The Hill functions, [Formula: see text], have been widely used in biology for over a century but, with the exception of [Formula: see text], they have had no justification other than as a convenient fit to empirical data. Here, we show that they are the universal limit for the sharpness of any input-output response arising from a Markov process model at thermodynamic equilibrium. Models may represent arbitrary molecular complexity, with multiple ligands, internal states, conformations, coregulators, etc, under core assumptions that are detailed in the paper. The model output may be any linear combination of steady-state probabilities, with components other than the chosen input ligand held constant. This formulation generalizes most of the responses in the literature. We use a coarse-graining method in the graph-theoretic linear framework to show that two sharpness measures for input-output responses fall within an effectively bounded region of the positive quadrant, [Formula: see text], for any equilibrium model with [Formula: see text] input binding sites. [Formula: see text] exhibits a cusp which approaches, but never exceeds, the sharpness of [Formula: see text], but the region and the cusp can be exceeded when models are taken away from thermodynamic equilibrium. Such fundamental thermodynamic limits are called Hopfield barriers, and our results provide a biophysical justification for the Hill functions as the universal Hopfield barriers for sharpness. Our results also introduce an object, [Formula: see text], whose structure may be of mathematical interest, and suggest the importance of characterizing Hopfield barriers for other forms of cellular information processing.

Hepatocyte lipid and glucose metabolism is regulated not only by major hormones like insulin and glucagon but also by many other factors, including calcium ions. Recently, mitochondria-associated membrane (MAM) dysfunction combined with incorrect IP3-receptor regulation has been shown to result in abnormal calcium signaling in hepatocytes. This dysfunction could further lead to hepatic metabolism pathology. However, the exact contribution of MAM dysfunction, incorrect IP3-receptor regulation and insulin resistance to the calcium-insulin-glucagon interplay is not understood yet. In this work, we analyze the role of abnormal calcium signaling and insulin dysfunction in hepatocytes by proposing a model of hepatocyte metabolic regulatory network with a detailed focus on the model construction details besides the biological aspect. In this work, we analyze the role of abnormal calcium signaling and insulin dysfunction in hepatocytes by proposing a model of hepatocyte metabolic regulatory network. We focus on the model construction details, model validation, and predictions. We describe the dynamic regulation of signaling processes by sigmoid Hill function. In particular, we study the effect of both the Hill function slope and the distance between Hill function extremes on metabolic processes in hepatocytes as a model of nonspecific insulin dysfunction. We also address the significant time difference between characteristic time of glucose hepatic processing and a typical calcium oscillation period in hepatocytes. Our modeling results show that calcium signaling dysfunction results in an abnormal increase in postprandial glucose levels, an abnormal glucose decrease in fasting, and a decreased amount of stored glycogen. An insulin dysfunction of glucose phosphorylation, glucose dephosphorylation, and glycogen breakdown also cause a noticeable effect. We also get some insight into the so-called hepatic insulin resistance paradox, confirming the hypothesis regarding indirect insulin action on hepatocytes via dysfunctional adipocyte lipolysis.

The ultrasensitivity of a dose response function can be quantifiably defined using the generalized Hill coefficient of the function. We examined an upper bound for the Hill coefficient of the composition of two functions, namely the product of their individual Hill coefficients. We proved that this upper bound holds for compositions of Hill functions, and that there are instances of counterexamples that exist for more general sigmoidal functions. Additionally, we tested computationally other types of sigmoidal functions, such as the logistic and inverse trigonometric functions, and we provided computational evidence that in these cases the inequality also holds. We show that in large generality there is a limit to how ultrasensitive the composition of two functions can be, which has applications to understanding signaling cascades in biochemical reactions.

