Cause-effect modeling as a general method of description and study of phenomena in complex hierarchical systems

The definition of the cause-effect model of a phenomenon and the rules of presenting these models in the form of cause-effect diagrams have been formulated. The relationship between cause-effect modeling and traditional methods of mathematical modeling has been analyzed. Examples of cause-effect models (diagrams) of phenomena of different physical nature are given, and the application of these models in studies of some problems is demonstrated. In particular, the mechanism of renormalizing the rate constans of chemical reactions in terms of dissipative resonance is considered. In addition, the renormalization of the climate sensitivity parameters and the relaxation time of the Earth climate system in terms of the two-component (CO2 + H2O) greenhouse effect is considered.     

Phase Diagrams for DNA Crystallization Systems

Abstract

Phase diagrams for several oligonucleotide duplex -spermine systems have been constructed. These diagrams characterize the duplex and spermine concentrations ranges in which crystalline precipitates are formed. All of them are wedge-like form. The slope of the upper branch of the diagram is determined by the oligonucleotide length. The position of the lower branch depends on both the nucleotide sequence and its length. The position of the lower branch depends on both the nucleotide sequence and its length. It has been shown that the addition to the system ofMgCl₂ and NaCl salts and MPD results in specific changes in the diagrams. A model for oligonucleotide duplex-spermine system has been suggested which explains the main characteristic features of the obtained phase diagrams. The experimental phase diagrams for the (pGpT)_n · (pApC)_n-spermine system (n = 2,3,4) have been analyzed ion terms of this model and the values of the binding constants of spermine and Mg²⁺ions binding to duplexes have been determined. It permitted to identify the complexes that precipitated in different regions of the phase diagrams under various conditions. The diagram obtained in the presence of a cobalt hexammine counterion is also considered. It has been shown that this phase diagram, in general, is similar to those obtained for the oligonucleotide duplex-spermine system.     

Merged mechanisms for hydride transfer from 1,4-dihydronicotinamides

Recent work on the reduction of heteroaromatic cations by 1,4-dihydronicotinamides and related reducing agents is reviewed. Extensive correlations are presented between the second-order rate constants (k₂) for these reactions and the second-order rate constants (k_OH) and equilibrium constants (pK_R⁺) for hydroxide ion attack on these cations. Close correlations of log k₂ with the electron affinities and one-electron reduction potentials of these cations are also presented. These relationships are considered in the context of a direct hydride transfer from donor to acceptor and also in terms of SET mechanisms which are also commonly discussed for such reactions. It is shown that the interpretation of these formal hydride transfer reactions in terms of an imbalanced development of electronic charge and C---H bond fission within the transition state species leads to a rational merging of the single-step hydride transfer mechanism and the SET mechanisms. The structures of the transition state species are expected to be highly variable and quite dependent upon the nature of the hydride donor and acceptor species, with considerable contribution from charge-transfer interactions. Such imbalanced transition state species are analyzed in terms of two different types of reaction coordinate diagrams and also in terms of the valence bond configuration mixing theory.     

Climate model sensitivity to atmospheric CO2 levels in the Early-Middle Paleogene

This study examines the sensitivity of the slab ocean version of the National Center for Atmospheric Research Climate System Model with revised Eocene geography, orography, and vegetation to changing carbon dioxide (CO₂) levels. We compare model results with temperature proxies from the geologic record for the Early-Middle Paleogene. We ran three modeling experiments with CO₂ levels at 500, 1000, and 2000 ppm, and all with atmospheric methane levels of 3.5 ppm. Surface temperatures in the two higher CO₂ scenarios are warmer than those of the 500 ppm scenario. The largest warming with increasing CO₂ occurred in the high latitudes, particularly in the Northern Hemisphere, during the wintertime. Compared to the 500 ppm case, Arctic wintertime temperatures increased by ∼10°C for the 1000 ppm scenario, and ∼20°C for the 2000 ppm scenario. The 1000 and 2000 ppm scenarios produced mean annual and cold month mean temperatures in mid- and high latitudes that are much more compatible with the climate interpretations from Eocene flora, especially for data from the Southern Hemisphere. Tropical sea surface temperatures (SSTs) in the 2000 ppm scenario, however, are still ∼4°C higher than the warmest temperatures inferred from proxy data. The better match between temperatures in the high CO₂ modeling scenario and high latitude climate interpretations is consistent with the idea that the CO₂ levels during the Eocene were high, at least 3-4 times the pre-industrial value of 280 ppm, but the discrepancies in the tropics suggest that SST estimates from proxies are too low or that the models lack some tropical cooling mechanism that was important at this time.     

