Noise-driven stem cell and progenitor population dynamics

Background

The balance between maintenance of the stem cell state and terminal differentiation is influenced by the cellular environment. The switching between these states has long been understood as a transition between attractor states of a molecular network. Herein, stochastic fluctuations are either suppressed or can trigger the transition, but they do not actually determine the attractor states.

Methodology/Principal Findings

We present a novel mathematical concept in which stem cell and progenitor population dynamics are described as a probabilistic process that arises from cell proliferation and small fluctuations in the state of differentiation. These state fluctuations reflect random transitions between different activation patterns of the underlying regulatory network. Importantly, the associated noise amplitudes are state-dependent and set by the environment. Their variability determines the attractor states, and thus actually governs population dynamics. This model quantitatively reproduces the observed dynamics of differentiation and dedifferentiation in promyelocytic precursor cells.

Conclusions/Significance

Consequently, state-specific noise modulation by external signals can be instrumental in controlling stem cell and progenitor population dynamics. We propose follow-up experiments for quantifying the imprinting influence of the environment on cellular noise regulation.     

Precritical State Transition Dynamics in the Attractor Landscape of a Molecular Interaction Network Underlying Colorectal Tumorigenesis

From the perspective of systems science, tumorigenesis can be hypothesized as a critical transition (an abrupt shift from one state to another) between proliferative and apoptotic attractors on the state space of a molecular interaction network, for which an attractor is defined as a stable state to which all initial states ultimately converge, and the region of convergence is called the basin of attraction. Before the critical transition, a cellular state might transit between the basin of attraction for an apoptotic attractor and that for a proliferative attractor due to the noise induced by the inherent stochasticity in molecular interactions. Such a flickering state transition (state transition between the basins of attraction for alternative attractors from the impact of noise) would become more frequent as the cellular state approaches near the boundary of the basin of attraction, which can increase the variation in the estimate of the respective basin size. To investigate this for colorectal tumorigenesis, we have constructed a stochastic Boolean network model of the molecular interaction network that contains an important set of proteins known to be involved in cancer. In particular, we considered 100 representative sequences of 20 gene mutations that drive colorectal tumorigenesis. We investigated the appearance of cancerous cells by examining the basin size of apoptotic, quiescent, and proliferative attractors along with the sequential accumulation of gene mutations during colorectal tumorigenesis. We introduced a measure to detect the flickering state transition as the variation in the estimate of the basin sizes for three-phenotype attractors from the impact of noise. Interestingly, we found that this measure abruptly increases before a cell becomes cancerous during colorectal tumorigenesis in most of the gene mutation sequences under a certain level of stochastic noise. This suggests that a frequent flickering state transition can be a precritical phenomenon of colorectal tumorigenesis.     

A new principle of enzyme catalysis: coupled vibrations facilitate conformational changes

The coupling between the molecular vibrations in chymotrypsinogen, alpha-chymotrypsin and tosyl-alpha-chymotrypsin, as expressed by the temperature factors of individual amino acid sidechains and by a flexibility parameter calculated from the masses and co-ordinates of the atoms, has been analyzed by calculation of the integral correlation coefficient, the autocorrelation coefficient, the Poincaré projection, the first Liapunov coefficient and the power spectra. The agreement between the results obtained with the temperature factors and the flexibility parameter as well as the correct display by the latter of known structural features support the validity of the approach. The localization and extent of the conformational change in the enzyme following its binding of a specific substrate is detected in the difference plot between the enzyme and the acylenzyme of the distribution of the flexibility parameter over the peptide chain. As many as about 70% of the aminoacids participate in this rearrangement. An attractor of low dimensionality, two, i.e. a limit cycle, is detected both in the total enzyme and in its domain which is mobilized by the specific substrate. A simple model based on a known prominent structural feature, which is common to the trypsin family of serine proteases, two extensive coaxial halfcylinders of beta-sheets, to which previously no mechanistic function could be assigned, is proposed to account for the role of the attractor in the catalytic process: (1) control of the entry of a specific substrate to the catalytic site by co-ordinated disentanglement of the interlocking sidechains; (2) correct positioning of the functional groups in the active site; (3) lowering of the activation energy of the formation of the transition state complex.     

