A one-dimensional map to study multi-seasonal coffee infestation by the coffee berry borer.

The coffee berry borer (CBB, Hypothenemus hampei) is the most serious insect pest of coffee worldwide; understanding the dynamics of its reproduction is essential for pest management. The female CBB penetrates the coffee berry, eats the seed, and reproduces inside it. A mathematical model of the infestation progress of the coffee berry by the CBB during several coffee seasons is formulated. The model represents the interaction among five populations: uninfested, slightly infested, and severely infested coffee berries, and free and encapsulated CBBs. Coffee harvesting is also included in the model. A one-dimensional map is derived for tracking the population dynamics subject to certain coffee harvesting percentages over several seasons. Stability analysis of the map's fixed points shows that CBB infestation could be eliminated or controlled to a specific level over multiple seasons of coffee harvesting. However, the percent of coffee harvesting required is determined by the level of CBB infestation at the beginning of the first season and in some cases it is impossible to achieve that percentage.

Parameter estimation for mathematical models of a nongastric H+(Na+)-K(+)(NH4+)-ATPase.

The role of nongastric H(+)-K(+)-ATPase (HKA) in ion homeostasis of macula densa (MD) cells is an open question. To begin to explore this issue, we developed two mathematical models that describe ion fluxes through a nongastric HKA. One model assumes a 1H(+):1K(+)-per-ATP stoichiometry; the other assumes a 2H(+):2K(+)-per-ATP stoichiometry. Both models include Na+ and NH4+ competitive binding with H+ and K+, respectively, a characteristic observed in vitro and in situ. Model rate constants were obtained by minimizing the distance between model and experimental outcomes. Both 1H(+)(1Na(+)):1K(+)(1NH4 (+))-per-ATP and 2H(+)(2Na(+)):2K(+)(2NH4 (+))-per-ATP models fit the experimental data well. Using both models, we simulated ion net fluxes as a function of cytosolic or luminal ion concentrations typical for the cortical thick ascending limb and MD region. We observed that (1) K+ and NH4+ flowed in the lumen-to-cytosol direction, (2) there was competitive behavior between luminal K+ and NH4+ and between cytosolic Na+ and H+, 3) ion fluxes were highly sensitive to changes in cytosolic Na+ or H+ concentrations, and 4) the transporter does mostly Na+ / K+ exchange under physiological conditions. These results support the concept that nongastric HKA may contribute to Na+ and pH homeostasis in MD cells. Furthermore, in both models, H+ flux reversed at a luminal pH that was <5.6. Such reversal led to Na+ / H+ exchange for a luminal pH of <2 and 4 in the 1:1-per-ATP and 2:2-per-ATP models, respectively. This suggests a novel role of nongastric HKA in cell Na+ homeostasis in the more acidic regions of the renal tubules.

Immune response to a pathogen in corals.

The sea fan coral (Gorgonia ventalina), one of the most abundant gorgonians in the tropical and subtropical Atlantic waters, have suffered several diseases that have diminished its abundance throughout their range. In this study, we present a model that analyzes the capacity of G. ventalina to eradicate a micro-pathogen under three immune responses: strong, moderate, and very weak. The model assumes that: (1) polyps are the main unit of the coral; (2) the population of polyps is homogeneously distributed; and (3) the immune system is activated by a signal. When an endosymbiont exceeds a density threshold, it becomes pathogenic, increasing polyp mortality. As a consequence, the colony emits a signal to its stem cells to differentiate into phagocytic and humoral cells, both of which combat the pathogen. Given a strong immune response, the pathogen is rapidly eradicated by the immune cells, and the coral polyp population returns to an equilibrium state. With a moderate immune response, polyps and pathogen coexist, but the maximum capacity of polyp density is never reached. An immunologically compromised colony offering a weak immune response is unable to stop pathogen growth, and the colony dies. This analysis suggests an alternative explanation for the spatial and temporal variability in disease incidence and mortality, which is based on the strength of the immune system of hosts rather than the virulence of the pathogen.

Fluid dilution and efficiency of Na(+) transport in a mathematical model of a thick ascending limb cell.

Thick ascending limb (TAL) cells are capable of reducing tubular fluid Na(+) concentration to as low as ~25 mM, and yet they are thought to transport Na(+) efficiently owing to passive paracellular Na(+) absorption. Transport efficiency in the TAL is of particular importance in the outer medulla where O(2) availability is limited by low blood flow. We used a mathematical model of a TAL cell to estimate the efficiency of Na(+) transport and to examine how tubular dilution and cell volume regulation influence transport efficiency. The TAL cell model represents 13 major solutes and the associated transporters and channels; model equations are based on mass conservation and electroneutrality constraints. We analyzed TAL transport in cells with conditions relevant to the inner stripe of the outer medulla, the cortico-medullary junction, and the distal cortical TAL. At each location Na(+) transport efficiency was computed as functions of changes in luminal NaCl concentration ([NaCl]), [K(+)], [NH(4)(+)], junctional Na(+) permeability, and apical K(+) permeability. Na(+) transport efficiency was calculated as the ratio of total net Na(+) transport to transcellular Na(+) transport. Transport efficiency is predicted to be highest at the cortico-medullary boundary where the transepithelial Na(+) gradient is the smallest. Transport efficiency is lowest in the cortex where luminal [NaCl] approaches static head.