Methane emissions from wetlands: biogeochemical,microbial, and modeling perspectives from local to global scales

Understanding the dynamics of methane (CH₄) emissions is of paramount importance because CH₄ has 25 times the global warming potential of carbon dioxide (CO₂) and is currently the second most important anthropogenic greenhouse gas. Wetlands are the single largest natural CH₄ source with median emissions from published studies of 164 Tg yr^?1, which is about a third of total global emissions. We provide a perspective on important new frontiers in obtaining a better understanding of CH₄ dynamics in natural systems, with a focus on wetlands. One of the most exciting recent developments in this field is the attempt to integrate the different methodologies and spatial scales of biogeochemistry, molecular microbiology, and modeling, and thus this is a major focus of this review. Our specific objectives are to provide an up‐to‐date synthesis of estimates of global CH₄ emissions from wetlands and other freshwater aquatic ecosystems, briefly summarize major biogeophysical controls over CH₄ emissions from wetlands, suggest new frontiers in CH₄ biogeochemistry, examine relationships between methanogen community structure and CH₄ dynamics in situ, and to review the current generation of CH₄ models. We highlight throughout some of the most pressing issues concerning global change and feedbacks on CH₄ emissions from natural ecosystems. Major uncertainties in estimating current and future CH₄ emissions from natural ecosystems include the following: (i) A number of important controls over CH₄ production, consumption, and transport have not been, or are inadequately, incorporated into existing CH₄ biogeochemistry models. (ii) Significant errors in regional and global emission estimates are derived from large spatial‐scale extrapolations from highly heterogeneous and often poorly mapped wetland complexes. (iii) The limited number of observations of CH₄ fluxes and their associated environmental variables loosely constrains the parameterization of process‐based biogeochemistry models.     

Mathematical models of the photosynthetic response of tree stands to rising CO2 concentrations and temperatures

Two published models of canopy photosynthesis, MAESTRO and BIOMASS, are simulated to examine the response of tree stands to increasing ambient concentrations of carbon dioxide (C_a) and temperatures. The models employ the same equations to described leaf gas exchange, but differ considerably in the level of detail employed to represent canopy structure and radiation environment. Daily rates of canopy photosynthesis simulated by the two models agree to within 10% across a range of CO₂ concentrations and temperatures. A doubling of C_a leads to modest increases of simulated daily canopy photosynthesis at low temperatures (10% increase at 10°C), but larger increases at higher temperatures (60% increase at 30°C). The temperature and CO₂ dependencies of canopy photosynthesis are interpreted in terms of simulated contributions by quantum-saturated and non-saturated foliage. Simulations are presented for periods ranging from a diurnal cycle to several years. Annual canopy photosynthesis simulated by BIOMASS for trees experiencing no water stress is linearly related to simulated annual absorbed photosynthetically active radiation, with light utilization coefficients for carbon of ?= 1.66 and 2.07g MJ^?1 derived for C_a of 350 and 700 μmol mol^?1, respectively.     

Thermodynamic Potential for the Abiotic Synthesis of Adenine, Cytosine, Guanine, Thymine, Uracil, Ribose, and Deoxyribose in Hydrothermal Systems

The thermodynamic potential for the abiotic synthesis of the five common nucleobases (adenine, cytosine, guanine, thymine, and uracil) and two monosaccharides (ribose and deoxyribose) from formaldehyde and hydrogen cyanide has been quantified under temperature, pressure, and bulk composition conditions that are representative of hydrothermal systems. The activities of the precursor molecules (formaldehyde and hydrogen cyanide) required to evaluate the thermodynamics of biomolecule synthesis were computed using the concentrations of aqueous N₂, CO, CO₂ and H₂ reported in the modern Rainbow hydrothermal system. The concentrations of precursor molecules that can be synthesized are strongly dependent on temperature with larger concentrations prevailing at lower temperatures. Similarly, the thermodynamic drive to synthesize nucleobases, ribose and deoxyribose varies considerably as a function of temperature: all of the biomolecules considered in this study are thermodynamically favored to be synthesized throughout the temperature range from 0°C to between 150°C and 250°C, depending on the biomolecule. Furthermore, activity diagrams have been generated to illustrate that activities in the range of 10⁻²– 10⁻⁶ for nucleobases, ribose and deoxyribose can be in equilibrium with a range of precursor molecule activities at 150°C and 500 bars. The results presented here support the notion that hydrothermal systems could have played a fundamental role in the origin of life, and can be used to plan and constrain experimental investigation of the abiotic synthesis of nucleic-acid related biomolecules.     