The Hill functions, ℋh(x)=xh/1+xh, have been widely used in biology for over a century but, with the exception of ℋ1, they have had no justification other than as a convenient fit to empirical data. Here, we show that they are the universal limit for the sharpness of any input-output response arising from a Markov process model at thermodynamic equilibrium. Models may represent arbitrary molecular complexity, with multiple ligands, internal states, conformations, co-regulators, etc, under core assumptions that are detailed in the paper. The model output may be any linear combination of steady-state probabilities, with components other than the chosen input ligand held constant. This formulation generalises most of the responses in the literature. We use a coarse-graining method in the graph-theoretic linear framework to show that two sharpness measures for input-output responses fall within an effectively bounded region of the positive quadrant, Ωm⊂ℝ+2, for any equilibrium model with m input binding sites. Ωm exhibits a cusp which approaches, but never exceeds, the sharpness of ℋm but the region and the cusp can be exceeded when models are taken away from thermodynamic equilibrium. Such fundamental thermodynamic limits are called Hopfield barriers and our results provide a biophysical justification for the Hill functions as the universal Hopfield barriers for sharpness. Our results also introduce an object, Ωm, whose structure may be of mathematical interest, and suggest the importance of characterising Hopfield barriers for other forms of cellular information processing.

In this paper we analyse stochastic expression of a single gene with its dynamics given by the classical Goodwin model with mRNA and protein contribution. We compare the effect of the presence of positive and negative feedback on the transcription regulation. In such cases we observe two qualitatively different types of asymptotic behaviour. In the case of a negative feedback loop, under sufficient conditions, one can find a stationary density for mRNA and protein molecules. In the case of a positive feedback loop we observe extinction of both types of molecules with time.

There are four cone morphologies in zebrafish, corresponding to UV (U), blue (B), green (G), and red (R)-sensing types; yet genetically, eight cone opsins are expressed. How eight opsins are physiologically siloed in four cone types is not well understood, and in larvae, cone physiological spectral peaks are unstudied. We use a spectral model to infer cone wavelength peaks, semisaturation irradiances, and saturation amplitudes from electroretinogram (ERG) datasets composed of multi-wavelength, multi-irradiance, aspartate-isolated, cone-PIII signals, as compiled from many 5- to 12-day larvae and 8- to 18-month-old adult eyes isolated from wild-type (WT) or roy orbison (roy) strains. Analysis suggests (in nm) a seven-cone, U-360/B1-427/B2-440/G1-460/G3-476/R1-575/R2-556, spectral physiology in WT larvae but a six-cone, U-349/B1-414/G3-483/G4-495/R1-572/R2-556, structure in WT adults. In roy larvae, there is a five-cone structure: U-373/B2-440/G1-460/R1-575/R2-556; in roy adults, there is a four-cone structure, B1-410/G3-482/R1-571/R2-556. Existence of multiple B, G, and R types is inferred from shifts in peaks with red or blue backgrounds. Cones were either high or low semisaturation types. The more sensitive, low semisaturation types included U, B1, and G1 cones [3.0-3.6 log(quanta·μm-2·s-1)]. The less sensitive, high semisaturation types were B2, G3, G4, R1, and R2 types [4.3-4.7 log(quanta·μm-2·s-1)]. In both WT and roy, U- and B- cone saturation amplitudes were greater in larvae than in adults, while G-cone saturation levels were greater in adults. R-cone saturation amplitudes were the largest (50-60% of maximal dataset amplitudes) and constant throughout development. WT and roy larvae differed in cone signal levels, with lesser UV- and greater G-cone amplitudes occurring in roy, indicating strain variation in physiological development of cone signals. These physiological measures of cone types suggest chromatic processing in zebrafish involves at least four to seven spectral signal processing pools.

We present here a software program dedicated to the fitting of experimental dose-response data, which integrates seamlessly with Excel and allows curve fitting plots and results to reside alongside data within Excel spreadsheets. The program, named eeFit, for Excel-Embedded Fitting software, requires no advanced knowledge of Excel or non-linear least-squares fitting. Any experimental data present in an Excel file, such as dose-effect data obtained with membrane receptor or ion channel ligands, can be graphed and fitted interactively with standard Hill models for activation or inhibition, or with more complex models for biphasic effects resulting from combinations of activation and inhibition. When benchmarked against the commercial program Origin, eeFit yielded equivalent or better results, in terms of accuracy and convergence, and proved much easier to learn and use.