Maximum urine concentrating capability in a mathematical model of the inner medulla of the rat kidney.

In a mathematical model of the urine concentrating mechanism of the inner medulla of the rat kidney, a nonlinear optimization technique was used to estimate parameter sets that maximize the urine-to-plasma osmolality ratio (U/P) while maintaining the urine flow rate within a plausible physiologic range. The model, which used a central core formulation, represented loops of Henle turning at all levels of the inner medulla and a composite collecting duct (CD). The parameters varied were: water flow and urea concentration in tubular fluid entering the descending thin limbs and the composite CD at the outer-inner medullary boundary; scaling factors for the number of loops of Henle and CDs as a function of medullary depth; location and increase rate of the urea permeability profile along the CD; and a scaling factor for the maximum rate of NaCl transport from the CD. The optimization algorithm sought to maximize a quantity E that equaled U/P minus a penalty function for insufficient urine flow. Maxima of E were sought by changing parameter values in the direction in parameter space in which E increased. The algorithm attained a maximum E that increased urine osmolality and inner medullary concentrating capability by 37.5% and 80.2%, respectively, above base-case values; the corresponding urine flow rate and the concentrations of NaCl and urea were all within or near reported experimental ranges. Our results predict that urine osmolality is particularly sensitive to three parameters: the urea concentration in tubular fluid entering the CD at the outer-inner medullary boundary, the location and increase rate of the urea permeability profile along the CD, and the rate of decrease of the CD population (and thus of CD surface area) along the cortico-medullary axis.

Parameter estimation for mathematical models of NKCC2 cotransporter isoforms.

An optimization problem, formulated using a nonlinear least-squares approach, was used to estimate parameters for kinetic models of the three isoforms of the kidney-specific Na-K-2Cl (NKCC2) cotransporter. Specifically, the optimization problem estimates the magnitude of model parameters (i.e., off-binding and translocation rate constants) by minimizing the distance between model unidirectional fluxes and published unidirectional (86)Rb(+) uptake curves for the A, B, and F isoforms of the NKCC2 cotransporter obtained in transfected Xenopus oocytes. By using different symmetry assumptions, NKCC2 models with five, six, seven, or eight parameters were evaluated. The optimization method identified parameter sets that yielded computed unidirectional fluxes consistent with the uptake data. However, the parameter values were not unique, in that systematic exploration of the parameter space revealed alternative parameter sets that fit the data with similar accuracy. Finally, we demonstrate that the optimization method can identify parameter sets for the three transporter isoforms that differ only in ion binding affinities, a result that is consistent with a published mutagenesis analysis of the molecular and structural bases for the differences in (86)Rb(+) uptake among the A, B, and F isoforms. These NKCC2 cotransporter models will facilitate the development of larger scale models of ion transport by thick ascending limb cells.

An optimization algorithm for a distributed-loop model of an avian urine concentrating mechanism.

To better understand how the avian kidney's morphological and transepithelial transport properties affect the urine concentrating mechanism (UCM), an inverse problem was solved for a mathematical model of the quail UCM. In this model, a continuous, monotonically decreasing population distribution of tubes, as a function of medullary length, was used to represent the loops of Henle, which reach to varying levels along the avian medullary cones. A measure of concentrating mechanism efficiency - the ratio of the free-water absorption rate (FWA) to the total NaCl active transport rate (TAT) - was optimized by varying a set of parameters within bounds suggested by physiological experiments. Those parameters include transepithelial transport properties of renal tubules, length of the prebend enlargement of the descending limb (DL), DL and collecting duct (CD) inflows, plasma Na(+) concentration, length of the cortical thick ascending limbs, central core solute diffusivity, and population distribution of loops of Henle and of CDs along the medullary cone. By selecting parameter values that increase urine flow rate (while maintaining a sufficiently high urine-to-plasma osmolality ratio (U/P)) and that reduce TAT, the optimization algorithm identified a set of parameter values that increased efficiency by approximately 60% above base-case efficiency. Thus, higher efficiency can be achieved by increasing urine flow rather than increasing U/P. The algorithm also identified a set of parameters that reduced efficiency by approximately 70% via the production of a urine having near-plasma osmolality at near-base-case TAT. In separate studies, maximum efficiency was evaluated as selected parameters were varied over large ranges. Shorter cones were found to be more efficient than longer ones, and an optimal loop of Henle distribution was found that is consistent with experimental findings.