Dynamics of commensalistic systems with self- and cross-inhibition

Up to three stable steady states are possible in a simple commensalistic system, taking place in an open-loop mixed reactor when the growth rates of the two species are inhibited by the substrates they prey on (Self-inhibition). Two stable states are possible in a system with noncompetitive inhibition of the species by the substrate they are not preying on (cross-inhibition). A large number of steady states as well as oscillatory states are possible when both self- and cross-inhibition are strong. Multiplicity of steady states is also possible in a reactor with biomas recirculation for these kinetics. Yet, the latter is more stable than the open-loop reactor in the sense that the domain of steady-state multiplicity is narrower. The stability of steady states and the dynamics of the systems for each of the investigated kinetics are summarized in a qualitative phase plane. The importance of the analysis for improving the selectively and yield of the system and for predicting the response of the system to changes in the operating conditions, is discussed.     

Nonlinear control of bioreactors with input multiplicities—an experimental work

The phrase input multiplicities means that an input variable with more than one value produces the same output value as if there were a single input–single output process. With input multiplicities, the value of the process gain changes as the manipulated variable changes, and beyond a certain input value, the sign of the gain also changes. A conventional PI controller for processes with input multiplicities may give unstable, less economical, or oscillatory responses. In the present work, control problems of a continuous bioreactor exhibiting two input multiplicities in the dilution rate on productivity were experimentally analyzed. A regulatory problem for the evaluation of controllers was taken up, i.e. a step change was made in the feed substrate concentration from 20 to 25 g/l at steady state conduction at lower (0.09386 h⁻¹) and higher (0.2278 h⁻¹) dilution rates for the same productivity of 2.9 g/l h. The nonlinear PI controller gave a more stable and fast response at both input dilution rates. The linear PI controller designed for a lower input dilution rate was stable, with some oscillations at the lower dilution rate, but the response was unstable at a higher dilution rate due to the input multiplicity behaviour of the process. Thus, nonlinear PI controller performance was found to be superior to that of the linear controller, and earlier reported theoretical results have been validated by the present experimental work.     

Feedfoward inhibition in biosynthetic pathways: inhibition of the aminoacyl-tRNA synthetase by intermediates of the pathway.

The stability of an amino acid biosynthetic pathway controlled by end-product inhibition is significantly improved if, in addition, the corresponding aminoacyl-tRNA synthetase is inhibited by an intermediate in the pathway. The more proximal the feedforward modifier is to the initial substrate, the more stable is the system. The temporal responsiveness of a system having both feedback and feedforward inhibition also is improved by having the feedforward modifier located at the beginning of the pathway. According to all other criteria that have been used previously to determine the functional effectiveness of biosynthetic pathways, the behavior of such a system essentially is unaffected by the position of the feedforward modifier in the pathway.     

Secretory process regularity monitors neuroendocrine feedback and feedforward signaling strength in humans

The present experiments examine the neuroregulatory hypothesis that the degree of sample-by-sample regularity of hormone output by an interlinked hypothalamopituitary target-organ system monitors the strength of feedback and/or feedforward signaling. To test this postulate and assess its generality, we implemented a total of nine thematically complementary perturbation experiments. In particular, we altered feedback or feedforward signaling selectively in two distinct neuroendocrine systems; namely, the growth hormone (GH) insulin-like growth factor type I (IGF-I) and the luteinizing hormone-testosterone axes. Four experimental paradigms comprised preferential reduction vs. enhancement of IGF-I or testosterone feedback signal strength; and, conversely, five others entailed selective attenuation vs. augmentation of GH-releasing hormone and gonadotropin-releasing hormone feedforward signal intensity. In these independent interventions, quantitation of subordinate (nonpulsatile) secretory pattern reproducibility via the approximate entropy statistic unmasked salient changes (P values typically <10(-3)) in the conditional regularity of serial hormone output with high consistency (96-100%). In particular, approximate entropy quantified degradation of secretory subpattern orderliness under either muted feedback restraint or heightened feedforward drive. Assuming valid interpretation of the biological constraints imposed, these experimental observations coincide with earlier reductionist mathematical predictions, wherein increased irregularity of coupled parameter output mirrors attenuated feedback and/or augmented feedforward coupling within an integrative system.     