A dynamical systems hypothesis of schizophrenia

We propose a top-down approach to the symptoms of schizophrenia based on a statistical dynamical framework. We show that a reduced depth in the basins of attraction of cortical attractor states destabilizes the activity at the network level due to the constant statistical fluctuations caused by the stochastic spiking of neurons. In integrate-and-fire network simulations, a decrease in the NMDA receptor conductances, which reduces the depth of the attractor basins, decreases the stability of short-term memory states and increases distractibility. The cognitive symptoms of schizophrenia such as distractibility, working memory deficits, or poor attention could be caused by this instability of attractor states in prefrontal cortical networks. Lower firing rates are also produced, and in the orbitofrontal and anterior cingulate cortex could account for the negative symptoms, including a reduction of emotions. Decreasing the GABA as well as the NMDA conductances produces not only switches between the attractor states, but also jumps from spontaneous activity into one of the attractors. We relate this to the positive symptoms of schizophrenia, including delusions, paranoia, and hallucinations, which may arise because the basins of attraction are shallow and there is instability in temporal lobe semantic memory networks, leading thoughts to move too freely round the attractor energy landscape.     

Intercellular communication via intracellular calcium oscillations

In this letter, we present the results of a simple model for intercellular communication via calcium oscillations, motivated in part by a recent experimental study. The model describes two cells (a "donor" and "sensor") whose intracellular dynamics involve a calcium-induced, calcium release process. The cells are coupled by assuming that the input of the sensor cell is proportional to the output of the donor cell. As one varies the frequency of calcium oscillations of the donor cell, the sensor cell passes through a sequence of N : M phase-locked regimes and exhibits a "Devil's staircase" behavior. Such a phase-locked response has been seen experimentally in pulsatile stimulation of single cells. We also study a stochastic version of the coupled two-cell model. We find that phase locking holds for realistic choices for the cell volume.     

Distinguishing the noise and attractor strength of coordinated limb movements using recurrence analysis

The variability of coupled rhythmic limb movements is assumed to be a consequence of the strength of a movement’s attractor dynamic and a constant stochastic noise process that continuously perturbs the movement system away from this dynamic. Recently, it has been suggested that the nonlinear technique of recurrence analysis can be used to index the effects of noise and attractor strength on movement variability. To test this, three experiments were conducted in which the attractor strength of bimanual wrist-pendulum movements (using coordination mode, movement frequency and detuning), as well as the magnitude of stochastic perturbations affecting the variability of these movements (using a temporally fluctuating visual metronome) was manipulated. The results of these experiments demonstrate that recurrence analysis can index parametric changes in the attractor strength of coupled rhythmic limb movements and the magnitude of metronome induced stochastic perturbations independently. The results of Experiments 1 and 2 also support the claim that differences between the variability of inphase and antiphase coordination, and between slow and fast movement frequencies are due to differences in attractor strength. In contrast to the standard assumption that the noise that characterizes interlimb coordination remains constant for different magnitudes of detuning (Δ ω) the results of Experiment 3 suggest that the magnitude of noise increases with increases in |Δ ω|.     

Are binary synapses superior to graded weight representations in stochastic attractor networks?

Synaptic plasticity is an underlying mechanism of learning and memory in neural systems, but it is controversial whether synaptic efficacy is modulated in a graded or binary manner. It has been argued that binary synaptic weights would be less susceptible to noise than graded weights, which has impelled some theoretical neuroscientists to shift from the use of graded to binary weights in their models. We compare retrieval performance of models using both binary and graded weight representations through numerical simulations of stochastic attractor networks. We also investigate stochastic attractor models using multiple discrete levels of weight states, and then investigate the optimal threshold for dilution of binary weight representations. Our results show that a binary weight representation is not less susceptible to noise than a graded weight representation in stochastic attractor models, and we find that the load capacities with an increasing number of weight states rapidly reach the load capacity with graded weights. The optimal threshold for dilution of binary weight representations under stochastic conditions occurs when approximately 50% of the smallest weights are set to zero.     

Transport catalysis

Carrier linked solute transport through biomembranes is analysed with the viewpoint of catalysis. Different from enzymes, in carriers the unchanged substrate induces optimum fit in the transition state. The enhanced intrinsic binding energy pays for the energy required of the global conformation changes, thus decreasing the activation energy barrier. This "induced transition fit" (ITF) explains several phenomena of carrier transport, e.g., high or low affinity substrate requirements for unidirectional versus exchange, external energy requirement for "low affinity" transport, the existence of side specific inhibitors to ground states of the carrier, the requirement of external energy in active transport to supplement catalytic energy in addition to generate electrochemical gradients.     

Can noise induce chaos?