Modeled interactive effects of precipitation, temperature, and [CO2] on ecosystem carbon and water dynamics in different climatic zones

Interactive effects of multiple global change factors on ecosystem processes are complex. It is relatively expensive to explore those interactions in manipulative experiments. We conducted a modeling analysis to identify potentially important interactions and to stimulate hypothesis formulation for experimental research. Four models were used to quantify interactive effects of climate warming (T), altered precipitation amounts [doubled (DP) and halved (HP)] and seasonality (SP, moving precipitation in July and August to January and February to create summer drought), and elevated [CO₂] (C) on net primary production (NPP), heterotrophic respiration (R_h), net ecosystem production (NEP), transpiration, and runoff. We examined those responses in seven ecosystems, including forests, grasslands, and heathlands in different climate zones. The modeling analysis showed that none of the three‐way interactions among T, C, and altered precipitation was substantial for either carbon or water processes, nor consistent among the seven ecosystems. However, two‐way interactive effects on NPP, R_h, and NEP were generally positive (i.e. amplification of one factor's effect by the other factor) between T and C or between T and DP. A negative interaction (i.e. depression of one factor's effect by the other factor) occurred for simulated NPP between T and HP. The interactive effects on runoff were positive between T and HP. Four pairs of two‐way interactive effects on plant transpiration were positive and two pairs negative. In addition, wet sites generally had smaller relative changes in NPP, R_h, runoff, and transpiration but larger absolute changes in NEP than dry sites in response to the treatments. The modeling results suggest new hypotheses to be tested in multifactor global change experiments. Likewise, more experimental evidence is needed for the further improvement of ecosystem models in order to adequately simulate complex interactive processes.     

The greenhouse effect: the impacts of carbon dioxide (CO2), ultraviolet-B (UV-B) radiation and ozone (O3) on vegetation (crops)

Man's influence on the greenhouse effect, the heating of the atmosphere due to increasing concentrations of tropospheric trace gases, is of much international concern. Among the climatic variables, elevated levels of carbon dioxide (CO₂), ultraviolet-B (UV-B) radiation and ozone (O₃) are known to have a direct effect on vegetation. Our current knowledge of these effects is mainly based on studies involving single stress mode. Thus, the joint effects of CO₂, UV-B and O₃ on vegetation are poorly understood. Nevertheless, based on the literature analysis of plant response to individual stress factors, it can be concluded that sorghum, pea, bean, potato, oat, lettuce, cucumber, rice and tomato are among the crop species potentially sensitive to the joint effects of the aforementioned three variables. Similar information for tree species is essentially lacking.At least with some climatic variables such as O₃, present modeling efforts of cause-effect relationships have proven to be controversial. While at a regional geographic scale ambient CO₂ concentrations appear to be relatively homogeneous, ambient concentrations of O₃ exhibit significant temporal and spatial variability. Because of the protective action of O₃ against UV-B, similar but inverse temporal and spatial variability is expected in the surface levels of UV-B. Thus, future experimental designs should consider these exposure dynamics and modeling cuase-effect relationships should be directed to stochastic processes.     

Uncertainties in predicting rice yield by current crop models under a wide range of climatic conditions

Predicting rice (Oryza sativa) productivity under future climates is important for global food security. Ecophysiological crop models in combination with climate model outputs are commonly used in yield prediction, but uncertainties associated with crop models remain largely unquantified. We evaluated 13 rice models against multi‐year experimental yield data at four sites with diverse climatic conditions in Asia and examined whether different modeling approaches on major physiological processes attribute to the uncertainties of prediction to field measured yields and to the uncertainties of sensitivity to changes in temperature and CO₂ concentration [CO₂]. We also examined whether a use of an ensemble of crop models can reduce the uncertainties. Individual models did not consistently reproduce both experimental and regional yields well, and uncertainty was larger at the warmest and coolest sites. The variation in yield projections was larger among crop models than variation resulting from 16 global climate model‐based scenarios. However, the mean of predictions of all crop models reproduced experimental data, with an uncertainty of less than 10% of measured yields. Using an ensemble of eight models calibrated only for phenology or five models calibrated in detail resulted in the uncertainty equivalent to that of the measured yield in well‐controlled agronomic field experiments. Sensitivity analysis indicates the necessity to improve the accuracy in predicting both biomass and harvest index in response to increasing [CO₂] and temperature.     