Delayed feedback control for a chemostat model

In this paper, we discuss asymptotic properties and numerical simulations of a chemostat model with delayed feedback control. A chemostat model with two organisms can be made coexistent by feedback control of the dilution rate which depends affinely on the concentrations of two organisms [P. De Leenher, H.L. Smith, Feedback control for chemostat models, J. Math. Biol. 46 (2003) 48]. Then the coexistence takes its simplest form; the equilibrium point in the non-negative orthant is globally asymptotically stable. We show that stability of the equilibrium point is changed by 'time-delay' caused in controlling the dilution rate after measuring the concentrations of two organisms.     

A minimal mechanistic model for temporal signal processing in the lateral geniculate nucleus

The receptive fields of cells in the lateral geniculate nucleus (LGN) are shaped by their diverse set of impinging inputs: feedforward synaptic inputs stemming from retina, and feedback inputs stemming from the visual cortex and the thalamic reticular nucleus. To probe the possible roles of these feedforward and feedback inputs in shaping the temporal receptive-field structure of LGN relay cells, we here present and investigate a minimal mechanistic firing-rate model tailored to elucidate their disparate features. The model for LGN relay ON cells includes feedforward excitation and inhibition (via interneurons) from retinal ON cells and excitatory and inhibitory (via thalamic reticular nucleus cells and interneurons) feedback from cortical ON and OFF cells. From a general firing-rate model formulated in terms of Volterra integral equations, we derive a single delay differential equation with absolute delay governing the dynamics of the system. A freely available and easy-to-use GUI-based MATLAB version of this minimal mechanistic LGN circuit model is provided. We particularly investigate the LGN relay-cell impulse response and find through thorough explorations of the model’s parameter space that both purely feedforward models and feedback models with feedforward excitation only, can account quantitatively for previously reported experimental results. We find, however, that the purely feedforward model predicts two impulse response measures, the time to first peak and the biphasic index (measuring the relative weight of the rebound phase) to be anticorrelated. In contrast, the models with feedback predict different correlations between these two measures. This suggests an experimental test assessing the relative importance of feedforward and feedback connections in shaping the impulse response of LGN relay cells.     

Transitional states in a chemostat culture of Candida utilis

Transient states of the chemostat Candida utilis 1668-3-37 culture were studied when its growth was limited by ethanol and an abrupt acidification of the medium from pH 5.0 to 2.2 was done or when the dilution rate was rapidly changed from D = 0.1 to 0.3 h-1 and back to 0.07 h-1. The pH shock was found to cause stronger oscillations in a number of parameters (the weight of dry biomass, the content of residual ethanol, the content of RNA in the cells) than a change in the dilution rate. In the latter case the population density changed more smoothly than the content of RNA did. DNA content remained at one and the same level in all of the experiments. All of the oscillations were observed only in the first generation after a shock; there upon, the culture remained for a long time (7 to 10 generations) in a very stable state typical of chemostat cultures. The oscillations induced by the unfavourable pH of the medium were compared with those caused by an abrupt change in the dilution rate. The pH shock brought about multiple damping oscillations of the parameters whereas a change in the dilution rate resulted, most often, in a merely one oscillation.     