An important component of the mathematical definition of chaos is sensitivity to initial conditions. Sensitivity to initial conditions is usually measured in a deterministic model by the dominant Lyapunov exponent (LE), with chaos indicated by a positive LE. The sensitivity measure has been extended to stochastic models; however, it is possible for the stochastic Lyapunov exponent (SLE) to be positive when the LE of the underlying deterministic model is negative, and vice versa. This occurs because the LE is a long-term average over the deterministic attractor while the SLE is the long-term average over the stationary probability distribution. The property of sensitivity to initial conditions, uniquely associated with chaotic dynamics in deterministic systems, is widespread in stochastic systems because of time spent near repelling invariant sets (such as unstable equilibria and unstable cycles). Such sensitivity is due to a mechanism fundamentally different from deterministic chaos. Positive SLE's should therefore not be viewed as a hallmark of chaos. We develop examples of ecological population models in which contradictory LE and SLE values lead to confusion about whether or not the population fluctuations are primarily the result of chaotic dynamics. We suggest that "chaos" should retain its deterministic definition in light of the origins and spirit of the topic in ecology. While a stochastic system cannot then strictly be chaotic, chaotic dynamics can be revealed in stochastic systems through the strong influence of underlying deterministic chaotic invariant sets.     

A study of some diffusion models of population growth

The stochastic differential equations of many diffusion processes which arise in studies of population growth in random environments can be transformed, if the Stratonovich stochastic calculus is employed, to the equation of the Wiener process. If the transformation function has certain properties then the transition probability density function and quantities relating to the time to first attain a given population size can be obtained from the known results for the Wiener process. Some other random growth processes can be derived from the Ornstein-Uhlenbeck process. These transformation methods are applied to the random processes of Malthusian growth, Pearl-Verhulst logistic growth and a recent model of density independent growth due to Levins.     

Evidence for progressive phosphorus limitation over long-term ecosystem development: Examination of a biogeochemical paradigm

Aims

To test predictions of ecosystem theory for changes in P cycling over primary succession, we determined soil phosphorus (P) in labile, primary mineral, organic, and occluded forms along a chronosequence of five wave cut terraces known as the “Ecological Staircase”. The Ecological Staircase terraces (T1-T5) transition naturally from fertile native coastal forests in California, USA, to diminutive pygmy vegetation over the span of?>?500,000 years of pedogenesis.

Methods

Soil P fractions were quantified to a depth of 40 cm on T1-T5 using a modified Hedley P fractionation procedure.

Results

Overall results confirmed the Walker and Syers Model of Phosphorus Transformations During Pedogenesis: total P declined from youngest (194 mg/kg P) to oldest (127 mg/kg P) sites; primary-mineral P decreased sharply from T1 to older sites; and occluded P dominated P pools at the oldest pygmy sites (T3-T5). In addition, foliar P concentrations declined markedly in the pygmy forest, and N/P of vegetation (T1: 6.03, T5: 14.4) and N/P_organic of mineral soils (T1: 6.10, T5: 25.3) increased significantly over time.

Conclusions

Results point to P as the primary limiting nutrient in the pygmy forest, exemplifying the terminal steady-state of ecosystem retrogression that underlies the persistence of this unique ecosystem.     

Free-energy landscape of enzyme catalysis

The concept is developed that enzyme mechanisms should be viewed as "catalytic networks" with multiple conformations that occur serially and in parallel in the mechanism. These coupled ensembles of conformations require a multi-dimensional standard free-energy surface that is very "rugged", containing multiple minima and transition states. Experimental and theoretical evidence is presented to support this concept.     

A Signature of Attractor Dynamics in the CA3 Region of the Hippocampus

The notion of attractor networks is the leading hypothesis for how associative memories are stored and recalled. A defining anatomical feature of such networks is excitatory recurrent connections. These “attract” the firing pattern of the network to a stored pattern, even when the external input is incomplete (pattern completion). The CA3 region of the hippocampus has been postulated to be such an attractor network; however, the experimental evidence has been ambiguous, leading to the suggestion that CA3 is not an attractor network. In order to resolve this controversy and to better understand how CA3 functions, we simulated CA3 and its input structures. In our simulation, we could reproduce critical experimental results and establish the criteria for identifying attractor properties. Notably, under conditions in which there is continuous input, the output should be “attracted” to a stored pattern. However, contrary to previous expectations, as a pattern is gradually “morphed” from one stored pattern to another, a sharp transition between output patterns is not expected. The observed firing patterns of CA3 meet these criteria and can be quantitatively accounted for by our model. Notably, as morphing proceeds, the activity pattern in the dentate gyrus changes; in contrast, the activity pattern in the downstream CA3 network is attracted to a stored pattern and thus undergoes little change. We furthermore show that other aspects of the observed firing patterns can be explained by learning that occurs during behavioral testing. The CA3 thus displays both the learning and recall signatures of an attractor network. These observations, taken together with existing anatomical and behavioral evidence, make the strong case that CA3 constructs associative memories based on attractor dynamics.     

