An Application of Dimensional Analysis in Cultural Anthropology

Lawlike generalizations can be developed only for variables exhibiting strict constancy in concept formulation. By considering anthropology a space and its basic concepts as dimensions and borrowing from ecology, cultural materialism, and social, political, and cultural anthropology, we develop a 14-dimensional space defined by 42 scaled variables. Dimensional analysis performed yields 14 equations and 28 dimensionless expressions that satisfy them. Additionally, we derive a set of 28 ethnological expressions. Entering scaled data from five societies, we solve for each. Twenty-one expressions yield significant order-ings. Examination of their performance confirms that they function holistically.     

Continuous measurements of a binding reaction using a capacitive biosensor

A capacitive biosensor with polyclonal antibodies raised against human serum albumin (HSA) immobilized on a gold transducer has been developed for continuous measurement of HSA in the muM-range. A mathematical model has been refined to describe integral HSA-binding curves assuming that (i) binding is essentially irreversible under the conditions used, (ii) the signal is scaled as the number of non-occupied binding sites and (iii) the rate of disappearance of available binding sites is scaled as the number of available binding sites and analyte concentration in solution. Deconvolution of the curves using the mathematical model indicates clearly that it is possible to retrieve concentration profiles (isocratic, linearly or exponentially increasing gradients) of the analyte in the continuous sample flow from the normalized integral binding (NIB) curves. The data presented constitutes the theoretical background and the first step towards the development of an analytical system allowing on-line detection of the concentration profile of the analyte from NIB-curves. Since the system can be used for extended time periods between regeneration steps, a low frequency of regeneration steps can be expected.     

A simple expression for quantifying bacterial chemotaxis using capillary assay data: application to the analysis of enhanced chemotactic responses from growth-limited cultures.

An individual cell-based mathematical model of Rivero et al. provides a framework for determining values of the chemotactic sensitivity coefficient chi 0, an intrinsic cell population parameter that characterizes the chemotactic response of bacterial populations. This coefficient can theoretically relate the swimming behavior of individual cells to the resulting migration of a bacterial population. When this model is applied to the commonly used capillary assay, an approximate solution can be obtained for a particular range of chemotactic strengths yielding a very simple analytical expression for estimating the value of chi 0, [formula: see text] from measurements of cell accumulation in the capillary, N, when attractant uptake is negligible. A0 and A infinity are the dimensionless attractant concentrations initially present at the mouth of the capillary and far into the capillary, respectively, which are scaled by Kd, the effective dissociation constant for receptor-attractant binding. D is the attractant diffusivity, and mu is the cell random motility coefficient. NRM is the cell accumulation in the capillary in the absence of an attractant gradient, from which mu can be determined independently as mu = (pi/4t)(NRM/pi r2bc)2, with r the capillary tube radius and bc the bacterial density initially in the chamber. When attractant uptake is significant, a slightly more involved procedure requiring a simple numerical integration becomes necessary. As an example, we apply this approach to quantitatively characterize, in terms of the chemotactic sensitivity coefficient chi 0, data from Terracciano indicating enhanced chemotactic responses of Escherichia coli to galactose when cultured under growth-limiting galactose levels in a chemostat.     

Characterization,scaling, and partial representation of diffuse and discrete input junctions to CA3 hippocampus

This paper applies a general mathematical system for characterizing and scaling functional connectivity and information flow across the diffuse (EC) and discrete (DG) input junctions to the CA3 hippocampus. Both gross connectivity and coordinated multiunit informational firing patterns are quantitatively characterized in terms of 32 defining parameters interrelated by 17 equations, and then scaled down according to rules for uniformly proportional scaling and for partial representation. The diffuse EC-CA3 junction is shown to be uniformly scalable with realistic representation of both essential spatiotemporal cooperativity and coordinated firing patterns down to populations of a few hundred neurons. Scaling of the discrete DG-CA3 junction can be effected with a two-step process, which necessarily deviates from uniform proportionality but nonetheless produces a valuable and readily interpretable reduced model, also utilizing a few hundred neurons in the receiving population. Partial representation produces a reduced model of only a portion of the full network where each model neuron corresponds directly to a biological neuron. The mathematical analysis illustrated here shows that although omissions and distortions are inescapable in such an application, satisfactorily complete and accurate models the size of pattern modules are possible. Finally, the mathematical characterization of these junctions generates a theory which sees the DG as a definer of the fine structure of embedded traces in the hippocampus and entire coordinated patterns of sequences of 14-cell links in CA3 as triggered by the firing of sequences of individual neurons in DG.     