Development of spatial coarse-to-fine processing in the visual pathway

The sequential analysis of information in a coarse-to-fine manner is a fundamental mode of processing in the visual pathway. Spatial frequency (SF) tuning, arguably the most fundamental feature of spatial vision, provides particular intuition within the coarse-to-fine framework: low spatial frequencies convey global information about an image (e.g., general orientation), while high spatial frequencies carry more detailed information (e.g., edges). In this paper, we study the development of cortical spatial frequency tuning. As feedforward input from the lateral geniculate nucleus (LGN) has been shown to have significant influence on cortical coarse-to-fine processing, we present a firing-rate based thalamocortical model which includes both feedforward and feedback components. We analyze the relationship between various model parameters (including cortical feedback strength) and responses. We confirm the importance of the antagonistic relationship between the center and surround responses in thalamic relay cell receptive fields (RFs), and further characterize how specific structural LGN RF parameters affect cortical coarse-to-fine processing. Our results also indicate that the effect of cortical feedback on spatial frequency tuning is age-dependent: in particular, cortical feedback more strongly affects coarse-to-fine processing in kittens than in adults. We use our results to propose an experimentally testable hypothesis for the function of the extensive feedback in the corticothalamic circuit.     

Abrupt regime shifts in space and time along rivers and connected lake systems

Regime shifts between clear and turbid water states are commonly found in shallow lakes. These shifts are attributed to a positive feedback between water clarity and submerged macrophytes (underwater plants). Altering the retention time of the water may influence these interactions and thus potentially reduce the probability of alternative stable states. Here we assessed the effect of water retention time on the occurrence of alternative states in water quality of flushed lakes, chains of lakes and rivers using a spatially explicit simple model. Our results indicate that increased flushing of lakes rapidly decreases the range of parameters with alternative stable states up to their total disappearance at a flushing rate of about 50% the algal growth rate. Similarly, in a chain of lakes or in rivers with low flowing velocity, our model predicts that alternative stable states can only occur for systems with a high retention time. Despite the lack of hysteresis at lower water retention times, we predict that abrupt changes between clear and turbid states are still possible both in time and in space. Over a wide range of parameters, the equilibrium state of the chain of lakes shows a steep gradient of vegetation cover. Further, the transient dynamics of the model often include rapid shifts in time. For example, a local regime shift that occurs upstream may propagate through the whole lake chain or river due to a domino effect. All results of the simple model could qualitatively be reproduced with a more mechanistic model. The abrupt rather than gradual response of submerged macrophytes to reduced turbidity levels still makes river systems rather resilient to management measure. The importance of the initial turbidity and the observed domino effect suggest that restoration measures should start upstream and that these measures should eventually trigger regime shifts downstream.     

Interspecific interactions affect species and community responses to climate shifts

The response of individual species to climate change may alter the composition and dynamics of communities. Here, we show that the impacts of environmental change on communities can depend on the nature of the interspecific interactions: mutualistic communities typically respond differently than commensalistic or parasitic communities. We model and analyse the geographic range shifting of metapopulations of two interacting species – a host and an obligate species. Different types of interspecific interactions are implemented by modifying local extinction rates according to the presence/absence of the other species. We distinguish and compare three fundamentally different community types: mutualism, commensalism and parasitism. We find that community dynamics during geographic range shifting critically depends on the type of interspecific interactions. Parasitic interactions exacerbate the negative effect of environmental change whereas mutualistic interactions only partly compensate it. Commensalistic interactions exhibit an intermediate response. Based on these model outcomes, we predict that parasitic species interactions may be more vulnerable to geographic range shifting than commensalistic or mutualistic ones. However, we observe that when climate stabilises following a period of change, the rate of community recovery is largely independent of the type of interspecific interactions. These results emphasize that communities respond delicately to environmental change, and that local interspecific interactions can affect range shifting communities at large spatial scales.     

Feedback regulation of EGFR signalling: decision making by early and delayed loops

Human-made information relay systems invariably incorporate central regulatory components, which are mirrored in biological systems by dense feedback and feedforward loops. This type of system control is exemplified by positive and negative feedback loops (for example, receptor endocytosis and dephosphorylation) that enable growth factors and receptor Tyr kinases of the epidermal growth factor receptor (EGFR)/ERBB family to regulate cellular function. Recent studies show that the collection of feedback regulatory loops can perform computational tasks - such as decoding ligand specificity, transforming graded input signals into a digital output and regulating response kinetics. Aberrant signal processing and feedback regulation can lead to defects associated with pathologies such as cancer.     