Theoretical Models of Chronotherapy: I. Periodic Perturbations in Oscillating Chemical Reactions

Dynamics resulting from periodic perturbations in oscillating chemical reaction, as theoretical models of chronotherapy techniques are studied. Replication reactions and fluctuations take place in these models giving rise to new species with autocatalitic properties. Oscillating chemical reaction models due to Hanusse (H) with three intermediaries and Bourceanu (E₁, E₂, E₃ ) with four, five and six intermediaries, respectively, are taken for this study and are manipulated numerically using a Gear algorithm. The bifurcation diagrams are obtained applying non-linear dynamic techniques, such as, power spectrum, stroboscopic method, Lyapunov exponents and correlation dimension D 2. The dynamics of Hanusse and Bourceanu perturbed models are very similar. The main dynamic behaviours observed after perturbation are quasiperiodicity, entrainment, some Arnold tongues and situations without chemical meaning. Chaos is not observed. This study contributes to understand theoretical and practical aspects of the chronotherapy techniques.     

Persistence in models of three interacting predator-prey populations

This paper considers a class of deterministic models of three interacting populations with a view towards determining when all of the populations persist. In analytical terms persistence means that liminf_t→∞x(t)> 0 for each population x(t); in geometric terms, that each trajectory of the modeling system of differential equations is eventually bounded away from the coordinate planes. The class of systems considered allows three level food webs, two competing predators feeding on a single prey, or a single predator feeding on two competing prey populations. As a corollary to the last case it is shown that the addition of a predator can lead to persistence of a three population system where, without a predator, the two competing populations on the lower trophic level would have only one survivor. The basic models are of Kolmogorov type, and the results improve several previous theorems on persistence.     

Effects of inorganic salts on the structural heterogeneity of serum albumin solutions

The formation of protein clusters or a protein-rich phase in undersaturated solutions of biopolymers is considered theoretically on the basis of phase diagrams of a water-protein-salt system. Concentrated (50–200 mg/ml of protein) water-salt solutions of human serum albumin molecules modified by a maleimide spin-label have been studied experimentally using the ESR technique to characterize the significant general features of the system behaviour suggested by the model phase diagrams. The inorganic ion content (NaCl, KSCN, MgCl₂, and CaCl₂) was varied in the range of 10⁻³–4 M. Salt-induced changes in different experimental ESR spin-label parameters based on relations between spectral line widths and amplitudes were determined and compared with the same parameters in salt-free solution. The data on dipole-dipole interactions of spin labels obtained at 77 K and on spin exchange at normal temperatures are indicative of local protein concentration inhomogeneities. The results have been described in terms of salt-induced dissociation of stabilized supramolecular structures in protein solution–protein clusters, liquid-liquid phase transition between the hydration water of clusters and that of individual proteins, and a rise in surface tension which results in protein stabilization. Received: 22 December 1998 / Revised version: 23 August 1999 / Accepted: 24 September 1999     

In silico optimization of pharmacokinetic properties and receptor binding affinity simultaneously: a ‘parallel progression approach to drug design’ applied to β-blockers

The present work exploits the potential of in silico approaches for minimizing attrition of leads in the later stages of drug development. We propose a theoretical approach, wherein ‘parallel’ information is generated to simultaneously optimize the pharmacokinetics (PK) and pharmacodynamics (PD) of lead candidates. β-blockers, though in use for many years, have suboptimal PKs; hence are an ideal test series for the ‘parallel progression approach’. This approach utilizes molecular modeling tools viz. hologram quantitative structure activity relationships, homology modeling, docking, predictive metabolism, and toxicity models. Validated models have been developed for PK parameters such as volume of distribution (log V_d) and clearance (log Cl), which together influence the half-life (t_1/2) of a drug. Simultaneously, models for PD in terms of inhibition constant pK_i have been developed. Thus, PK and PD properties of β-blockers were concurrently analyzed and after iterative cycling, modifications were proposed that lead to compounds with optimized PK and PD. We report some of the resultant re-engineered β-blockers with improved half-lives and pK_i values comparable with marketed β-blockers. These were further analyzed by the docking studies to evaluate their binding poses. Finally, metabolic and toxicological assessment of these molecules was done through in silico methods. The strategy proposed herein has potential universal applicability, and can be used in any drug discovery scenario; provided that the data used is consistent in terms of experimental conditions, endpoints, and methods employed. Thus the ‘parallel progression approach’ helps to simultaneously fine-tune various properties of the drug and would be an invaluable tool during the drug development process.     