A cometabilic kinetics model incorporating enzyme inhbition, inactivation, and recovery: I. Model development, analysis, and testing

Cometabolic biodegradation prcesses are important for bioremediation of hazardous waste sites. However, these proceeses are not well understood and have not been modeled thoroughly. Traditional Michaelis-Menten kinetics models often are used, but toxic effects and bacterial responses to toxicity may cause changes in enzyme levels, rendering such models inappropriate. In this article, a conceptual and mathematical model of cometabolic enzyme kinetics i described. Model derivation is based on enzyme/growth-substrate/nongrowth-substrate interaction and incorporates enzyme inhibition (caused by the presence of a cometabolic compound), inactivation (resulting from toxicity of a cometabolic product), and recovery (associated with bacterial synthesis of new enbzyme in response to inactivation). The mathematical model consists of a system of two, nonlinear ordinary differential equations that can be solved implicitly using numerical methods, providing estimates of model parameters. Model analysis shows that growth substraate adn nongrowth substrate oxidation rates are related by a dimensionless constant. Reliability of tehy model solution prcedure is verifiedl by abnalyzing data ses, containing random error, from simulated experimentss with trichhloroethyylene (TCE) degradation by ammonia-oxidizing bacterialunder various conditions. Estimation of the recovery rate contant is deterimined to be sensitive to intial TCE concentration. Model assumptions are evaluated in a companion article using data from TCE degradation experiments with amoniaoxidizing bacteria. (c) 1995 John Wiley & Sons, Inc.     

Design and construction of a preparative-scale dynamic field gradient focusing apparatus

A linear model is used to show that dynamic field gradient focusing (DFGF) can be scaled to preparative capacity, approximately O (10 mgs). This paper explains how the preparative-scale DFGF apparatus was designed and fabricated. Scaled-down experiments and mathematical modeling guided material selection and design changes during construction to increase the probability that the prototype preparative-scale DFGF apparatus would perform as intended. The finished prototype successfully focused bovine hemoglobin from an initial concentration of 6.82 to 15 mg/mL and allowed for 86% recovery of injected protein.     

Optimal biocatalyst loading in a fixed bed

The optimal distribution of biocatalyst in a fixed bed operating at steady state was determined to minimize the length of the bed for a fixed conversion of 95%. The distribution in terms of the biocatalyst loading on an inert support depends upon the axial distance from the bed entrance (continuous solution) as well as a set of dimensionless parameters that reflect the bed geometry, the bulk flow, reaction kinetics and diffusional characteristics. A mathematical model of the system with the following features was used to obtain the results: (1) convective mass transfer and dispersion in the bulk phase; (2) mass transfer from the bulk phase to the surface of the catalyst particle; and (3) simultaneous diffusion and chemical reaction in the catalyst particle with Michaelis–Menton kinetics and a reliable diffusion model (Zhao and DeLancey in Biotechnol Bioeng 64:434–441, 1999, 2000). The solution to the mathematical model was obtained with Mathematica utilizing the Runge Kutta 4–5 method. The dimensionless length resulting from the continuous solution was compared with the optimal length restricted to a uniform or constant cell loading across the entire bed. The maximum difference in the dimensionless length between the continuous and uniform solutions was determined to be 6.5%. The model was applied to published conversion data for the continuous production of ethanol that included cell loading (Taylor et al. in Biotechnol Prog 15:740–751, 2002). The data indicated a minimum production cost at a catalyst loading within 10% of the optimum predicted by the mathematical model. The production rate versus cell loading in most cases displayed a sufficiently broad optimum that the same (non-optimal) rate could be obtained at a significantly smaller loading such as a rate at 80% loading being equal to the rate at 20% loading. These results are particularly important because of the renewed interest in ethanol production (Novozymes and BBI International, Fuel ethanol: a technological evolution, 2004).     

