A Simple Phenomenological Theory of Tissue Growth

A simple phenomenological framework for modeling growth of living tissues is proposed. Growth is defined as a change of mass and configuration of the tissue. Tissue is considered as an open system where mass conservation is violated and the full-scale mass balance is applied. A possible structure of constitutive equations is discussed with reference tosimple growing materials. 'Thermoelastic' formulation of the simple growing material is specified. Within this framework traction free growth of cylindrical and spherical bodies is examined. It is shown that the theory accommodates the case where stresses are not generated in uniform volumetric growth. It is also found that surface growth corresponds to aboundary layersolution of the governing equations. This finding proves the ability of continuum mechanics to describe surface growth. The latter is contrary to the usual use of purely kinematical theories, which do not involve balance and constitutive equations, for treating surface growth.     

The role of mass balance equations in growth mechanics illustrated in surface and volume dissolutions

Growth mechanics problems require the solution of mass balance equations that include supply terms and account for mass exchanges among constituents of a mixture. Though growth may often be accompanied by a variety of concomitant phenomena that increase modeling complexity, such as solid matrix deformation, evolving traction-free configurations, cell division, and active cell contraction, it is important to distinguish these accompanying phenomena from the fundamental growth process that consists of deposition or removal of mass from the solid matrix. Therefore, the objective of this study is to present a canonical problem of growth, namely, dissolution of a rigid solid matrix in a solvent. This problem illustrates a case of negative growth (loss of mass) of the solid in a mixture framework that includes three species, a solid, a solvent, and a solute, where the solute is the product of the solid dissolution. By analyzing both volumetric and surface dissolutions, the two fundamental modes of growth are investigated within the unified framework of mixture theory.     

A reliable simulator for dynamic flux balance analysis

Dynamic flux balance analysis (DFBA) provides a platform for detailed design, control and optimization of biochemical process technologies. It is a promising modeling framework that combines genome‐scale metabolic network analysis with dynamic simulation of the extracellular environment. Dynamic flux balance analysis assumes that the intracellular species concentrations are in equilibrium with the extracellular environment. The resulting underdetermined stoichiometric model is solved under the assumption of a biochemical objective such as growth rate maximization. The model of the metabolism is coupled with the dynamic mass balance equations of the extracellular environment via expressions for the rates of substrate uptake and product excretion, which imposes additional constraints on the linear program (LP) defined by growth rate maximization of the metabolism. The linear program is embedded into the dynamic model of the bioreactor, and together with the additional constraints this provides an accurate model of the substrate consumption, product secretion, and biomass production during operation. A DFBA model consists of a system of ordinary differential equations for which the evaluation of the right‐hand side requires not only function evaluations, but also the solution of one or more linear programs. The numerical tool presented here accurately and efficiently simulates large‐scale dynamic flux balance models. The main advantages that this approach has over existing implementation are that the integration scheme has a variable step size, that the linear program only has to be solved when qualitative changes in the optimal flux distribution of the metabolic network occur, and that it can reliably simulate behavior near the boundary of the domain where the model is defined. This is illustrated through large‐scale examples taken from the literature. Biotechnol. Bioeng. 2013; 110: 792–802. © 2012 Wiley Periodicals, Inc.     

Acclimation in dynamic models based on structural relationships

1. A number of recent tree growth models have been based on the assumption that tree structure follows certain empirical rules and carbon allocation is performed so as to maintain these rules when growth and senescence occur simultaneously. This paper introduces a method of combining structural rules with adaptive regulation in a carbon balance model.
2. The method is based on regulating gross growth but situations are analysed where increased senescence of tissue functions as an alternative means of re-establishing the balance.
3. The model does not aim to be mechanistic but merely to describe the process of regulation, or acclimation to changing environments and situations, as constrained by the carbon balance. The method incorporates parameters that describe the rate of returning to the balance after disturbance or after a change in the goal, which can be determined empirically.
4. Two models are given as examples of the use of the method, and the requirements and limitations of control in a carbon balance framework are discussed.
5. The method is best applied as a technical tool to describe variations and disturbance in balanced growth models when the variations are present but not very large. In addition, it can be used as a theoretical framework for the analysis of regulation as constrained by the conservation of mass.     

Simple model for the energetics of growth on substrates with different degrees of reduction

A simple model is developed for the energy transformation in growing microbial systems. The model is based on a linear equation for ATP consumption in the processes of growth and maintenance. A combination of this equation with macroscopic balances for the various components, the systems exchanges with the environment, and application of the concepts of the elementary balance allow the derivation of linear equations for the exchange of substrate, oxygen, and carbon dioxide with the environment. For growth on one sole carbon and energy source the model allows the definition of a critical substrate yield are expected and below which is decreasing substrate yield and energy supply growth limitation are expected. This restriction can be interpreted in a variety of other ways. It supplies a rationale for non-energy-production-coupled transfer of hydrogen to oxygen or wasteful expenditure of ATP in growth on highly reduced substrates. It also allows the formulation of a limit to the maximum yield on oxygen that can never be exceeded in growth on highly reduced substrates.     

The role of stress in the growth of a multicell spheroid

Rather recent experimental results demonstrate the non–negligible role of mechanical stress in the growth of a multicell spheroid. In this paper we discuss a theoretical framework for volumetric growth suitable for modeling the growth of soft tissues exhibiting the properties of a solid. After a proper kinematic decomposition, balance equations for mass, momentum and energy are discussed together with constitutive relationships. The mathematical model is then applied to avascular tumor growth. We show by numerical simulation that, under assumption of spherical symmetry, the mathematical model is able to reproduce the experimental data with a satisfying qualitative agreement.     