Divergent response of seasonally dry tropical vegetation to climatic variations in dry and wet seasons

Interannual variations of photosynthesis in tropical seasonally dry vegetation are one of the dominant drivers to interannual variations of atmospheric CO₂ growth rate. Yet, the seasonal differences in the response of photosynthesis to climate variations in these ecosystems remain poorly understood. Here using Normalized Difference Vegetation Index (NDVI), we explored the response of photosynthesis of seasonally dry tropical vegetation to climatic variations in the dry and the wet seasons during the past three decades. We found significant (p < 0.01) differences between dry and wet seasons in the interannual response of photosynthesis to temperature (γ^int) and to precipitation (δ^int). γ^int is ~1% °C^?1 more negative and δ^int is ~8% 100 mm^?1 more positive in the dry season than in the wet season. Further analyses show that the seasonal difference in γ^int can be explained by background moisture and temperature conditions. Positive γ^int occurred in wet season where mean temperature is lower than 27°C and precipitation is at least 60 mm larger than potential evapotranspiration. Two widely used Gross Primary Productivity (GPP) estimates (empirical modeling by machine‐learning algorithm applied to flux tower measurements, and nine process‐based carbon cycle models) were examined for the GPP–climate relationship over wet and dry seasons. The GPP derived from empirical modeling can partly reproduce the divergence of γ^int, while most process models cannot. The overestimate by process models on negative impacts by warmer temperature during the wet season highlights the shortcomings of current carbon cycle models in representing interactive impacts of temperature and moisture on photosynthesis. Improving representations on soil water uptake, leaf temperature, nitrogen cycling, and soil moisture may help improve modeling skills in reproducing seasonal differences of photosynthesis–climate relationship and thus the projection for impacts of climate change on tropical carbon cycle.     

Comparison of Models for IP3 Receptor Kinetics Using Stochastic Simulations

Inositol 1,4,5-trisphosphate receptor (IP₃R) is a ubiquitous intracellular calcium (Ca²⁺) channel which has a major role in controlling Ca²⁺ levels in neurons. A variety of computational models have been developed to describe the kinetic function of IP₃R under different conditions. In the field of computational neuroscience, it is of great interest to apply the existing models of IP₃R when modeling local Ca²⁺ transients in dendrites or overall Ca²⁺ dynamics in large neuronal models. The goal of this study was to evaluate existing IP₃R models, based on electrophysiological data. This was done in order to be able to suggest suitable models for neuronal modeling. Altogether four models (Othmer and Tang, 1993; Dawson et al., 2003; Fraiman and Dawson, 2004; Doi et al., 2005) were selected for a more detailed comparison. The selection was based on the computational efficiency of the models and the type of experimental data that was used in developing the model. The kinetics of all four models were simulated by stochastic means, using the simulation software STEPS, which implements the Gillespie stochastic simulation algorithm. The results show major differences in the statistical properties of model functionality. Of the four compared models, the one by Fraiman and Dawson (2004) proved most satisfactory in producing the specific features of experimental findings reported in literature. To our knowledge, the present study is the first detailed evaluation of IP₃R models using stochastic simulation methods, thus providing an important setting for constructing a new, realistic model of IP₃R channel kinetics for compartmental modeling of neuronal functions. We conclude that the kinetics of IP₃R with different concentrations of Ca²⁺ and IP₃ should be more carefully addressed when new models for IP₃R are developed.     

