Transitions between oscillatory modes in a glycolytic model system

Dissecting cell fate dynamics in pediatric glioblastoma through the lens of complex systems and cellular cybernetics

Cancers are complex dynamic ecosystems. Reductionist approaches to science are inadequate in characterizing their self-organized patterns and collective emergent behaviors. Since current approaches to single-cell analysis in cancer systems rely primarily on single time-point multiomics, many of the temporal features and causal adaptive behaviors in cancer dynamics are vastly ignored. As such, tools and concepts from the interdisciplinary paradigm of complex systems theory are introduced herein to decode the cellular cybernetics of cancer differentiation dynamics and behavioral patterns. An intuition for the attractors and complex networks underlying cancer processes such as cell fate decision-making, multiscale pattern formation systems, and epigenetic state-transitions is developed. The applications of complex systems physics in paving targeted therapies and causal pattern discovery in precision oncology are discussed. Pediatric high-grade gliomas are discussed as a model-system to demonstrate that cancers are complex adaptive systems, in which the emergence and selection of heterogeneous cellular states and phenotypic plasticity are driven by complex multiscale network dynamics. In specific, pediatric glioblastoma (GBM) is used as a proof-of-concept model to illustrate the applications of the complex systems framework in understanding GBM cell fate decisions and decoding their adaptive cellular dynamics. The scope of these tools in forecasting cancer cell fate dynamics in the emerging field of computational oncology and patient-centered systems medicine is highlighted.

Synchronization of Bioelectric Oscillations in Networks of Nonexcitable Cells: From Single-Cell to Multicellular States

Biological networks use collective oscillations for information processing tasks. In particular, oscillatory membrane potentials have been observed in nonexcitable cells and bacterial communities where specific ion channel proteins contribute to the bioelectric coordination of large populations. We aim at describing theoretically the oscillatory spatiotemporal patterns that emerge at the multicellular level from the single-cell bioelectric dynamics. To this end, we focus on two key questions: (i) What single-cell properties are relevant to multicellular behavior? (ii) What properties defined at the multicellular level can allow an external control of the bioelectric dynamics? In particular, we explore the interplay between transcriptional and translational dynamics and membrane potential dynamics in a model multicellular ensemble, describe the spatiotemporal patterns that arise when the average electric potential allows groups of cells to act as a coordinated multicellular patch, and characterize the resulting synchronization phenomena. The simulations concern bioelectric networks and collective communication across different scales based on oscillatory and synchronization phenomena, thus shedding light on the physiological dynamics of a wide range of endogenous contexts across embryogenesis and regeneration.

Gene therapy--why can it fail?

The success of reductionism in medicine has enabled the experimental expression of individual genes in complex living systems. The promise of gene therapy, permanent reversal or amelioration of disease symptoms without dependence on a long-lasting intake of drugs, has come within reach because of these conceptual and technical advances in molecular biology. However, there have been setbacks posing serious questions for the medical community. The incidents came at a time when technical advances in the manipulation of DNA had led to wide-spread testing of gene based therapies. In fact, the major limiting factor of this approach had been perceived to be gene delivery rather than toxicity. Here we discuss the hypothesis that knowledge of DNA sequences for relevant genes alone will not be sufficient to allow this promise to come to fruition, unless additional factors are recognized and addressed. The physiologic consequences of gene expression depend on gene dosage, transcriptional regulation by promoters, posttranscriptional editing, and interdependence among gene products, all of which vary among cells. The success of gene therapy will depend, in part, on insight into the factors summarized here, very much like successful drug therapy has depended on an understanding of the manifold influences of pharmacokinetics and pharmacodynamics. In principle, these considerations apply to all transfections, gene disruptions, and transgenic approaches and to potential clinical applications derived from them. Gaining insight and control over those factors may allow gene therapy to live up to current expectations.