Adaptive Responses Limited by Intrinsic Noise

Sensory systems have mechanisms to respond to the external environment and adapt to them. Such adaptive responses are effective for a wide dynamic range of sensing and perception of temporal change in stimulus. However, noise generated by the adaptation system itself as well as extrinsic noise in sensory inputs may impose a limit on the ability of adaptation systems. The relation between response and noise is well understood for equilibrium systems in the form of fluctuation response relation. However, the relation for nonequilibrium systems, including adaptive systems, are poorly understood. Here, we systematically explore such a relation between response and fluctuation in adaptation systems. We study the two network motifs, incoherent feedforward loops (iFFL) and negative feedback loops (nFBL), that can achieve perfect adaptation. We find that the response magnitude in adaption systems is limited by its intrinsic noise, implying that higher response would have higher noise component as well. Comparing the relation of response and noise in iFFL and nFBL, we show that whereas iFFL exhibits adaptation over a wider parameter range, nFBL offers higher response to noise ratio than iFFL. We also identify the condition that yields the upper limit of response for both network motifs. These results may explain the reason of why nFBL seems to be more abundant in nature for the implementation of adaption systems.     

Stability of a model of human granulopoiesis using continuous maturation

A class of mathematical models involving a convection-reaction partial differential equation (PDE) is introduced with reference to recovering human granulopoiesis after high dose chemotherapy with stem cell support. The stability properties of the model are addressed by means of numerical investigations and analysis. A simplified model with proliferation rate and mobilization rate independent of maturity shows that the model is stable as the maturation rate grows without bounds, but may go through stable and non-stable regimens as the maturation rate varies. It is also shown that the system is stable when parameters are chosen to approximate a real physiological situation. System characteristics do not change profoundly by introduction of a maturity-dependent proliferation and mobilization rate, as is necessary to make the model operate more in accordance with hematological observations. However, by changing the system mitotic responsiveness with respect to changes in cytokine level, the system is still stable but may show persistent oscillations much resembling clinical observations of cyclic neutropenia. Furthermore, in these cases, changes in the model feedback signal caused by, for instance, an impaired effective cytokine elimination by cell receptors may enforce these oscillations markedly.     

Ventilatory control during exercise with increased respiratory dead space in goats

Our objectives were to determine 1) the effects of increased respiratory dead space (VD) on the ventilatory response to exercise and 2) whether changes in the ventilatory response are due to changes in chemoreceptor feedback (rest to exercise) vs. changes in the feedforward exercise stimulus. Steady-state ventilation (VI) and arterial blood gas responses to mild or moderate hyperoxic exercise in goats were compared with and without increased VD. Responses were compared using a simple mathematical model with the following assumptions: 1) steady state, 2) linear CO2 chemoreceptor feedback, 3) linear feedforward exercise stimulus proportional to CO2 production (VCO2) and characterized by an exercise gain (Gex), and 4) additive exercise stimulus and CO2 feedback producing the system gain (Gsys = delta VI/delta VCO2). Model predictions at constant Gex [assuming VD-to-tidal volume (VT) ratio independent of VCO2] are that increased VD/VT will 1) increase arterial PCO2 (PaCO2) and VI at rest and 2) increase Gsys via changes in chemoreceptor feedback due to a small increase in the PaCO2 vs. VCO2 slope. Experimental results indicate that increased VD increased VD/VT, PaCO2, and VI at rest and increased Gsys during exercise. However, measurable changes in the PaCO2 vs. VCO2 slope occurred only at high VD/VT or running speeds. Gex was estimated at each VD for each goat by using the model in conjunction with experimental measurements. With 0.2 liter VD, Gex increased 40% (P less than 0.01); with 0.6 liter VD, Gex increased 110% between 0 and 2.4 km/h and 5% grade (P less than 0.01) but not between 2.4 and 4.8 km/h. Thus, Gex is increased by VD through a limited range. In goats, increases in Gsys with increased VD result from increases in both Gex and CO2 chemoreceptor feedback. These results are consistent with other experimental treatments that increase the exercise ventilatory response, maintaining constant relative PaCO2 regulation, and suggest that a common mechanism linked to resting ventilatory drive modulates Gex.     

