment time in the model is shifted by. In the following we show how the change of decay rates induces changes from one cell-type to another.

In particular we show how changes of the overall strength of the decay rates results in differentiated dynamics, i. This allows to understand recent experimental observations which indicate correlations between cell-type, expressed sets of agents, and decay-rates [27] — [29] , [31] — [33]. For a fixed interaction network temporal self-organization can be maintained for a wide range of decay rates.

We show this in the same -node system considered in Fig.

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Figure 3 a shows the Lyapunov exponent as a function of. A plateau, where , is clearly visible. If the decay rate is larger than a critical value , the Lyapunov exponent becomes negative and the system stable. If the decay rate is smaller than a critical value of , temporal balance can not be achieved any more, refocusing breaks down, and the system becomes chaotic and trajectories diverge exponentially with.

Self-Organized Criticality

In Fig. Figure 3 b also shows that at several critical values of in the plateau region the sequences of active regulatory sub-networks changes when temporal balance can no longer be established merely by adapting the switching times of a sequence. Sequences do not usually change completely at critical values of and are only expanded by additional active subsets. This can be seen clearly in the 3D Poincare map of the dynamics Fig. The Lyapunov exponent of the four node system, Eq. In b the length of the periodic sequence of domains is plotted in green triangles and the number of different active sets as red squares.

In c the sequences of active sets are shown for decay rates , and. The limit circles for decay rates short sequence and long sequence are visualized in d in a Poincare map using three out of four phase-space dimensions. With decreasing the radius of the limit circle becomes wider and additional sets marked with colors become active. In e the spectra of eigenvalues are shown for all the appearing active sets with. The mathematical reason why such critical decay rates exist is that changes of shift the eigenvalue spectra of the active interaction matrix , shown in Fig.

The real part of the leading eigenvalues, , is becoming smaller larger than zero and becomes an attractor repellor of. The stable fixed point then either is accessible and the dynamic changes from periodic to stationary or inaccessible and the dynamic changes qualitatively but remains periodic. Which agents become active in a given active set is depicted in Fig. If node is active in active set then the associated field is white and black otherwise. The number of expressed agents is the number of agents that are active at least once during a period of the dynamics.

To demonstrate that not only the periodic activation of agents depends on but also the number of expressed nodes itself, we consider a larger sequentially linear system with agents.

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  7. The interaction matrix of the system is a random matrix with average connectivity , meaning for each node interactions with other agents have been randomly chosen with equal probability. Each non-zero entry, describing such an interaction, is drawn from a normal distribution with mean zero and a standard deviation of.

    This means that the interaction strength is of magnitude on average and has positive or negative sign with equal probability. For large decay rates the system is stable and is a fixed-point of the dynamics. As decreases becomes unstable for. However for the system ends up in some stable accessible fixed point so that approaches a stationary state and.

    In this range increases with. The plateau with stable self-organized critical dynamics only emerges in the range where number of active sets and expressed network size vary strongly. A small window of stability exists for see inset. Example of a SL system with and and identical initial conditions for all values of expressing different portions of the regulatory networks. For is stable. In the range the has become unstable but the plateau can not form since the dynamic finds active sets with stable and accessible.

    The inset in b shows that in the plateau region a small window, , exists where again an active set contains an accessible attracting the dynamics. In the range the plateau forms and dynamics gets periodic. For the system gets unstable. The strong dependence of on the decay-rate up to of the total regulatory network demonstrates clearly that decay-rates alone massively influence sequences of active systems without changing the interaction strength between agents in the regulatory network at all.

    Moreover, decay rates can also cause switches between fixed-point dynamics and periodic dynamics. While fixed points favor larger decay-rates in the example there can also exist fixed points for smaller decay rates window of stability where systems favor periodic dynamics.


    We presented a model which de-composes the dynamics of molecular concentrations — governed by the full molecular regulatory networks — into a temporal sequence of active sub-networks. This novel type of model allows not only to reduce the vast complexity of the full regulatory network into sub-networks of manageable size but further to approximate the complicated dynamics by linear methods. The intrinsic nonlinearities in the system which lead to alternating dynamics in concentrations as found in countless experiments are absorbed into switching events, where the dynamics of one linear system switches to another one.

    In this view different cell types correspond to different sequences of active sub-networks over time. These sequentially linear models allow not only for the first time to describe all the relevant dynamical features of the GNR homeostasis, multi-stability, periodic dynamics, alternating activity, differentiability, and self-organized criticality , but also offers the handle to understand the role of molecular decay rates.