Effects of inert volume-excluding macromolecules on protein fiber formation. I. Equilibrium models

The equilibrium Oosawa-Asakura model for nucleated assembly of rod-like protein fibers is recast in terms of dimensionless (scaled) quantities. The model is then generalized to treat arbitrarily large deviations from thermodynamic ideality arising from high fractional volume occupancy by an inert protein or polymer. Each state of association of the self-associating protein is modeled as an equivalent rigid convex particle (sphere or spherocylinder) and the crowding species is modeled either as an equivalent sphere or cylindrical rod. The resulting conservation of mass relation is readily solved to yield the fractional abundance of monomer, from which the entire equilibrium distribution of oligomeric species can be calculated, either directly or through the use of an additional scaling relationship. Results indicating the potential effect of volume occupancy on the equilibrium solubility of the self-associating protein and upon the equilibrium distribution of polymer size are presented. It is found that the fractional (logarithmic) change in both solubility and in the breadth of the polymer size distribution scale almost linearly with the fractional (logarithmic) change in the thermodynamic activity of monomer.     

Gas absorption in pulmonary airways at low Peclet number

A mathematical model is presented that investigates the mass transport of a diffusible and soluble gas contaminant through a liquid-lined tube when the Peclet number is small. The transport is determined by four dimensionless parameters: lambda, the tube aspect ratio; d, the relative difference in end concentrations; gamma, the radial transport coefficient; and Pe, the Peclet number. The problem is formulated for arbitrary gamma, but in the case of ozone and nitrous oxides the value of gamma is small. An asymptotic analysis for Pe much less than 1 and gamma much less than 1 is presented which yields the concentration field and transport characteristics we seek. It also provides a low Peclet number analysis for the conjugate problem of mass and heat transfer that is not currently available in the literature. The application to transport in the small airways of the lung is discussed, particularly the radial absorption differences in inspiratory and expiratory flow. Depending on the relative sizes of gamma and Pe, fractional uptake decreases with increasing Pe during inspiration but can increase during expiration.     

Steric hindrance effects on surface reactions: applications to BIAcore

Because surface-volume reactions occur in many biological and industrial processes, understanding the rate of such reactions is important. The BIAcore surface plasmon resonance (SPR) biosensor for measuring rate constants has such a geometry. Though several models of the BIAcore have been presented, few take into account that large ligand molecules can block multiple receptor sites, thus skewing the sensogram data. In this paper some general mathematical principles are stated for handling this phenomenon, and a surface-reaction model is presented explicitly. An integro-partial differential equation results, which can be simplified greatly using perturbation techniques, yielding linear and nonlinear integrodifferential equations. Explicit and asymptotic solutions are constructed for cases motivated by experimental design. The general analysis can provide insight into surface-volume reactions occurring in various contexts. In particular, the steric hindrance effect can be quantified with a single dimensionless parameter. This work was supported in part by NIGMS Grant 1R01GM067244-01.     

Approximation of continuous fermentation by semicontinuous cultures

Semicontinuous fermentations, in which a fraction of a culture is replaced with fresh media at regular intervals, have been previously used as a means of approximating continuous growth. In most cases deviations from continuous operation were erroneously estimated using Fencl's model, which is only valid when the specific growth rate is independent of the substrate concentration. An approach to modeling Semicontinuous growth that incorporates the same kinetics followed in batch and continuous growth was developed and tested for Monod's expression for the specific growth rate. A dimensionless form of the model was used to simulate Semicontinuous fermentations for comparison to continuous growth. Differences between Semicontinuous and continuous growth were found to depend on three dimensionless variables: feed concentration, replacement rate, and time between replacements. For given values of the dimensionless feed concentration and time between replacements, a range of dimensionless replacement rates can be determined over which semi-continuous cultures are approximately continuous.     

Analysis of miRNA regulation suggests an explanation for 'unexpected' increase in target protein levels

MicroRNA (miRNA) has been mostly associated with decrease in target protein expression levels. Recently, 'unexpected' observations of increase in target protein expression attributed to microRNA regulation have been reported. We formulate a comprehensive model for regulation by miRNA that includes both reversible mRNA-miRNA binding and selective return of RNA. We use this mathematical model incorporating multiple individual steps in the regulation process to study the simultaneous effects of these steps on the target protein level. We show that four dimensionless numbers obtained from 12 rate constants are sufficient to define the relative change in steady state target protein levels. We quantify the range of these numbers for which such pleiotropic increase in protein levels is possible, and interpret the experimental findings in the framework of our model such that the results are no longer unexpected. Finally, we show through stochastic simulation that the nature of the target protein distribution remains unchanged and the relative steady state noise levels are also completely defined by the values of these dimensionless numbers, irrespective of the individual reaction rate constants.     