Microtubule-like properties of the bacterial actin homolog ParM-R1

In preparation for mammalian cell division, microtubules repeatedly probe the cytoplasm to capture chromosomes and assemble the mitotic spindle. Critical features of this microtubule system are the formation of radial arrays centered at the centrosomes and dynamic instability, leading to persistent cycles of polymerization and depolymerization. Here, we show that actin homolog, ParM-R1 that drives segregation of the R1 multidrug resistance plasmid from Escherichia coli, can also self-organize in vitro into asters, which resemble astral microtubules. ParM-R1 asters grow from centrosome-like structures consisting of interconnected nodes related by a pseudo 8-fold symmetry. In addition, we show that ParM-R1 is able to perform persistent microtubule-like oscillations of assembly and disassembly. In vitro, a whole population of ParM-R1 filaments is synchronized between phases of growth and shrinkage, leading to prolonged synchronous oscillations even at physiological ParM-R1 concentrations. These results imply that the selection pressure to reliably segregate DNA during cell division has led to common mechanisms within diverse segregation machineries.

DEPENDENCE OF FLUX DISTRIBUTION AND SYSTEM COORDINATION ON DYNAMICAL STATES FOR BIOCHEMICAL SYSTEMS WITH MULTIPLE COEXISTING STATES

Flux distribution and system coordination in branched biochemical systems with a source producing multiple coexisting states have been studied both theoretically and numerically. For such systems, flux distribution depends sensitively on the dynamical states and the parameter values of enzymatic kinetics at branching point. Various dynamical states including noise-induced new states can be located by superimposing noisy fluctuations on a branched biochemical system with multiple coexisting states. Once noise induces transitions or new dynamical states, the flux through a specific branch may increase, maintain or decrease, depending on parameter values of enzymatic kinetics at branching point. Furthermore, system coordination can be destroyed by noise-induced dynamical changes. When at least one state cannot be coordinated in the absence of noise, noise-induced transitions may destroy system coordination. When all coexisting states are coordinated in the absence of noise, noise-induced new states may be still able to destroy system coordination. It is revealed that destruction of system coordination is due to the interaction of enzyme saturation and noise-induced dynamical changes. Finally, control of flux distribution and maintenance of system coordination for biochemical systems with multiple coexisting states are discussed.

Systems biology of microbial communities

Microbes exist naturally in a wide range of environments in communities where their interactions are significant, spanning the extremes of high acidity and high temperature environments to soil and the ocean. We present a practical discussion of three different approaches for modeling microbial communities: rate equations, individual-based modeling, and population dynamics. We illustrate the approaches with detailed examples. Each approach is best fit to different levels of system representation, and they have different needs for detailed biological input. Thus, this set of approaches is able to address the operation and function of microbial communities on a wide range of organizational levels.

Properties of strange attractors in yeast glycolysis

The properties of periodic and aperiodic glycolytic oscillations observed in yeast extracts under sinusoidal glucose input were analyzed by the following methods. (1) Spectral analysis, rendering sharp peaks for periodic responses and enhanced broad-band noise for aperiodic oscillations. (2) Phase plane analysis, leading to closed and to open trajectories for periodic and aperiodic oscillations, respectively. (3) Rotation of a phase plane proportionally to time, revealing strange attractors associated with the aperiodic oscillations. (4) Stroboscopic plot on the phase plane, showing that the strange attractors follow a stretch-fold-press process, if the stroboscoping phase is varied. (5) Stroboscopic transfer plot, admitting a period of three transfer processes and thus implying chaos according to the Li-Yorke theorem. (6) Determination of the rate of information production by differentiation of the transfer plot, yielding approx. 0.21 bits per min for the chaotically glycolyzing yeast extract.

Controlling tissue microenvironments: biomimetics, transport phenomena, and reacting systems

The reconstruction of tissues ex vivo and production of cells capable of maintaining a stable performance for extended time periods in sufficient quantity for synthetic or therapeutic purposes are primary objectives of tissue engineering. The ability to characterize and manipulate the cellular microenvironment is critical for successful implementation of such cell-based bioengineered systems. As a result, knowledge of fundamental biomimetics, transport phenomena, and reaction engineering concepts is essential to system design and development. Once the requirements of a specific tissue microenvironment are understood, the biomimetic system specifications can be identified and a design implemented. Utilization of novel membrane systems that are engineered to possess unique transport and reactive features is one successful approach presented here. The limited availability of tissue or cells for these systems dictates the need for microscale reactors. A capstone illustration based on cellular therapy for type 1 diabetes mellitus via encapsulation techniques is presented as a representative example of this approach, to stress the importance of integrated systems.

