On the use of non-linear regression with the logistic equation for changes with time of percentage root length colonized by arbuscular mycorrhizal fungi

For the regression of sigmoid-shaped responses with time t of colonization C of roots by arbuscular mycorrhizal fungi, C=C _p/1+[e ^−k(t−ti) ] is the most useful form of the logistic equation. At the time of inflection t _i the slope is maximal and directly proportional to the product of the colonization plateau C _p and the abruptness k of the curve. Coefficient k has a high value when the curve rises abruptly following and preceding long shallow phases. The logistic equation has a curve that is symmetrical about t _i such that C=C _p/2 at inflection. Although the logistic equation can generate a good fit to many data sets for changes in colonization with time, there are cases that are not sigmoid and the logistic equation does not apply. For sigmoid curves, the lag in the development of colonization is directly related to both t _i and k but not to the plateau and not to the value of the maximum slope. Higher values of k or t _i reflect longer lag. When considered alone, t _i and k do not fully summarize the lag in colonization, and so a numerical method to combine them is presented here which allows lag to be compared between curves. In this method, the lag is evaluated by calculating the time during early colonization when the slope equals half of the value of the maximum slope. In summary, use of the logistic equation for regression of sigmoid curves of colonization with time allows numerical comparison between curves of the lag, the period of steep ascent, and the plateau. The logistic equation does not model directly the fundamental processes at work in the development of the mycorrhizae. Instead, it can be used as described here to gain insight into the colonization process by comparing the dynamics of that colonization for different species under various conditions. Accepted: 20 October 2000     

Bacterial growth in space flight: logistic growth curve parameters for Escherichia coli and Bacillus subtilis

Previous investigations have reported that bacterial suspension cultures grow to higher stationary concentrations in space flight than on Earth; however, none of these investigations included extensive ground controls under varied inertial conditions. This study includes extensive controls and cell-growth data taken at several times during lag phase, log phase, and stationary phase of Escherichia coli and Bacillus subtilis. The Marquardt-Levenberg, least-squares fitting algorithm was used to calculate kinetic growth parameters from the logistic bacterial growth equations for space-flight and control growth curves. Space-flight cultures grew to higher stationary-phase concentrations and had shorter lag-phase durations. Also, evidence was found for increased exponential growth rate in space. Received: 27 February 1998 / Received revision: 21 August 1998 / Accepted: 3 September 1998     

A stochastic computer model for simulating population growth

A model is described for investigating the interactions of age-specific birth and death rates, age distribution and density-governing factors determining the growth form of single-species populations. It employs Monte Carlo techniques to simulate the births and deaths of individuals while density-governing factors are represented by simple algebraic equations relating survival and fecundity to population density. In all respects the model's behavior agrees with the results of more conventional mathematical approaches, including the logistic model andLotka's Law, which predicts a relationship betwen age-specific rates, rate of increase and age distribution. Situations involving exponential growth, three different age-independent density functions affecting survival, three affecting fecundity and their nine combinations were tested. The one function meeting the assumptions of the logistic model produced a logistic growth curve embodying the correct values or r_m and K. The others generated sigmoid curves to which arbitrary logistic curves could be fitted with varying success. Because of populational time lags, two of the functions affecting fecundity produced overshoots and damped oscillations during the initial approach to the steady state. The general behavior of age-dependent density functions is briefly explored and a complex example is described that produces population fluctuations by an egg cannibalism mechanism similar to that found in the flour beetle Tribolium. The model is free of inherent time lags found in other discrete time models yet these may be easily introduced. Because it manipulates separate individuals, the model may be combined readily with the Monte Carlo simulation models of population genetics to study eco-genetic phenomena.     

Growth, gas exchange, leaf nitrogen and carbohydrate concentrations in NAD‐ME and NADP‐ME C4 grasses grown in elevated CO2