A Design Principle of Group-level Decision Making in Cell Populations

Populations of cells often switch states as a group to cope with environmental changes such as nutrient availability and cell density. Although the gene circuits that underlie the switches are well understood at the level of single cells, the ways in which such circuits work in concert among many cells to support group-level switches are not fully explored. Experimental studies of microbial quorum sensing show that group-level changes in cellular states occur in either a graded or an all-or-none fashion. Here, we show through numerical simulations and mathematical analysis that these behaviors generally originate from two distinct forms of bistability. The choice of bistability is uniquely determined by a dimensionless parameter that compares the synthesis and the transport of the inducing molecules. The role of the parameter is universal, such that it not only applies to the autoinducing circuits typically found in bacteria but also to the more complex gene circuits involved in transmembrane receptor signaling. Furthermore, in gene circuits with negative feedback, the same dimensionless parameter determines the coherence of group-level transitions from quiescence to a rhythmic state. The set of biochemical parameters in bacterial quorum-sensing circuits appear to be tuned so that the cells can use either type of transition. The design principle identified here serves as the basis for the analysis and control of cellular collective decision making.     

Asymptotic and dimensionless analysis of the response of living tissue to surgical pulsed CO2 lasers

The mathematical model of the interaction between the radiation of pulsed CO2 lasers and tissue was revisited. Asymptotic calculations were employed to determine upper and lower bounds for the evaporated volume and crater depth. Dimensionless time variables for conduction, beam attenuation by tissue vapors and for damage were introduced. Optimal exposure parameters were identified through a dimensionless analysis.     

Modeling and testing treated tumor growth using cubic smoothing splines

Human tumor xenograft models are often used in preclinical study to evaluate the therapeutic efficacy of a certain compound or a combination of certain compounds. In a typical human tumor xenograft model, human carcinoma cells are implanted to subjects such as severe combined immunodeficient (SCID) mice. Treatment with test compounds is initiated after tumor nodule has appeared, and continued for a certain time period. Tumor volumes are measured over the duration of the experiment. It is well known that untreated tumor growth may follow certain patterns, which can be described by certain mathematical models. However, the growth patterns of the treated tumors with multiple treatment episodes are quite complex, and the usage of parametric models is limited. We propose using cubic smoothing splines to describe tumor growth for each treatment group and for each subject, respectively. The proposed smoothing splines are quite flexible in modeling different growth patterns. In addition, using this procedure, we can obtain tumor growth and growth rate over time for each treatment group and for each subject, and examine whether tumor growth follows certain growth pattern. To examine the overall treatment effect and group differences, the scaled chi-squared test statistics based on the fitted group-level growth curves are proposed. A case study is provided to illustrate the application of this method, and simulations are carried out to examine the performances of the scaled chi-squared tests.     

A model to predict canine pelvic limb musculoskeletal geometry

We developed a model to predict the three-dimensional canine pelvic limb muscular geometry (i.e., all muscle moment arms during any instant in gait). Forty-one muscle origins and insertions, as well as external landmarks (to obtain anthropometric dimensions) were marked on both pelvic limbs of five dogs and digitized on biplanar radiographs. Reference frames in the pelvis, femur, and tibia established the three-dimensional coordinates of each origin, insertion, and landmark. A set of dimensionless 'scaled coordinates' was created by dividing the actual origin and insertion coordinates by selected anthropometric dimensions of each animal. Combining scaled coordinates from all ten limbs produced an averaged 'template' of scaled coordinates. To provide limited validation of the scaling procedure, we measured the anthropometric dimensions between externally palpable landmarks of two additional pelvic limbs. The anthropometric dimensions were multiplied by the averaged template coordinates to calculate two new sets of hindlimb muscle coordinates within the three bony reference frames. The two limbs then were dissected, muscle endpoints were marked, and biplanar radiographs of each of the limb segments were digitized. The actual coordinates so obtained were similar to those predicted by the template and anthropometric measures.