Joint synchrony of reciprocal hormonal signaling in human paradigms of both ACTH excess and cortisol depletion

The hypothalamo-pituitary-adrenal axis is a stress-adaptive neuroendocrine ensemble, in which adrenocorticotropin (ACTH) drives cortisol secretion (feedforward) and cortisol restrains ACTH outflow (feedback). Quantifying direction- and pathway-specific adjustments within this and other interlinked systems by noninvasive means remains difficult. The present study tests the hypothesis that forward and reverse cross-approximate entropy (X-ApEn), a lag-, scale-, and model-independent measure of two-signal synchrony, would allow quantifiable discrimination of feedforward (ACTH --> cortisol) and feedback (cortisol --> ACTH) control. To this end, forward X-ApEn was defined by employing serial ACTH concentrations as a template to appraise pair-wise synchrony with cortisol secretion rates and vice versa for reverse X-ApEn. Coupled hormone profiles included normal ACTH-normal cortisol, high ACTH-high cortisol, and high ACTH-low cortisol concentrations in 35 healthy subjects, 21 patients with tumoral ACTH secretion, and 9 volunteers given placebo and a steroidogenic inhibitor, respectively. We used forward and reverse X-ApEn analyses to identify marked and equivalent losses of feedforward and feedback linkages (both P < 0.001) in patients with tumoral ACTH secretion. An identical analytical strategy revealed that ACTH --> cortisol feedforward synchrony decreases (P < 0.001), whereas cortisol --> ACTH feedback synchrony increases (P < 0.001), in response to hypocortisolemia. The collective outcomes establish precedence for pathway-specific adaptations in a major neurohormonal system. Thus quantification of directionally defined joint synchrony of biologically coupled signals offers a noninvasive strategy to dissect feedforward- and feedback-selective adaptations in an interactive axis.     

Transient Response of Continuously Cultured Heterogeneous Populations to Changes in Temperature

Completely mixed, once-through continuous culture systems of heterogeneous microbial populations of sewage origin were systematically examined for response to changes in reactor temperature. Systems were operated at two dilution rates of 0.125 and 0.25 per h. "Steady state" conditions of the systems were assessed with the reactors operating at 25 C. From this base line, temperature was decreased to as low as 8 C and increased to as high as 57.5 C. Response was assessed in the ensuing transient phase as the system approached a new "steady state." The response was measured by changes in amount and type of carbon source in the reactor effluent as determined by the chemical oxygen demand test, the anthrone test, and gas chromatography. Biological solids concentration and cell composition (protein, carbohydrate, ribonucleic acid and deoxyribonucleic acid) were also determined. These systems responded more favorably to increases than to decreases in temperature. Regardless of the direction of change, the system with the lowest dilution rate (D = 0.125 per h) responded more successfully; i.e., there was less leakage of carbon source in the effluent and less dilute-out of cells during the transient phase.     

Continuous ethanol production by a flocculating strain of Kluyveromyces marxianus: bioreactor performance

A flocculating strain of Kluyveromyces marxianus was used for alcoholic fermentation in a continuous bioreactor working with zero residual concentration in effluent. Specific kinetic parameters were improved by increasing dilution rate, which is similar to results obtained with ultrafiltration systems. Specific biomass accumulation rate had always a value greater than 92.5% of specific biomass growth rate and was independent of the dilution rate. Productivity is shown to be 12.5 times greater than in conventional continuous operation and is directly proportional to dilution rate. Maximum biomass concentration also presents a linear relationship with dilution rate. The largest obtained biomass concentration is 8 times greater than in a conventional continuous fermentor.     

