P.R. Shorten

The distribution of urine deposited on a pasture from grazing animals

Intensive agricultural production practices are known to cause far-reaching effects on water quality. The current paper addresses and quantifies these effects caused by high stocking rates. A set of stochastic difference equations describing the development of the proportion of a grazed field either unaffected by urine deposition, or affected by multiple (1, 2, ...) urine depositions is described. A solution to this set of equations is found for the expected value of multiple (0, 1, 2, ...) urine depositions, and the variances of these depositions. It is assumed that an animal voids urine with a Poisson probability distribution, and that each urine deposition covers a random area with a Gaussian probability density. Given these reasonable assumptions, the probability distributions for each multiplicity of patch distribution can be found numerically. The utility of the results obtained is illustrated for a problem in assessing the nitrogen (N) pollution of ground water from different grazing strategies. It is demonstrated quantitatively that mob stocking (typical of winter management regimes in New Zealand) is often caused by rotational grazing. The latter is often used to optimize grass growth and intake, especially in winter. This increases (more than linearly) the level of N pollution in ground water. This is because of the increased frequency of multiple urine depositions, i.e. more than one urine deposition on the same patch of land in a short time.

A mathematical model of fatigue in skeletal muscle force contraction

The ability for muscle to repeatedly generate force is limited by fatigue. The cellular mechanisms behind muscle fatigue are complex and potentially include breakdown at many points along the excitation–contraction pathway. In this paper we construct a mathematical model of the skeletal muscle excitation–contraction pathway based on the cellular biochemical events that link excitation to contraction. The model includes descriptions of membrane voltage, calcium cycling and crossbridge dynamics and was parameterised and validated using the response characteristics of mouse skeletal muscle to a range of electrical stimuli. This model was used to uncover the complexities of skeletal muscle fatigue. We also parameterised our model to describe force kinetics in fast and slow twitch fibre types, which have a number of biochemical and biophysical differences. How these differences interact to generate different force/fatigue responses in fast- and slow- twitch fibres is not well understood and we used our modelling approach to bring new insights to this relationship.

A mathematical analysis of obstructed diffusion within skeletal muscle.

Molecules are transported through the myofilament lattice of skeletal muscle fibers during muscle activation. The myofilaments, along with the myosin heads, sarcoplasmic reticulum, t-tubules, and mitochondria, obstruct the diffusion of molecules through the muscle fiber. In this work, we studied the process of obstructed diffusion within the myofilament lattice using Monte Carlo simulation, level-set and homogenization theory. We found that these intracellular obstacles significantly reduce the diffusion of material through skeletal muscle and generate diffusion anisotropy that is consistent with experimentally observed slower diffusion in the radial than the longitudinal direction. Our model also predicts that protein size has a significant effect on the diffusion of material through muscle, which is consistent with experimental measurements. Protein diffusion on the myofilament lattice is also anomalous (i.e., it does not obey Brownian motion) for proteins that are close in size to the myofilament spacing. The obstructed transport of Ca2+ and ATP-bound Ca2+ through the myofilament lattice also generates smaller Ca2+ transients. In addition, we used homogenization theory to discover that the nonhomogeneous distribution in the troponin binding sites has no effect on the macroscopic Ca2+ dynamics. The nonuniform sarcoplasmic reticulum Ca2+-ATPase pump distribution also introduces small asymmetries in the myoplasmic Ca2+ transients.

A mathematical model for mammary fatty acid synthesis and triglyceride assembly: the role of stearoyl CoA desaturase (SCD)

An increase in the proportion of unsaturated fatty acids in milk is considered desirable for human health. A prerequisite for the manipulation of milk fat composition is a co-ordinated understanding of the complex interactions in its biosynthesis. It has been suggested that an increase in the expression of mammary stearoyl-CoA-desaturase (SCD) would enrich mono-unsaturated fatty acids in milk, and therefore improve its nutritional properties. To investigate the potential effects of changes in expression of mammary enzymes and substrate availability on milk fat composition, we constructed, parameterized and evaluated a mechanistic mathematical model of fatty acid biosynthesis and milk-fat triglyceride assembly. The objective was to describe changes in the amount and composition of milk fat produced by bovine mammary cells due to changes in nutrition. Using the model we found that a 50% up-regulation in SCD activity increased the molar fraction of milk triglyceride 18:1 from 0.30 to 0.33 and 16:1 from 0.04 to 0.06. Up-regulation of SCD therefore did not appear to be the optimal method for increasing the content of unsaturated fatty acids in milk fat. The model was also used to determine the likely rate-limiting processes for the incorporation of unsaturated fatty acids into milk fat. Halving the concentration of glycerol 3-phosphate increased the molar fraction of milk triglyceride 18:1 from 0.30 to 0.35 and decreased the molar fraction of milk triglyceride 16:0 from 0.30 to 0.22. This achieved the desirable outcome of producing more unsaturated low-fat milk. Our model also predicted that a K232A mutation in the bovine mammary DGAT1 gene that is linked with an increase in milk fat yield would be consistent with a 120% increase in the DGAT acylation rate and also would be associated with a decrease in milk mono-unsaturated fatty acids.