Complexity science and homeopathy: a synthetic overview

Homeopathy is founded on 'holistic' and 'vitalistic' paradigms, which may be interpreted--at least in part--in terms of a framework provided by the theory of dynamic systems and of complexity. The conceptual models and some experimental findings from complexity science may support the paradoxical claims of similia principle and of dilution/dynamization effects. It is argued that better appreciation of three main properties of complex systems: non-linearity, self-organization, and dynamicity, will not only add to our basic understanding of homeopathic phenomena but also illuminate new directions for experimental investigations and therapeutic settings.

The rhythm of yeast

Although yeast are unicellular and comparatively simple organisms, they have a sense of time which is not related to reproduction cycles. The glycolytic pathway exhibits oscillatory behaviour, i.e. the metabolite concentrations oscillate around phosphofructokinase. The frequency of these oscillations is about 1 min when using intact cells. Also a yeast cell extract can oscillate, though with a lower frequency. With intact cells the macroscopic oscillations can only be observed when most of the cells oscillate in concert. Transient oscillations can be observed upon simultaneous induction; sustained oscillations require an active synchronisation mechanism. Such an active synchronisation mechanism, which involves acetaldehyde as a signalling compound, operates under certain conditions. How common these oscillations are in the absence of a synchronisation mechanism is an open question. Under aerobic conditions an oscillatory metabolism can also be observed, but with a much lower frequency than the glycolytic oscillations. The frequency is between one and several hours. These oscillations are partly related to the reproductive cycle, i.e. the budding index also oscillates; however, under some conditions they are unrelated to the reproductive cycle, i.e. the budding index is constant. These oscillations also have an active synchronisation mechanism, which involves hydrogen sulfide as a synchronising agent. Oscillations with a frequency of days can be observed with yeast colonies on plates. Here the oscillations have a synchronisation mechanism which uses ammonia as a synchronising agent.

State selection in coupled identical biochemical systems with coexisting stable states

This work examines state selection for coupled biochemical systems with coexisting stable states. For biochemically identical biochemical systems, different coupled systems are examined for the coexistence of (a) one steady state and one oscillatory state or (b) two oscillatory states. For case (a), it is revealed that state selection is always governed by two key factors: the values of kinetic parameters and the coupling strength. When the coupling strength is small, the coupled systems remain in the basin of attraction of their original states. When it is sufficiently large, all coupled systems are always entrained, independently of their original states. Furthermore, for the entrainment, which of the two coexisting states is selected depends sensitively on the activity of recycling enzyme (one of kinetic parameters). It is shown that this is because changing the activity of recycling enzyme alters the size of basin of attraction of each state. When both systems in the same oscillatory state are coupled, an additional factor, namely phase shift between two oscillations, may also affect state selection, and coupling may cause the systems to select either the original oscillatory state or the coexisting steady state. In addition to the features of case (a), case (b) also supports quasiperiodic oscillations and synchronisation of two periodic oscillations. Implications of the results for understanding state selection during the evolution of coupled biochemical systems with coexisting stable states are discussed.

