H. D. Landahl

A three stage population model with cannibalism

A simple population model consisting of one adult and two larval stages with cannibalism or competition among the larval stages is presented. The solutions are found to be either periodic or of a steady state nature depending on the ratios of fertility and cannibalism among the larvae. Two similar cannibalism pressure functions are compared and the conditions that lead to steady or periodic solutions, or to extinction, are examined.

Some conditions for sustained oscillations in biochemical chains with feedback inhibition

A chain ofn reactions is considered in which the last substance inhibits the production of the first with degreep, p being the order of the inhibition. Maintained oscillations are possible for certain values of the parameters under the following conditions: (1) If there is no time delay, then there must be at least three compartments (n=3) and either the degree of inhibition is sufficiently large (p>8 forn=3) or there must be enzymatic removal from the first compartment, in which casep≥1. (2) If there are time delays, but there is no enzymatic removal, the degree of the inhibition must be greater than or equal to 2 for any value ofn. (3) If there is a time delay in addition to enzymatic removal, one compartment with simple first order inhibition is sufficient. Conditions on the parameters necessary for maintained oscillations are given for many of the cases discussed.

Comparison of models of insulin release

A model for insulin secretion with a storage and a labile compartment, as well as a provisionary factor, is combined with a signal model in which the signal can be the difference between an excitation and an inhibition, or the difference in concentrations inside and outside some cell components. The model, using a single set of values for the parameters, accounts in a semiquantitative manner for all of the regularly appearing features of the insulin secretion from thein vitro perfused pancreas to a wide range of patterns of glucose and tolbutamide stimulation. Among the features which can be accounted for are: early and late secretion of insulin as a function of glucose in terms of a single parameter; the apparent depletion and recovery during a pulsed pattern of stimulation by tolbutamide; the hypersecretion following a short period of rest during a prolonged stimulation by glucose; the negative spike which occurs when the concentration of glucose, which has been maintained for a period of time, is suddenly reduced to a lower level; and the appropriate responses to slow and fast ramp functions of glucose concentration.

IN VITRO STUDIES SUGGESTING A TWO-COMPARTMENTAL MODEL FOR INSULIN SECRETION

SUMMARY Our previous studies with the in vitro perfused pancreas of the rat have shown that insulin release in response to a constant stimulation is multiphasic. Response to glucose or tolbutamide is characterized by a rapid release of insulin followed by a rapid decrease in secretion rate. During this later phase, the pancreas can be relatively refractive to further stimulation. After 15 min, glucose but not tolbutamide produces a second rise in the rate of insulin release which increases during subsequent perfusion. When, after prolonged glucose stimulation, the pancreas is rested and then restimulated with a glucose pulse, a super-normal secretion of insulin occurs. Thus glucose produces either a refractory or hypersensitive state in the pancreas depending on the phase studied. If glucose concentration is increased slowly from 0 to 250 mg per 100 ml, the multiphasic response no longer occurs. A mathematical model was devised which closely duplicated the various multiphasic patterns of insulin release. The model assumes two compartments of stored insulin which are concentration related. The smaller labile compartment is more sensitive to agents stimulating insulin release and is the primary site of insulin secretion. The model also includes an action of agents such as glucose to increase the production of theoretical potentiator which in turn activates the provision of newly available insulin to the small compartment. Examination of the constants derived from the model for perfused rat pancreas shows: (1) the small compartment is only a few % of total extractable insulin; (2) exchange rates between the large and the small compartment are slow, thus a rapid stimulation of insulin release results in a partial depletion of the small compartment; (3) glucose and tolbutamide probably act on the same small compartment though not necessarily by the same mechanism; (4) the rate of non-specific degradation of the potentiator is slow, thus for some time after stopping glucose, addition of insulin to the small compartment continues (producing a hypersensitive pancreas). The model suggests, but does not prove, the novel hypothesis that newly provided insulin directly enters the small labile compartment and is the insulin preferentially excreted. It is also easily modified to incorporate our previons suggestion of a stimulator-induced feedback inhibition.

Effects of Tolbutamide Pretreatment on the Rate of Conversion of Newly Synthesized Proinsulin to Insulin and the Compartmental Characteristics of Insulin Storage in Isolated Rat Islets