Plants with the C₄ photosynthetic pathway have predominantly one of three decarboxylation enzymes in their bundle sheath cells. Within the grass family (Poaceae) bundle sheath leakiness to CO₂ is purported to be lowest in the nicotinamide adenine dinucleotide phosphate-malic enzyme (NADP-ME, EC 1.1.1.40) group, highest in the NAD-ME (EC 1.1.1.39) group and intermediate in the phosphoenolpyruvate carboxykinase (PCK, EC 4.1.1.32) group. We investigated the hypothesis that growth and photosynthesis of NAD-ME C₄ grasses would respond more to elevated CO₂ treatment than NADP-ME grasses. Plants were grown in 8-1 pots in growth chambers with ample water and fertilizer for 39 days at a continuous CO₂ concentration of either 350 or 700 µl l^?1. NAD-ME species included Bouteloua gracilis Lag. ex Steud (Blue grama), Buchloe dactyloides (Nutt.) Engelm. (Buffalo grass) and Panicum virgatum L. (Switchgrass) and the NADP-ME species were Andropogon gerardii Vittman (Big bluestem), Schizachyrium scoparium (Michx.) Nash (Little bluestem), and Sorghastrum nutans (L.) Nash (Indian grass). Contrary to our hypothesis, growth of the NADP-ME grasses was generally greater under elevated CO₂ (significant for A. gerardii and S. nutans), while none of the NAD-ME grasses had a significant growth response. Increased leaf total non-structural carbohydrate (TNC) was associated with greater growth responses of NADP-ME grasses. Decreased leaf nitrogen in NADP-ME species grown at elevated CO₂ was found to be an artifact of TNC dilution. Assimilation (A) vs intercellular CO₂ (C_i) curves revealed that leaf photosynthesis was not saturated at 350 µl l^?1 CO₂ in any of these C₄ grasses. Assimilation of elevated CO₂-grown A. gerardii was higher than in plants grown in ambient CO₂. In contrast, B. gracilis grown in elevated CO₂ displayed lower A, a trait more commonly reported in C₃ plants. Photosynthetic acclimation in B. gracilis was not related to leaf TNC or nitrogen concentrations, but A:C_i curves suggest a reduction in activity of both phosphoenolpyruvate (PEP) carboxylase (EC 4.1.1.31) and ribulose-1,5-bisphosphate carboxylase/oxygenase (Rubisco, EC 4.1.1.39). Some adaptation of stomatal functioning was also seen in B. gracilis and A. gerardii leaves grown in elevated CO₂. Our study shows that C₄ grasses have the capacity for increased growth and photosynthesis under elevated CO₂ even when water and nutrients are non-limiting. While it was the NADP-ME species which had significant responses in the present study, we have previously reported significant growth increases in elevated CO₂ for B. gracilis.     

Effect of prey concentration on growth of piscivorous Japanese Spanish mackerel Scomberomorus niphonius larvae in the Seto Inland Sea, Japan

Japanese Spanish mackerel, Scomberomorus niphonius, larvae feed almost exclusively on fish larvae from the first‐feeding stage. The relationship between the growth of S. niphonius larvae and concentration of major prey organisms of the larvae, clupeid larvae, was investigated in the Sea of Hiuchi, the central Seto Inland Sea, Japan, from 28 to 29 May 1997. Water temperature, salinity, and the concentration of clupeid larvae had no significant effect on the S. niphonius larval concentration. Mean growth rate of S. niphonius larvae varied between 0.38 and 0.64 mm day^?1. The temperature and salinity had no significant effect on the mean larval growth rate while there existed prey concentration‐dependent growth at lower prey concentration. The relationship between the mean larval growth rate (G_m) and concentration of clupeid larvae (C_C) was expressed by a logarithmic equation: G_m = 0.037 log C_C + 0.441 (n = 16, r² = 0.519, P < 0.01).     

Thermochemical Characterization of the Growth Metabolism of the 2,4-Dichlorophenol-Degrading Strain Pseudomonas GT241-1 by Microcalorimetry

By using an LKB-2277 Bioactivity Monitor and the ampoule method, the heat output of the growth metabolism of a 2,4-dichlorophenol-degrading bacterial strain, Pseudomonas strain GT241-1, has been determined at 30°C. From the thermogenic curves, it can be established that the thermokinetic equation of their growth metabolism is P _t = P _{t = 0} exp(k _m t), dP/dt = k _m P ¹, with the order of growth metabolism n = 1. The experimental results indicate that the relationship between the metabolic power (P) and the cell concentration (C) and the relationship between the metabolic power of each cell (P ₀) and the cell concentration can be characterized by the following thermal equations, respectively: C = a + kP and lnC = a+kP ₀ or d dC/dP ₀ = KC ¹. The order of the P ₀ –C equation n is also 1. These results are very significant for environmental sciences, biology, and thermochemistry.     