Molecular switches, such as the protein kinase CaMKII, play a fundamental role in cell signaling by decoding inputs into either high or low states of activity; because the high activation state can be turned on and persist after the input ceases, these switches have earned a reputation as \"digital.\" Although this on/off, binary perspective has been valuable for understanding long timescale synaptic plasticity, accumulating experimental evidence suggests that the CaMKII switch can also control plasticity on short timescales. To investigate this idea further, a non-autonomous, nonlinear ordinary differential equation, representative of a general bistable molecular switch, is analyzed. The results suggest that switch activity in regions surrounding either the high- or low-stable states of activation could act as a reliable analog signal, whose short timescale fluctuations relative to equilibrium track instantaneous input frequency. The model makes intriguing predictions and is validated against previous work demonstrating its suitability as a minimal representation of switch dynamics; in combination with existing experimental evidence, the theory suggests a multiplexed encoding of instantaneous frequency information over short timescales, with integration of total activity over longer timescales.

The haematopoietic lineages with leukaemia lineages are considered in this paper. In particular, we mainly consider that haematopoietic lineages are tightly controlled by negative feedback inhibition of end-product. Actually, leukemia has been found 100 years ago. Up to now, the exact mechanism is still unknown, and many factors are thought to be associated with the pathogenesis of leukemia. Nevertheless, it is very necessary to continue the profound study of the pathogenesis of leukemia. Here, we propose a new mathematical model which include some negative feedback inhibition from the terminally differentiated cells of haematopoietic lineages to the haematopoietic stem cells and haematopoietic progenitor cells in order to describe the regulatory mechanisms mentioned above by a set of ordinary differential equations. Afterwards, we carried out detailed dynamical bifurcation analysis of the model, and obtained some meaningful results.
In this work, we mainly perform the analysis of the mathematic model by bifurcation theory and numerical simulations. We have not only incorporated some new negative feedback mechanisms to the existing model, but also constructed our own model by using the modeling method of stem cell theory with probability method. Through a series of qualitative analysis and numerical simulations, we obtain that the weak negative feedback for differentiation probability is conducive to the cure of leukemia. However, with the strengthening of negative feedback, leukemia will be more difficult to be cured, and even induce death. In contrast, strong negative feedback for differentiation rate of progenitor cells can promote healthy haematopoiesis and suppress leukaemia.
These results demonstrate that healthy progenitor cells are bestowed a competitive advantage over leukaemia stem cells. Weak g1, g2, and h1 enable the system stays in the healthy state. However, strong h2 can promote healthy haematopoiesis and suppress leukaemia.

Characterizations of the dynamics of hybrid zones in space and time can give insights about traits and processes important in population divergence and speciation. We characterized a hybrid zone between tanagers in the genus Ramphocelus (Aves, Thraupidae) located in southwestern Colombia. We evaluated whether this hybrid zone originated as a result of secondary contact or of primary differentiation, and described its dynamics across time using spatial analyses of molecular, morphological, and coloration data in combination with paleodistribution modeling.
Models of potential historical distributions based on climatic data and genetic signatures of demographic expansion suggested that the hybrid zone likely originated following secondary contact between populations that expanded their ranges out of isolated areas in the Quaternary. Concordant patterns of variation in phenotypic characters across the hybrid zone and its narrow extent are suggestive of a tension zone, maintained by a balance between dispersal and selection against hybrids. Estimates of phenotypic cline parameters obtained using specimens collected over nearly a century revealed that, in recent decades, the zone appears to have moved to the east and to higher elevations, and may have become narrower. Genetic variation was not clearly structured along the hybrid zone, but comparisons between historical and contemporary specimens suggested that temporal changes in its genetic makeup may also have occurred.
Our data suggest that the hybrid zone likey resulted from secondary contact between populations. The observed changes in the hybrid zone may be a result of sexual selection, asymmetric gene flow, or environmental change.