From simple to complex oscillatory behavior in metabolic and genetic control networks

We present an overview of mechanisms responsible for simple or complex oscillatory behavior in metabolic and genetic control networks. Besides simple periodic behavior corresponding to the evolution toward a limit cycle we consider complex modes of oscillatory behavior such as complex periodic oscillations of the bursting type and chaos. Multiple attractors are also discussed, e.g., the coexistence between a stable steady state and a stable limit cycle (hard excitation), or the coexistence between two simultaneously stable limit cycles (birhythmicity). We discuss mechanisms responsible for the transition from simple to complex oscillatory behavior by means of a number of models serving as selected examples. The models were originally proposed to account for simple periodic oscillations observed experimentally at the cellular level in a variety of biological systems. In a second stage, these models were modified to allow for complex oscillatory phenomena such as bursting, birhythmicity, or chaos. We consider successively (1) models based on enzyme regulation, proposed for glycolytic oscillations and for the control of successive phases of the cell cycle, respectively; (2) a model for intracellular Ca(2+) oscillations based on transport regulation; (3) a model for oscillations of cyclic AMP based on receptor desensitization in Dictyostelium cells; and (4) a model based on genetic regulation for circadian rhythms in Drosophila. Two main classes of mechanism leading from simple to complex oscillatory behavior are identified, namely (i) the interplay between two endogenous oscillatory mechanisms, which can take multiple forms, overt or more subtle, depending on whether the two oscillators each involve their own regulatory feedback loop or share a common feedback loop while differing by some related process, and (ii) self-modulation of the oscillator through feedback from the system's output on one of the parameters controlling oscillatory behavior. However, the latter mechanism may also be viewed as involving the interplay between two feedback processes, each of which might be capable of producing oscillations. Although our discussion primarily focuses on the case of autonomous oscillatory behavior, we also consider the case of nonautonomous complex oscillations in a model for circadian oscillations subjected to periodic forcing by a light-dark cycle and show that the occurrence of entrainment versus chaos in these conditions markedly depends on the wave form of periodic forcing. (c) 2001 American Institute of Physics.

ADP modulates the dynamic behavior of the glycolytic pathway of Escherichia coli

A mathematical model that includes biochemical interactions among the PTS system, phosphofructokinase (PFK), and pyruvate kinase (PK) is used to evaluate the dynamic behavior of the glycolytic pathway of Escherichia coli under steady-state conditions. The influence of ADP, phosphoenolpyruvate (PEP), and fructose-6-phosphate (F6P) on the dynamic regulation of this pathway is also analyzed. The model shows that the dynamic behavior of the system is affected significantly depending on whether ADP, PEP, or F6P is considered constant a steady state. Sustained oscillations are observed only when dADP/dt not equal 0 and completely suppressed if dADP/dt = 0 at any steady-state value. However, when PEP or F6P is constant, the system evolves toward the formation of stable limit cycles with periods ranging from 0.2 min to hours.

Diversity of temporal self-organized behaviors in a biochemical system

The numerical study of a glycolytic model formed by a system of three delay-differential equations revealed a notable richness of temporal structures which included the three main routes to chaos, as well as a multiplicity of stable coexisting states. The Feigenbaum, intermitency and quasiperiodicity routes to chaos can emerge in the biochemical oscillator. Moreover, different types of birhythmicity, trirhythmicity and hard excitation emerge in the phase space. For a single range of the control parameter it can be observed the coexistence of two quasiperiodicity routes to chaos, the coexistence of a stable steady state with a stable torus, and the coexistence of a strange attractor with different stable regimes such as chaos with different periodic regimes, chaos with bursting behavior, and chaos with torus. In most of the numerical studies, the biochemical oscillator has been considered under periodic input flux being the mean input flux rate 6 mM/h. On the other hand, several investigators have observed quasiperiodic time patterns and chaotic oscillations by monitoring the fluorescence of NADH in glycolyzing yeast under sinusoidal glucose input flux. Our numerical results match well with these experimental studies.