Effects of water on the mechanical properties of gelatin films

The mechanical properties of gelatin films were studied in relation to the effect of water, and compared with those of collagen films. The S-shaped sorption isotherm was separated into an adsorption curve C₁ and dissolution curve C₂. From the C₂ curve, the interaction parameter χ₁ of Flory–Huggins' equation was calculated. The χ₁ of gelatin were larger than those of collagen at low relative humidities (RH), while they coincided with each other at high RH. It was found that a composite curve was made by shifting stress relaxation curves obtained at different humidities along the log time axis. The shift factor for the formation of the composite curve was analyzed by Fujita–Kishimoto's equation, which was based on the free volume theory. The parameter β, which expressed the extent of the contribution of sorbed water to the increase in the free volume of the system, was 0.05 in the range of C₂ from 0 to 0.08 (0–65% RH). This value was much smaller than 0.16 for collagen. The value was 0.16 in the range of C₂ higher than 0.08, which was equal to that of the collagen. Dynamic shear modulus G′, loss modulus G″, and tan δ were determined as functions of RH. The gelatin film extended more than 100% at 73% RH under the very small stress of about 10⁷ dyn/cm². This corresponds to the region where β changes from 0.05 to 0.16, although such a phenomenon was not observed in the collagen film. The wide-angle X-ray pattern of extended gelatin was similar to that of renatured collagen fiber.     

Inhibitory effects of ethanol in alcoholic fermentation

The ethanol tolerance behaviour of the strain Saccharomyces cerevisiae Sc 5 regarding the growth is characterized by a threshold ethanol inhibitory concentration (P₁' = 42.5 g/l) and a linear relationship between the specific growth rate and the ethanol concentration within the limits P₁' < P < P′. The maximum ethanol concentration for growth amounts to P′ = 84 g/l. A general model for the inhibition of growth and alcohol production, respectively, caused by ethanol, is deduced from experimental and bibliographical data: If the inhibitory effects are linear, the exponents b, b' become 1.     

Kinetics of batch-wise enzymatic cycling system for mass production of chiral compound

Kinetics of batch-wise enzymatic cycling system (oxidoreductase-catalyzed reaction system involving enzyme-coupled cofactor regeneration) has been studied covering a broad range of the conserved total cofactor concentration, [C]₀ (=NAD(P)⁺?+?NAD(P)H), based on reasonable several assumptions. It is composed of two elementary reactions, i.e. product synthesis reaction and cofactor regeneration reaction, both of which have been expressed by Michaelis–Menten type rate equations. A novel dimensionless variable, r, has been introduced, which is defined as the concentration of one of the two cofactor components, [X] (NADH⁺ or NADPH⁺), divided by [C]₀, i.e. r .e[X]/[C]₀. The following results have been obtained. (1) The fundamental equation of the batch-wise enzymatic cycling system has been transformed to a differential equation whose formula is: dr/dT?=?N(r)/D(r) (N(r) and D(r) are quadratic equations of r having different coefficients). (2) It has been elucidated that the batch-wise enzymatic cycling system has two phases, an early short transient phase followed by a long phase in quasi-steady state (QSS). (3) In the enzymatic cycling system, r converges to a definite level regardless of any initial value of r. (4) In QSS, the definite level of r nearly equals the singular solution, r_singular, of the differential equation. (5) The actual rate of the targeted product (chiral compound) formation can be calculated by Michaelis–Menten equation in which the cofactor concentration is [C]₀×r_singular instead of [C]₀. r_singular has been proposed to name “redistribution factor”. (6) It is recommended that the “unit” of the cofactor regeneration enzyme be 2–3 times more used than the “unit” of the synthesis enzyme and that [C]₀ be 15–25 times more than the K_m value. Four special cases relating to the batch-wise enzymatic cycling system have been discussed.     

Modelling and predicting the effect of temperature,water activity and pH on growth of Streptococcus iniae in Tilapia

Aims: To evaluate and model the growth of Streptococcus iniae affect by temperatures (10–45°C), water activity (A_w; 0·995–0·957), and pH (5–8). Methods and Results: Temperatures, A_w, and pH were adjusted. The behaviour of S. iniae was studied and modelled. Growth curves were fitted by using logistic, Gompertz, and Baranyi models. The maximum growth rates obtained from the primary model were then modelled as a function of temperature, A_w, and pH using the Belehradek‐type models for secondary model. The optimum values for growth were found to be in the range of 35–40°C, A_w 0.995–1, and pH 6–7. The statistical characteristics of the models were validated by r², mean square error, bias, and accuracy factors. The results of validation indicated that Baranyi model performed the best. Conclusions: The effect of temperature, A_w/NaCl, pH control of S. iniae in tilapia could be satisfactorily predicted under current experimental conditions, and the proposed models could serve as a tool for this purpose. Significance and Impact of the Study: The suggested predictive model can be used for risk assessment concerning S. iniae in tilapia.     