Toibutamide (1 g/kg body wt) was administered to male rats for 3 days to determine the effects of this pretreatment on subsequent insulin biosynthesis and compartmental storage characteristics of freshly isolated islets. Islets were isolated 16 h after the last toibutamide administration, at a time when fed plasma glucose concentrations were normal. Islet glucagon was unchanged but insulin content was significantly reduced (38 ± 1.2 ng IRI/islet from seven untreated rats versus 7.9 ± 1.2 ng IRI/islet from eight treated rats). After toibutamide pretreatment, the rate of incorporation of 3Hleucine into islet proinsulin was unchanged, but the t1/2 of labeled proinsulin-to-insulin conversion was significantly (P < 0.001) decreased from 36 to 20 min. After treatment, actual rates of glucose-stimulated insulin secretion were 50% lower, however, because due to the proportionately greater depletion of islet insulin content, the fractional rate of secretion was increased twofold. After treatment, there was evidence of compartmental, heterogeneous insulin storage, and glucose still marked newly synthesized insulin for preferential release; however, the differential release of new and old insulin converged rapidly with time. Mathematical integration of the data suggested dilution of the newly synthesized insulin compartment with unlabeled insulin during the chase period, but additionally indicated more rapid mixing of newly synthesized with previously stored, unlabeled insulin. Thus, tolbutamide-treated rats partially compensated for acute insulin depletion by (1) increasing the rate of proinsulin-to-insulin conversion, but not increasing the rate of proinsulin biosynthesis; (2) doubling the glucose-stimulated fractional secretory rate of the depleted cellular insulin storage compartment; and (3) retaining compartmental storage characteristics but mixing newly synthesized insulin more rapidly with the compartment of previously stored, unlabeled insulin.

Glucose-stimulated 45Ca Uptake in Isolated Rat Islets

Net 45Ca uptake (in excess of the extracellular 3H-sucrose space) and insulin release were measured under low (2 mM) and high (20 mM) glucose conditions in collagenase-isolated rat islets. 45Ca uptake curves were mathematically fitted and subjected to compartmental analysis. Within the first 60 seconds after addition of trace 45Ca, islets showed a similar rapid uptake of 45Ca regardless of the glucose concentration or length of time of prior exposure to glucose. Net 45Ca uptake continued to increase for 30-60 minutes, and the islets in high glucose showed approximately twofold greater maximum uptake than islets in low glucose. Islets preincubated in low glucose and then incubated in high glucose showed a 5-15-minute delay in net 45Ca uptake as compared with islets that had been preincubated in high glucose. Insulin release was detectable by 10 minutes of incubation with high glucose. Mathematical modeling of the low- and high-glucose net 45Ca uptake curves suggests that there are at least two calcium “compartments” within the β-cell. One compartment is small, rapidly filled, and insensitive to glucose, while the other, larger compartment, is slowly filled and fills to a much greater extent in the presence of high glucose. A major proportion of the glucose-stimulated uptake is at the level of influx.

Relationship of Octanol/Water Partition Coefficient and Molecular Weight to Cellular Permeability and Partitioning in S49 Lymphoma Cells

We have used modified standard methods and derived new formulae to quantitate cell permeability (P), cell/media partitioning (λ), and intracellular sequestration or binding rate constants (m) for cultured S49 murine lymphoma cells in suspension. Using 15 standard compounds and anticancer drugs, we found quantitative relationships among log P, log PO (octanol/pH 7.4 buffer partition coefficient), and molecular weight (MW) such that logP = −4.5 + 0.56log (PO(MW)−1/2). A good correlation among P, λ, and MW was also determined with λ = 0.67 + 5890 gm1/2 cm−1 sec (P (MW)1/2). These studies show that there is a strong partitioning (λ) dependence to molecular weight and permeability that can be predicted even for known carrier-transported and biotransformable compounds. Furthermore, results of this study show that the slope of the plot of permeability and lipophilicity is not necessarily unity as has been postulated from the results of other studies.

A mathematical model for first degree block and the Wenckebach phenomenon

A simple mathematical model of first and second degree heart block is developed on the assumption that eachR wave, with its associated ventricular contraction, results in the release of a substance which raises the excitation threshold of the conducting tissue lying between theS-A andA-V nodes. The conduction pathway is represented by a chain of excitatory elements, the first member of which is acted upon by a regularly occurringP wave. The rate constant of the removal of this inhibitory substance, i.e., recovery rate, is the only constant necessary to be varied in order to pass from normal to first degree block to second degree.

Note on oscillations in a closed chain of compartments with linear, unidirectional transport

A closed chain of compartments in which there is unidirectional transport between adjacent members can exhibit damped oscillations. For a system ofn equivalent compartments, the value ofn which gives the greatest difference between the first maximum and first minimum isn=11, the difference being 1.57%. The greatest difference between the first maximum value and the steady state value is 4% and is obtained whenn=25. The results are illustrated graphically forn equal to 5, 10, 25 and 100.

Some conditions for sustained oscillations in chain processes with feedback inhibition and saturable removal.

A biochemical chain with feedback inhibition, enzymatic removal and catalyzed input is considered. Some conditions on the parameters are given for which the result is a simple limit cycle behavior or a limit cycle surrounding an unstable limit cycle and a stable point. In the latter case, the system in the steady state can either be at rest or oscillate around the point at rest, depending on the initial conditions.