Qualitative theory of the spread of a new gene into a resident population

This work addresses the problems that arise in the estimation of the risk of invader/unwanted GMO spread and the optimal release of desired genes into a population through the release of transgenic individuals. On the basis of a general model of the propagation of an advantageous allele through a population we analyze the thresholds and critical aggregations in gene frequency necessary for the spread of new gene carriers. It is shown that if the invader appears at one place in the ecosystem then it will not spread throughout the ecosystem unless it exceeds some critical threshold, where the critical threshold is defined in terms of both the amount and distribution of the invader. The value of the critical threshold will depend on the fitness of the invader relative to the fitness of the resident organisms in the ecosystem and the mechanism of its dispersion. It is also shown that typically an invader will not spread symmetrically, even if the environment is isotropic, but rather develops clusters that form filaments within the ecosystem. We also demonstrate that if the invader aggregation is sufficiently large then after an initial period the advance of the invader into the resident population takes the form of a traveling wave. The speed of this wave tends to a speed characteristic of the relative fitness and dispersive mechanisms of the invader.

Development of a formula for estimating plasma free cortisol concentration from a measured total cortisol concentration when elastase-cleaved and intact corticosteroid binding globulin coexist.

Cortisol bound to corticosteroid binding globulin (CBG) contributes up to 90% of the total cortisol concentration in circulation. Therefore, changes in the binding kinetics of cortisol to CBG can potentially impact on the concentration of free cortisol, the only form that is responsible for the physiological function of the hormone. When CBG is cleaved into elastase-cleaved CBG (eCBG) by the activity of neutrophil elastase, its affinity for cortisol is reduced. Therefore, when eCBG coexists with intact CBG (iCBG) in plasma, the calculation of free cortisol concentration based on the formulae that considers only one CBG pool with the same affinity for cortisol may be inappropriate. In this study, we developed in vivo and in vitro models of cortisol partitioning which considers two CBG pools, iCBG and eCBG, with different affinities for cortisol, and deduce a new formula for calculating plasma free cortisol concentration. The formula provides better estimates of free cortisol concentration than previously used formulae when measurements of the concentrations of the two CBG forms are available. The model can also be used to estimate the affinity of CBG and albumin for cortisol in different clinical groups. We found no significant difference in the estimated affinity of CBG and albumin for cortisol in normal, sepsis and septic shock groups, although free cortisol was higher in sepsis and septic shock groups. The in vivo model also demonstrated that the concentration of interstitial free cortisol is increased locally at a site of inflammation where iCBG is cleaved to form eCBG by the activity of elastase released by neutrophils. This supports the argument that the cleavage of iCBG at sites of inflammation leads to more lower-affinity eCBG and may be a mechanism that permits the local concentration of free cortisol to increase at these sites, while allowing basal free cortisol concentrations at other sites to remain unaffected.

Anomalous ion diffusion within skeletal muscle transverse tubule networks

BackgroundSkeletal muscle fibres contain transverse tubular (t-tubule) networks that allow electrical signals to rapidly propagate into the fibre. These electrical signals are generated by the transport of ions across the t-tubule membranes and this can result in significant changes in ion concentrations within the t-tubules during muscle excitation. During periods of repeated high-frequency activation of skeletal muscle the t-tubule K+ concentration is believed to increase significantly and diffusive K+ transport from the t-tubules into the interstitial space provides a mechanism for alleviating muscle membrane depolarization. However, the tortuous nature of the highly branched space-filling t-tubule network impedes the diffusion of material through the network. The effective diffusion coefficient for ions in the t-tubules has been measured to be approximately five times lower than in free solution, which is significantly different from existing theoretical values of the effective diffusion coefficient that range from 2–3 times lower than in free solution. To resolve this discrepancy, in this paper we study the process of diffusion within electron microscope scanned sections of the skeletal muscle t-tubule network using mathematical modelling and computer simulation techniques. Our model includes t-tubule geometry, tautness, hydrodynamic and non-planar network factors.ResultsUsing our model we found that the t-tubule network geometry reduced the K+ diffusion coefficient to 19–27% of its value in free solution, which is consistent with the experimentally observed value of 21% and is significantly smaller than existing theoretical values that range from 32–50%. We also found that diffusion in the t-tubules is anomalous for skeletal muscle fibres with a diameter of less than approximately 10–20 μm as a result of obstructed diffusion. We also observed that the [K+] within the interior of the t-tubule network during high-frequency activation is greater for fibres with a larger diameter. Smaller skeletal muscle fibres are therefore more resistant to membrane depolarization. Because the t-tubule network is anisotropic and inhomogeneous, we also found that the [K+] distribution generated within the network was irregular for fibres of small diameter.ConclusionOur model explains the measured effective diffusion coefficient for ions in skeletal muscle t-tubules.