Projected Effects of Climate Change on Patterns of Vertebrate and Tree Species Richness in the Conterminous United States

General circulation models (GCM) predict that increasing levels of atmospheric carbon dioxide (CO₂) and other greenhouse gases will lead to dramatic changes in climate. It is known that the spatial variability of species richness over continental spatial scales is strongly correlated with contemporary climate. Assuming that this relationship between species richness and climate persists under conditions of increased CO₂, what changes could we expect to occur in terms of species richness? To address this question, I used observed relationships between contemporary richness and climate, coupled with climate projections from five GCM, to project these future changes. These models predict that the richness of vertebrate ectotherms will increase over most of the conterminous United States. Mammal and bird richness are predicted to decrease in much of the southern US and to increase in cool, mountainous areas. Woody plant richness is likely to increase throughout the North and West and to decrease in the southwestern deserts. These projections represent changes that are likely to occur over long time scales (millennia); short-term changes are expected to be mainly negative.     

Forest water use and water use efficiency at elevated CO2: a model‐data intercomparison at two contrasting temperate forest FACE sites

Predicted responses of transpiration to elevated atmospheric CO₂ concentration (eCO₂) are highly variable amongst process‐based models. To better understand and constrain this variability amongst models, we conducted an intercomparison of 11 ecosystem models applied to data from two forest free‐air CO₂ enrichment (FACE) experiments at Duke University and Oak Ridge National Laboratory. We analysed model structures to identify the key underlying assumptions causing differences in model predictions of transpiration and canopy water use efficiency. We then compared the models against data to identify model assumptions that are incorrect or are large sources of uncertainty. We found that model‐to‐model and model‐to‐observations differences resulted from four key sets of assumptions, namely (i) the nature of the stomatal response to elevated CO₂ (coupling between photosynthesis and stomata was supported by the data); (ii) the roles of the leaf and atmospheric boundary layer (models which assumed multiple conductance terms in series predicted more decoupled fluxes than observed at the broadleaf site); (iii) the treatment of canopy interception (large intermodel variability, 2–15%); and (iv) the impact of soil moisture stress (process uncertainty in how models limit carbon and water fluxes during moisture stress). Overall, model predictions of the CO₂ effect on WUE were reasonable (intermodel μ = approximately 28% ± 10%) compared to the observations (μ = approximately 30% ± 13%) at the well‐coupled coniferous site (Duke), but poor (intermodel μ = approximately 24% ± 6%; observations μ = approximately 38% ± 7%) at the broadleaf site (Oak Ridge). The study yields a framework for analysing and interpreting model predictions of transpiration responses to eCO₂, and highlights key improvements to these types of models.     

Contribution of genetic effects to genetic variance components with epistasis and linkage disequilibrium

Background

Cockerham genetic models are commonly used in quantitative trait loci (QTL) analysis with a special feature of partitioning genotypic variances into various genetic variance components, while the F_∞ genetic models are widely used in genetic association studies. Over years, there have been some confusion about the relationship between these two type of models. A link between the additive, dominance and epistatic effects in an F_∞ model and the additive, dominance and epistatic variance components in a Cockerham model has not been well established, especially when there are multiple QTL in presence of epistasis and linkage disequilibrium (LD).

Results

In this paper, we further explore the differences and links between the F_∞ and Cockerham models. First, we show that the Cockerham type models are allelic based models with a special modification to correct a confounding problem. Several important moment functions, which are useful for partition of variance components in Cockerham models, are also derived. Next, we discuss properties of the F_∞ models in partition of genotypic variances. Its difference from that of the Cockerham models is addressed. Finally, for a two-locus biallelic QTL model with epistasis and LD between the loci, we present detailed formulas for calculation of the genetic variance components in terms of the additive, dominant and epistatic effects in an F_∞ model. A new way of linking the Cockerham and F_∞ model parameters through their coding variables of genotypes is also proposed, which is especially useful when reduced F_∞ models are applied.

Conclusion

The Cockerham type models are allele-based models with a focus on partition of genotypic variances into various genetic variance components, which are contributed by allelic effects and their interactions. By contrast, the F_∞ regression models are genotype-based models focusing on modeling and testing of within-locus genotypic effects and locus-by-locus genotypic interactions. When there is no need to distinguish the paternal and maternal allelic effects, these two types of models are transferable. Transformation between an F_∞ model's parameters and its corresponding Cockerham model's parameters can be established through a relationship between their coding variables of genotypes. Genetic variance components in terms of the additive, dominance and epistatic genetic effects in an F_∞ model can then be calculated by translating formulas derived for the Cockerham models.