Interfacial regulation of bacterial sphingomyelinase activity

The objective of this study was to define how the quality of the buffer/membrane interface influences the activity of bacterial sphingomyelinase acting at the interface. The enzyme reaction was carried out in a zero-order trough using a surface barostat. This approach allowed for proper control of the physico-chemical properties of the substrate molecules. Since the molecular area of ceramide is smaller than that of sphingomyelin, the hydrolysis reaction could be followed `on-line' from the monolayer area decrease at constant surface pressure. The hydrolysis reaction could be divided into two separate phases, the first being the lag-phase (time between enzyme addition and commencement of the monolayer area change), and the second phase being the actual hydrolysis reaction (from which a maximal degradation rate could be determined). The activity of sphingomyelinase (Staphylococcus aureus) toward bovine brain sphingomyelin (bb-SM) was markedly enhanced by Mg²⁺ (maximal activation at 5 mM). Mg²⁺ also influenced the lag-phase of the reaction (the lag-time increased markedly when the Mg²⁺ concentration decreased below 1 mM). Saturated sphingomyelins (bb-SM and N-palmitoyl sphingomyelin [N-P-SM]) were more slowly degraded than the mono-unsaturated N-oleoyl sphingomyelin (N-O-SM). Both bb-SM and N-P-SM monolayers underwent a phase-transition at room temperature, whereas the N-O-SM monolayer did not. The phase-transition (liquid-expanded to liquid-condensed) was observed to greatly increase the lag-time of the hydrolysis reaction. The activity of sphingomyelinase was also sensitive to the lateral surface pressure of the monolayer membrane. Maximal degradation rate was achieved at 20 mN/m (with bb-SM, 30°C); above this pressure the lag-time of the reaction increased sharply. The inclusion of 4 mol% of cholesterol into a [³H]sphingomyelin monolayer markedly increased the extent of [³H]sphingomyelin degradation, and shortened the lag-time of the reaction. The inclusion of 10 mol% of zwitterionic or negatively charged phospholipids to the [³H]sphingomyelin monolayer did not affect the sphingomyelinase reaction significantly. In conclusion, this study has demonstrated that the physico-chemical properties of the substrate molecules have a dominating influence on the activity of a bacterial sphingomyelinase acting at the buffer/membrane interface.     

A New Generalized Logistic Sigmoid Growth Equation Compared with the Richards Growth Equation

A new sigmoid growth equation is presented for curve-fitting,analysis and simulation of growth curves. Like the logisticgrowth equation, it increases monotonically, with both upperand lower asymptotes. Like the Richards growth equation, itcan have its maximum slope at any value between its minimumand maximum. The new sigmoid equation is unique because it alwaystends towards exponential growth at small sizes or low densities,unlike the Richards equation, which only has this characteristicin part of its range. The new sigmoid equation is thereforeuniquely suitable for circumstances in which growth at smallsizes or low densities is expected to be approximately exponential,and the maximum slope of the growth curve can be at any value.Eleven widely different sigmoid curves were constructed withan exponential form at low values, using an independent algorithm.Sets of 100 variations of sequences of 20 points along eachcurve were created by adding random errors. In general, thenew sigmoid equation fitted the sequences of points as closelyas the original curves that they were generated from. The newsigmoid equation always gave closer fits and more accurate estimatesof the characteristics of the 11 original sigmoid curves thanthe Richards equation. The Richards equation could not estimatethe maximum intrinsic rate of increase (relative growth rate)of several of the curves. Both equations tended to estimatethat points of inflexion were closer to half the maximum sizethan was actually the case; the Richards equation underestimatedasymmetry by more than the new sigmoid equation. When the twoequations were compared by fitting to the example dataset thatwas used in the original presentation of the Richards growthequation, both equations gave good fits. The Richards equationis sometimes suitable for growth processes that may or may notbe close to exponential during initial growth. The new sigmoidis more suitable when initial growth is believed to be generallyclose to exponential, when estimates of maximum relative growthrate are required, or for generic growth simulations.Copyright1999 Annals of Botany Company Asymptote,Cucumis melo,curve-fitting, exponential growth, intrinsic rate of increase, logistic equation, maximum growth rate, model, non-linear least-squares regression, numerical algorithm, point of inflexion, relative growth rate, Richards growth equation, sigmoid growth curve.     