Trade-off between responsiveness and noise suppression in biomolecular system responses to environmental cues

When living systems detect changes in their external environment their response must be measured to balance the need to react appropriately with the need to remain stable, ignoring insignificant signals. Because this is a fundamental challenge of all biological systems that execute programs in response to stimuli, we developed a generalized time-frequency analysis (TFA) framework to systematically explore the dynamical properties of biomolecular networks. Using TFA, we focused on two well-characterized yeast gene regulatory networks responsive to carbon-source shifts and a mammalian innate immune regulatory network responsive to lipopolysaccharides (LPS). The networks are comprised of two different basic architectures. Dual positive and negative feedback loops make up the yeast galactose network; whereas overlapping positive and negative feed-forward loops are common to the yeast fatty-acid response network and the LPS-induced network of macrophages. TFA revealed remarkably distinct network behaviors in terms of trade-offs in responsiveness and noise suppression that are appropriately tuned to each biological response. The wild type galactose network was found to be highly responsive while the oleate network has greater noise suppression ability. The LPS network appeared more balanced, exhibiting less bias toward noise suppression or responsiveness. Exploration of the network parameter space exposed dramatic differences in system behaviors for each network. These studies highlight fundamental structural and dynamical principles that underlie each network, reveal constrained parameters of positive and negative feedback and feed-forward strengths that tune the networks appropriately for their respective biological roles, and demonstrate the general utility of the TFA approach for systems and synthetic biology.     

Analysis of bidirectional pattern synchrony of concentration-secretion pairs: implementation in the human testicular and adrenal axes

The hypothalamo-pituitary-testicular and hypothalamo-pituitary-adrenal axes are prototypical coupled neuroendocrine systems. In the present study, we contrasted in vivo linkages within and between these two axes using methods without linearity assumptions. We examined 11 young (21-31 yr) and 8 older (62-74 yr) men who underwent frequent (every 2.5 min) blood sampling overnight for paired measurement of LH and testosterone and 35 adults (17 women and 18 men; 26-77 yr old) who underwent adrenocorticotropic hormone (ACTH) and cortisol measurements every 10 min for 24 h. To mirror physiological interactions, hormone secretion was first deconvolved from serial concentrations with a waveform-independent biexponential elimination model. Feedforward synchrony, feedback synchrony, and the difference in feedforward-feedback synchrony were quantified by the cross-approximate entropy (X-ApEn) statistic. These were applied in a forward (LH concentration template, examining pattern recurrence in testosterone secretion), reverse (testosterone concentration template, examining pattern recurrence in LH secretion), and differential (forward minus reverse) manner, respectively. Analogous concentration-secretion X-ApEn estimates were calculated from ACTH-cortisol pairs. X-ApEn, a scale- and model-independent measure of pattern reproducibility, disclosed 1) greater testosterone-LH feedback coordination than LH-testosterone feedforward synchrony in healthy men and significant and symmetric erosion of both feedforward and feedback linkages with aging; 2) more synchronous ACTH concentration-dependent feedforward than feedback drive of cortisol secretion, independent of gender and age; and 3) enhanced detection of bidirectional physiological regulation by in vivo pairwise concentration-secretion compared with concentration-concentration analyses. The linking of relevant biological input to output signals and vice versa should be useful in the dissection of the reciprocal control of neuroendocrine systems or even in the analysis of other nonendocrine networks.     

Learning combined feedback and feedforward control of a musculoskeletal system

The goal of this paper is the learning of neuromuscular control, given the following necessary conditions: (1) time delays in the control loop, (2) non-linear muscle characteristics, (3) learning of feedforward and feedback control, (4) possibility of feedback gain modulation during a task. A control system and learning methodology that satisfy those conditions is given. The control system contains a neural network, comprising both feedforward and feedback control. The learning method is backpropagation through time with an explicit sensitivity model. Results will be given for a one degree of freedom arm with two muscles. Good control results are achieved which compare well with experimental data. Analysis of the controller shows that significant differences in controller characteristics are found if the loop delays are neglected. During a control task the system shows feedback gain modulation, similar to experimentally found reflex gain modulation during rapid voluntary contraction. If only limited feedback information is available to the controller the system learns to co-contract the antagonistic muscle pair. In this way joint stiffness increases and stable control is more easily maintained. Received: 7 November 1995 / Accepted in revised form: 13 February 1996