Efficient transition to growth on fermentable carbon sources in Saccharomyces cerevisiae requires signaling through the Ras pathway

Strains carrying ras2(318S) as their sole RAS gene fail to elicit a transient increase in cAMP levels following addition of glucose to starved cells but maintain normal steady-state levels of cAMP under a variety of growth conditions. Such strains show extended delays in resuming growth following transition from a quiescent state to glucose-containing growth media, either in emerging from stationary phase or following inoculation as spores onto fresh media. Otherwise, growth of such strains is indistinguishable from that of RAS2(+) strains. ras2(318S) strains also exhibit a delay in glucose-stimulated phosphorylation and turnover of fructose-1,6-bisphosphatase, a substrate of the cAMP-dependent protein kinase A (PKA) and a key component of the gluconeogenic branch of the glycolytic pathway. Finally Tpk(w) strains, which fail to modulate PKA in response to fluctuations in cAMP levels, show the same growth delay phenotypes, as do ras2(318S) strains. These observations indicate that the glucose-induced cAMP spike results in a transient activation of PKA, which is required for efficient transition of yeast cells from a quiescent state to resumption of rapid growth. This represents the first demonstration that yeast cells use the Ras pathway to transmit a signal to effect a biological change in response to an upstream stimulus.

Coexistence of multiple propagating wave-fronts in a regulated enzyme reaction model: link with birhythmicity and multi-threshold excitability

We analyze the spatial propagation of wave-fronts in a biochemical model for a product-activated enzyme reaction with non-linear recycling of product into substrate. This model was previously studied as a prototype for the coexistence of two distinct types of periodic oscillations (birhythmicity). The system is initially in a stable steady state characterized by the property of multi-threshold excitability, by which it is capable of amplifying in a pulsatory manner perturbations exceeding two distinct thresholds. In such conditions, when the effect of diffusion is taken into account, two distinct wave-fronts are shown to propagate in space, with distinct amplitudes and velocities, for the same set of parameter values, depending on the magnitude of the initial perturbation. Such a multiplicity of propagating wave-fronts represents a new type of coexistence of multiple modes of dynamic behavior, besides the coexistence involving, under spatially homogeneous conditions, multiple steady states, multiple periodic regimes, or a combination of steady and periodic regimes.

Acetaldehyde mediates the synchronization of sustained glycolytic oscillations in populations of yeast cells

In the presence of cyanide, populations of yeast cells can exhibit sustained oscillations in the concentration of glycolytic metabolites, NADH and ATP. This study attempts to answer the long-standing question of whether and how oscillations of individual cells are synchronized. It shows that mixing two cell populations that oscillate 180 degrees out of phase only transiently abolishes the macroscopic oscillation. After a few minutes, NADH fluorescence of the mixed population resumes oscillations up to the original amplitude. At low cell densities, addition of acetaldehyde causes transient oscillations. At higher cell densities, where the oscillations are autonomous, 70 microM acetaldehyde causes phase shifts. Extracellular acetaldehyde is shown to oscillate around the 70 microM level. We conclude that acetaldehyde synchronizes the oscillations of the individual cells.

The role of fructose 2,6-bisphosphate in glycolytic oscillations in extracts and cells of Saccharomyces cerevisiae

Fructose 2,6-bisphosphate is physiologically one of the most potent activators of yeast 6-phosphofructo-1-kinase. The glycolytic oscillation observed in cell-free cytoplasmic extracts of the yeast Saccharomyces cerevisiae responds to the addition of fructose 2,6-bisphosphate in micromolar concentrations by showing a pronounced decrease of both the amplitude and the period. The oscillations can be suppressed completely by 10 microM and above of this activator but recovers almost fully (95%) to the unperturbed state after 3 h. Fructose 2,6-bisphosphate shifts the phases of the oscillations by a maximal +/- 60 degrees. Oscillations in concentration of endogenous fructose 2,6-bisphosphate in the extract were also observed. Fructose 2,6-bisphosphate alters the dynamic properties of 6-phosphofructo-1-kinase which are vital for its role as the 'oscillophore'. However, the minute amount (approximately 0.3 microM) of endogenous fructose 2,6-bisphosphate and the phase relationship of its oscillations compared with other metabolites indicate that this activator is not an essential component of the oscillatory mechanism. Further support for this conclusion is the observation of sustained oscillations in both the extracts and a population of intact cells of a mutant strain (YFA) of S. cerevisiae with no detectable fructose 2,6-bisphosphate (less than 5 nM).