Richards's equation and nonlinear mixed models applied to avian growth: why use them?

Postnatal growth is an important life‐history trait that varies widely across avian species, and several equations with a sigmoidal shape have been used to model it. Classical three‐parameter models have an inflection point fixed at a percentage of the upper asymptote which could be an unrealistic assumption generating biased fits. The Richards model emerged as an interesting alternative because it includes an extra parameter that determines the location of the inflection point which can move freely along the growth curve. Recently, nonlinear mixed models (NLMM) have been used in modeling avian growth because these models can deal with a lack of independence among data as typically occurs with multiple measurements on the same individual or on groups of related individuals. Here, we evaluated the usefulness of von Bertalanffy, Gompertz, logistic, U₄ and Richards's equations modeling chick growth in the imperial shag Phalacrocorax atriceps. We modelled growth in commonly used morphological traits, including body mass, bill length, head length and tarsus length, and compared the performance of models by using NLMM. Estimated adult size, age at maximum growth and maximum growth rates markedly differed across models. Overall, the most consistent performance in estimated adult size was obtained by the Richards model that showed deviations from mean adult size within 5%. Based on AICc values, the Richards equation was the best model for all traits analyzed. For tarsus length, both Richards and U₄ models provided indistinguishable fits because the relative inflection value estimated from the Richards model was very close to that assumed by the U₄ model. Our results highlight the bias incurred by three‐parameter models when the assumed inflection placement deviates from that derived from data. Thus, the application of the Richards equation using the NLMM framework represents a flexible and powerful tool for the analysis of avian growth.     

Rat C-6 Glioma Cells Maintained In an Organ Culture System: A Study of Kinetic Parameters

Rat C-6 glioma cells were grown on a sponge foam matrix in an organ culture system and the cell cycle parameters, including the growth fraction (GF), were assessed after autoradiography. the zones of growth consisted of a compact upper layer (UL) at the gaseous interface, a central necrotic layer and a deeper lower layer (LL) which invaded the matrix. the fraction of continuously labeled mitoses (FCLM) was similar in both the UL and LL cells. the derivatives of the FCLM curves obtained in three experiments gave an average modal T_G2 of 5 hr. A mathematical model relating GF, T_G2, T_C and labeling index as a function of time, LI(t), was devised for cells in a steady state exposed continuously to tritiated thymidine and was applied to data obtained from UL cells. A mean GF of 9% (range: 8–10%) and a mean cell cycle time (T_C') of 27 hr (range: 13–47 hr) were obtained. the mean T_S was calculated to be 11 hr (range: 8–16 hr) by the method of grain counts per mitotic figure or grain index (GI). Knowledge of T_S permitted alternative calculation of the cell cycle time from the equation T_S/T_C= LI(0)/GF: this gave a mean cell cycle time (T_C) of 29 hr (range: 20–45 hr). Except for the GF, the cell kinetics were comparable to those of the same cell line grown in monolayer culture. the GF in the in vitro system described is in the lower range reported in some human malignant gliomas in vivo.     

Adaptational changes in cellular phospholipids and fatty acid composition of the food pathogen Listeria monocytogenes as a stress response to disinfectant sanitizer benzalkonium chloride