A mathematical model of pregnancy recognition in mammals.

In this paper we develop a mathematical model of the luteal phase of the reproductive cycle in mammals with the aim to generate a systems understanding of pregnancy recognition. Pregnancy recognition is initiated by the production of interferon tau (IFNtau) by the growing conceptus. This ensures that the maternal corpus luteum (CL) remains viable to secrete progesterone, which is critical for providing a uterine microenvironment suitable for embryonic growth. Our mathematical model describes the interactions among the CL, the reproductive hormones and the hormone receptors in the uterus. It also characterises the complex interactions amongst the uterine oestrogen, progesterone and oxytocin receptors that control the sensitivity of the uterus to oestrogen, progesterone and oxytocin, respectively. The model is represented by a dynamical system and exhibits qualitative features consistent with the known experimental results in sheep. A key factor identified was a time-dependent threshold for the IFNtau signal below which the presence of the embryo might not be recognised and thus pregnancy would likely fail. Furthermore, the model indicated that if the IFNtau signal is later than around day 13 of the cycle, then pregnancy will not be recognised irrespective of the IFNtau concentration. The thresholds in the concentration and time of the IFNtau signal is a screening mechanism whereby only embryos of sufficient quality are able to prevent luteolysis (i.e. regression of the CL). The effect of progesterone secretion rate from the CL on pregnancy recognition was investigated. The model suggests that if the secretion rate is low then the initiation of the IFNtau signal is delayed, which in turn compromises the likelihood of a pregnancy being recognised by the CL. Furthermore, pregnancy recognition does not occur below a critical threshold in the progesterone secretion rate. In summary, the model can be used to identify the most favourable conditions for pregnancy recognition.

Cyclic glycine-proline regulates IGF-1 homeostasis by altering the binding of IGFBP-3 to IGF-1

The homeostasis of insulin-like growth factor-1 (IGF-1) is essential for metabolism, development and survival. Insufficient IGF-1 is associated with poor recovery from wounds whereas excessive IGF-1 contributes to growth of tumours. We have shown that cyclic glycine-proline (cGP), a metabolite of IGF-1, can normalise IGF-1 function by showing its efficacy in improving the recovery from ischemic brain injury in rats and inhibiting the growth of lymphomic tumours in mice. Further investigation in cell culture suggested that cGP promoted the activity of IGF-1 when it was insufficient, but inhibited the activity of IGF-1 when it was excessive. Mathematical modelling revealed that the efficacy of cGP was a modulated IGF-1 effect via changing the binding of IGF-1 to its binding proteins, which dynamically regulates the balance between bioavailable and non-bioavailable IGF-1. Our data reveal a novel mechanism of auto-regulation of IGF-1, which has physiological and pathophysiological consequences and potential pharmacological utility.

A mathematical model of the interaction between bovine blastocyst developmental stage and progesterone-stimulated uterine factors on differential embryonic development observed on Day 15 of gestation

A complex interaction between the developing bovine embryo and the growth potential of the uterine milieu it inhabits results in an embryo capable of developing past the maternal recognition stage and on to a successful pregnancy. Previously, we observed variation in the lengths of embryos recovered 8 d after bulk transfer of Day 7 in vitro-produced (IVP) blastocysts into the same uterus. Potential causes of the differential embryonic growth were examined and modeled using 2 rounds of bulk (n = 4-6) IVP transfers and recovery of these embryos 8 d later. Morphological and gene expression measurements of the embryos were determined and the progesterone concentration of the cows was measured throughout the reproductive cycle as a reflection of the status of the uterine environment. These data were used to develop and evaluate a model that describes the interaction between the uterine environment and the growth rate of the developing embryo. Expression of 6 trophectoderm genes (IFNT, TKDP1, PAG11, PTGS2, DKK1, and PDPN) was correlated with conceptus length. The model determined that if the embryo develops to blastocyst stage, the uterine environment, driven by progesterone, is a more important component than blastocyst size in the stimulation of embryonic growth rate to ensure adequate interferon tau (IFNT) for pregnancy recognition. We detected an effect of Day 7 progesterone on the expression of all 6 genes, embryonic disc size, and trophectoderm length on Day 15. We also found effects of embryo transfer size on trophectoderm length and expression of IFNT and PAG11 on Day 15. Lower energy balance over the period from transfer to recovery was associated with reduced embryo growth to Day 15, and this effect was independent of progesterone. Energy balance also affected expression of PDPN and TKDP1 on Day 15. We observed an effect of energy balance from transfer to recovery on embryo survival in cows with partial embryo losses, where embryo factors dominate embryo survival, with cows with greater energy balance having lower embryo losses. This effect was independent of energy balance 40 d before transfer and suggests that energy balance has direct, immediate effects on the embryo and maternal environment during this period. Furthermore, energy balance effects on embryo survival in cows with partial embryo losses were largely mediated by expression of TKDP1, PAG11, and PDPN. These results provide candidate signaling pathways for the effect of progesterone and energy balance on embryo growth and survival.