Density dependence,boundedness, and attraction: detecting stability in stochastic systems

Summary By analogy with deterministic stability, the stability of stochastic ecological systems can be viewed as a tendency for population densities to avoid dynamic boundaries (i.e. boundedness) or to approach a dynamic attractor (i.e. attraction). At the population level, these two views generate predictions consistent with density dependence. I therefore devised two new statistical tests of attraction, the random-walk attraction test and the randomized attraction test; I then used them successfully, along with randomization techniques that detect boundedness and two autocorrelation methods, to test for density dependence in published sequences of population densities. The attraction tests identify the apparent attractor, the band of densities toward which density tends to shift between generations. Locating the apparent attractor can generate a prediction of the next direction of density change; for data from a dragonfly assemblage, about 80% of these predictions were correct. From the single-population tests, I also developed two multispecies tests of attraction (the multispecies random-walk and randomized attraction tests) and two multispecies tests of boundedness (the multispecies permutation and randomization tests). These detected attraction and boundedness in the dragonfly assemblage and attraction in a collection of laboratory fruitfly populations. An evaluation of the statistical power of the new density attraction tests indicates a strong dependence on the sequence length n and on the number of populations m: power increases with n and particularly with m. Nevertheless, detecting attraction becomes likely even in populations with strong linear density-dependence only with n>30 or for shorter sequences in multispecies assemblages.     

The devil's in the details: genetic and phenotypic divergence between artificial and native populations of the endangered pupfish (Cyprinodon diabolis)

Propagation of threatened or endangered species in artificial habitats is a common strategy for reducing the probability of extinction by demographic or stochastic forces. Differential selection, founder effects and genetic drift can conspire to cause artificial populations to differ irreversibly from native populations for characters important for fitness, thereby compromising conservation efforts. Here we show that artificial propagation of the endangered Devil's Hole pupfish Cyprinodon diabolis resulted in rapid divergence for phenotypic and genetic characteristics despite attempts to replicate key characteristics of the species' native habitat when designing the artificial environments. Although differences in behavior and morphology between the native pool population and the two artificial pools may reflect phenotypic plasticity, the results underscore the need to monitor and control (to the extent possible) closely the evolutionary process when propagating native species in artificial pools for multiple generations.     

Stochastic dynamic study of optical transition properties of single GFP-like molecules

Due to high fluctuations and quantum uncertainty, the processes of single-molecules should be treated by stochastic methods. To study fluorescence time series and their statistical properties, we have applied two stochastic methods, one of which is an analytic method to study the off-time distributions of certain fluorescence transitions and the other is Gillespie’s method of stochastic simulations. These methods have been applied to study the optical transition properties of two single-molecule systems, GFPmut2 and a Dronpa-like molecule, to yield results in approximate agreement with experimental observations on these systems. Rigorous oscillatory time series of GFPmut2 before it unfolds in the presence of denaturants have not been obtained based on the stochastic method used, but, on the other hand, the stochastic treatment puts constraints on the conditions under which such oscillatory behavior is possible. Furthermore, a sensitivity analysis is carried out on GFPmut2 to assess the effects of transition rates on the observables, such as fluorescence intensities.     

Stable oscillations in mathematical models of biological control systems

Summary Oscillations in a class of piecewise linear (PL) equations which have been proposed to model biological control systems are considered. The flows in phase space determined by the PL equations can be classified by a directed graph, called a state transition diagram, on anN-cube. Each vertex of theN-cube corresponds to an orthant in phase space and each edge corresponds to an open boundary between neighboring orthants. If the state transition diagram contains a certain configuration called a cyclic attractor, then we prove that for the associated PL equation, all trajectories in the regions of phase space corresponding to the cyclic attractor either (i) approach a unique stable limit cycle attractor, or (ii) approach the origin, in the limitt→∞. An algebraic criterion is given to distinguish the two cases. Equations which can be used to model feedback inhibition are introduced to illustrate the techniques.     

Invasion dynamics and attractor inheritance

 We study the dynamics of a population of residents that is being invaded by an initially rare mutant. We show that under relatively mild conditions the sum of the mutant and resident population sizes stays arbitrarily close to the initial attractor of the monomorphic resident population whenever the mutant has a strategy sufficiently similar to that of the resident. For stochastic systems we show that the probability density of the sum of the mutant and resident population sizes stays arbitrarily close to the stationary probability density of the monomorphic resident population. Attractor switching, evolutionary suicide as well as most cases of ``the resident strikes back' in systems with multiple attractors are possible only near a bifurcation point in the strategy space where the resident attractor undergoes a discontinuous change. Away from such points, when the mutant takes over the population from the resident and hence becomes the new resident itself, the population stays on the same attractor. In other words, the new resident ``inherits' the attractor from its predecessor, the former resident. Received: 10 December 2000 / Revised version: 14 September 2001 / Published online: 17 May 2002