Aims: This study provides a first approach to observing the alterations of the cell membrane lipids in the adaptation response of Listeria monocytogenes to the sanitizer benzalkonium chloride. Methods and Results: A thorough investigation of the composition of polar and neutral lipids from L. monocytogenes grown when exposed to benzalkonium chloride is compared to cells optimally grown. The adaptation mechanism of L. monocytogenes in the presence of benzalkonium chloride caused (i) an increase in saturated‐chain fatty acids (mainly C_16:0 and C_18:0) and unsaturated fatty acids (mainly C_16:1 and C_18:1) at the expense of branched‐chain fatty acids (mainly C_a‐15:0 and C_a‐17:0) mainly because of neutral fatty acids; (ii) no alteration in the percentage of neutral and polar lipid content among total lipids; (iii) a decrease in lipid phosphorus and (iv) an obvious increase in the anionic phospholipids and a decrease in the amphiphilic phosphoaminolipid. Conclusions: These lipid changes could lead to decreased membrane fluidity and also to modifications of physicochemical properties of cell surface and thus changes in bacterial adhesion to abiotic surfaces. Significance and Impact of the Study: The adaptation and resistance of L. monocytogenes to disinfectants is able to change its physiology to allow growth in food‐processing plants. Understanding microbial stress response mechanisms would improve the effective use of disinfectants.     

Identifizierung prozeßspezifischer lag-Zeiten und Berücksichtigung bei der Berechnung phänomenologischer Wachstumsparameter

A significant difference between the maxima of the specific growth rate (μ) and the specific product formation (k_p) may indicate a retarded correlation model of biomass (X) and product (P). Therefore phenomenological growth parameters which connect X and P or their derivations (Y_p/x, k_p) have to take into consideration this kind of lag-time in the same way.     

Misconceptions about the comparison of intrinsic rates of natural increase

The intrinsic rate of natural increase (r_m) is a common measurement in entomology to describe and evaluate the growth and adaptation of a population of arthropods to certain environmental conditions. Following the method of Birch, the r_m is the solution of an exponential equation, which depends on the whole life cycle of each female and her survival time. A simplification of this equation was provided by Wyatt and White, which allows the study to be shortened as it does not depend on any survival times and only a part of the life cycle of the females. Therefore, this method has become quite popular among entomologists. As the r_m is a population parameter, it lacks any variance and thus a valid statistical comparison of r_ms for different populations is not straightforward. Hence, many approaches include statistical misconceptions. We discuss those approaches, apply them to real data and demonstrate some drawbacks of them. Furthermore, we present an easy to implement and consistent method for the comparison of r_ms.     

Photosynthetic pathway types of evergreen rosette plants (Liliaceae) of the Chihuahuan desert

Summary Diurnal patterns of CO₂ exchange and titratable acidity were monitored in six species of evergreen rosette plants growing in controlled environment chambers and under outdoor environmental conditions. These patterns indicated that two of the species, Yucca baccata and Y. torreyi, were constituitive CAM plants while the other species, Y. elata, Y. campestris, Nolina microcarpa and Dasylirion wheeleri, were C₃ plants. The C₃ species did not exhibit CAM when grown in any of several different temperature, photoperiod, and moisture regimes. Both photosynthetic pathway types appear adapted to desert environments and all species show environmentally induced changes in their photosynthetic responses consistent with desert adaptation. The results of this study do not indicate that changes in the photosynthetic pathway type are an adaptation in any of these species.     

Influence of base-pair changes and cooperativity parameters on the melting curves of short DNAs

Theoretical melting curves were calculated for four DNA restriction fragments, 157–257 base pairs (bp), and a series of hypothetical block DNAs with sequences d(C_2xA_xC_2x). d(C_2xT_xG_2x), 5 ? x ? 40. These DNAs provided a mixture of A·T/G·C sequence distributions with which to investigate the effects of parameters and base-pair changes on the melting of short DNAs. The sensitivity of DNA melting curves to changes in internal loop melting parameters σ and κ was examined. As Expected, theoretical melting curves of short DNAs with a quasirandom base-pair sequence vary little with changes in internal loop parameters. End melting dominates the transition behaviour of these moleucles. This was also observed for the block DNAs up to x = 22. Beyond this length, melting curves are highly sensitive to the internal loop parameters. Sensitivity is also predicted for a 157-bp fragment with a block distribution of A·T and G·C pairs. These results indicate that accurate evaluation of internal loop parameters is possible with short DNAs (100–200 bp) containing a G·C/A·T/G·C block distribution with at least 22 bp in each block. Duplex-to-single-strands dissociation parameters were reevaluated form experimental melting curve data of eight DNA fragments using a least squares fit approach. This analysis confirmed parameter values previously found with a simplified dissociation model. A Priori predictions are made on the effects of base-pair changes on the melting curves of three characterized DNA restriction fragments. Single base-pair changes are predicted to induce small but measurable changes in the melting curves. The characteristics of the altered melting curves depend on the location of the base-pair change.