Oscillatory insulin secretion in perifused isolated rat islets

After a step-function increase in glucose concentration, insulin secretion by perifused isolated rat islets of Langerhans showed oscillations superimposed on the well-known first- and second-phase secretory components. The oscillations were sustained for the length of the experiment and corresponded to at least four cycles. This established the existence of an oscillatory pacemaker with a narrow dispersion of periodicities intrinsic to the islets and showed that synchronization of islet action could be achieved by a step-function increase in glucose concentration. The observed period of 16 min is similar to the period of oscillatory insulin secretion in a number of intact organisms. This argues for identity of pacemakers in vivo and in isolated islets. This means that neural or other forms of interislet communication are not prerequisites for oscillatory insulin secretion. Theophylline increased the length of the oscillatory period, suggesting the periodicity of the pacemaker of insulin secretion can be metabolically regulated. This observation also provided a basis for explaining fine tuning of oscillatory periods by the nervous system.

Limit-cycle oscillations and chaos in reaction networks subject to conservation of mass

A cyclic network of autocatalytic reactions involving an unbuffered cofactor and a number of components subject to conservation of mass displays a surprising richness of dynamical behaviors. Limit-cycle oscillations are possible over a wide range of parameter values. Additionally, a cascade of period-doubling bifurcations leading to chaos can coexist with a multiplicity of stable steady states. These results draw attention to the role of unbuffering as a feedback in biochemical systems.

Finding complex oscillatory phenomena in biochemical systems. An empirical approach

Starting with a model for a product-activated enzymatic reaction proposed for glycolytic oscillations, we show how more complex oscillatory phenomena may develop when the basic model is modified by addition of product recycling into substrate or by coupling in parallel or in series two autocatalytic enzyme reactions. Among the new modes of behavior are the coexistence between two stable types of oscillations (birhythmicity), bursting, and aperiodic oscillations (chaos). On the basis of these results, we outline an empirical method for finding complex oscillatory phenomena in autonomous biochemical systems, not subjected to forcing by a periodic input. This procedure relies on finding in parameter space two domains of instability of the steady state and bringing them close to each other until they merge. Complex phenomena occur in or near the region where the two domains overlap. The method applies to the search for birhythmicity, bursting and chaos in a model for the cAMP signalling system of Dictyostelium discoideum amoebae.

Chaos in biological systems

Chaos is a widespread and easily recognizable phenomenon that hardly anybody took notice of until recently. The reason may be that chaos has something profoundly counterintuitive about it. It will not fit easily into any familiar cause–effect frame. The best introduction to chaos is by the way of an example. Consider a leaking faucet (Shaw, 1984). When the weight of the accumulating drop exceeds the surface tension the drop falls and a new drop begins to form. If the leak is small and the pressure in the faucet is constant, the time taken for the drop to reach the critical weight is constant. The dripping is perfectly periodic, the period depending on the leak rate. If the leak is slightly increased, the period of dripping will decrease slightly and vice versa. However, somewhere beyond this point the leaking faucet becomes a nuisance. When the leak is increased beyond a certain point the dripping looses its regularity. The time interval between the drops will first alternate periodically between a short and a long time interval. After a further increase of the leak this double periodic pattern will become unstable and change into a new pattern where four different time intervals between the drops alternate periodically. As the leak is further increased the period will double again and again and finally the dripping becomes completely irregular without any repeating pattern. When this occurs we are observing chaos. At the same time we are posed with the problem of understanding how such a ridiculously simple system can show random behaviour.

Chaotic dynamics in yeast glycolysis under periodic substrate input flux

The numerical analysis for a glycolytic model containing the enzymes phosphofructokinase and pyruvate kinase reveals different types of entrainment, as well as chaotic response under sinusoidal substrate input. Entrainment with response periods 1, 2, 3, 5 and 7-times the input flux period and aperiodic behaviour is verified by measurements of NADH fluorescence in extracts of Saccharomyces cerevisiae in the theoretically predicted range. The stroboscopic transfer function obtained from the aperiodic signal admits period 3, implying chaos according to the Li-Yorke theorem.