Pattern formation in reaction-diffusion models with nonuniform domain growth

Recent examples of biological pattern formation where a pattern changes qualitatively as the underlying domain grows have given rise to renewed interest in the reaction-diffusion (Turing) model for pattern formation. Several authors have now reported studies showing that with the addition of domain growth the Turing model can generate sequences of patterns consistent with experimental observations. These studies demonstrate the tendency for the symmetrical splitting or insertion of concentration peaks in response to domain growth. This process has also been suggested as a mechanism for reliable pattern selection. However, thus far authors have only considered the restricted case where growth is uniform throughout the domain. In this paper we generalize our recent results for reaction-diffusion pattern formation on growing domains to consider the effects of spatially nonuniform growth. The purpose is twofold: firstly to demonstrate that the addition of weak spatial heterogeneity does not significantly alter pattern selection from the uniform case, but secondly that sufficiently strong nonuniformity, for example where only a restricted part of the domain is growing, can give rise to sequences of patterns not seen for the uniform case, giving a further mechanism for controlling pattern selection. A framework for modelling is presented in which domain expansion and boundary (apical) growth are unified in a consistent manner. The results have implications for all reaction-diffusion type models subject to underlying domain growth.     

Application of macroscopic principles to microbial metabolism

General expressions for mass, elemental, energy, and entropy balances are derived and applied to microbial growth and product formation. The state of the art of the application of elemental balances to aerobic and heterotrophic growth is reviewed and extended somewhat to include the majority of the cases commonly encountered in biotechnology. The degree of reduction concept is extended to include nitrogen sources other than ammonia. The relationship between a number of accepted measures for the comparison of substrate yields is investigated. The theory is illustrated using a generalized correlation for oxygen yield data. The stoichiometry of anaerobic product formation is briefly treated, a limit to the maximum carbon conservation in product is derived, using the concept of elemental balance. In the treatment of growth energetics the correct statement of the second law of thermodynamics for growing organisms is emphasized. For aerobic heterotrophic growth the concept of thermodynamic efficiency is used to formulate a limit the substrate yield can never surpass. It is combined with a limit due to the fact that the maximum carbon conservation in biomass can obviously never surpass unity. It is shown that growth on substrates of a low degree of reduction is energy limited, for substrates of a high degree of reduction carbon limitation takes over. Based on a literature review concerning yield data some semiempirical notions useful for a preliminary evolution of aerobic heterotrophic growth are developed. The thermodynamic efficiency definition is completed by two other efficiency measures, which allow derivation of simple equations for oxygen consumption and heat production. The range of validity of the constancy of the rate of heat production to the rate of oxygen consumption is analyzed using these efficiency measures. The energetic of anaerobic growth are treated—it is shown that an approximate analysis in terms of an enthalpy balance is not valid for this case, the evaluation of the efficiency of growth has to be based on Gibbs free energy changes. A preliminary analysis shows the existence of regularities concerning the free energy conservation on anaerobic growth. The treatment is extended to include the effect of growth rate by the introduction of a linear relationship for substrate consumption. Aerobic and anaerobic growth are discussed using this relationship. A correlation useful in judging the potentialities for improvement in anaerobic product formation processes is derived. Finally the relevance of macroscopic principles to the modeling of bioengineering systems is discussed.     

Biomechanical models of hyphal growth in actinomycetes

The tip growth of filamentary actinomycetes is investigated within the framework of large deformation membrane theory in which the cell wall is represented as a growing elastic membrane with geometry-dependent elastic properties. The model exhibits realistic hyphal shapes and indicates a self-similar tip growth mechanism consistent with that observed in experiments. It also demonstrates a simple mechanism for hyphal swelling and beading that is observed in the presence of a lysing agent.     

Evolution of linkage disequilibrium of the founders in exponentially growing populations

Evolution of linkage disequilibrium of the founders in exponentially growing populations was studied using a time-inhomogeneous It? process model. The model is an extension of the diffusion approximation of the Wright-Fisher model. As a measure of linkage disequilibrium, the squared standard linkage deviation, which is defined by a ratio of the moments, was considered. A system of ordinary differential equations that these moments obey was obtained. This system can be solved numerically. By simulations, it was shown that the squared standard linkage deviation gives a good approximation of the expectation of the squared correlation coefficient of gamete frequencies. In addition, a perturbative solution was obtained when the growth rate is not large. By using the perturbation, an asymptotic formula for the squared standard linkage deviation after a large number of generations was obtained. According to the formula, the squared standard linkage deviation tends to be 1/(4Nc), where N is the current size of the population and c is the recombination fraction between two loci. It is dependent on neither the initial effective size, the growth rate, nor the mutation rate. In exponentially growing populations, linkage disequilibrium will be asymptotically the same as that in a constant size population, the effective size of which is the current effective size.     

Maintenance of butterfly populations with all-female broods under recurrent extinction and recolonization

A theory is considered which can explain the apparent persistence of butterfly populations containing a special type of females producing all-female broods. Under normal circumstances it would be expected that these populations should become extinct, but this does not happen in nature. The theory is based on the possibility of a balance between populations dying out and habitats being recolonized. It is shown that a stable equilibrium can exist in a simple model leading to a set of differential equations and also in a more realistic simulation model. The first approach may have a bearing on certain prey-predator systems as well.     

Urban Water Mass Balance Analysis

Planning for “water‐sensitive” cities has become a priority for sustainable urban development in Australia. There has been little quantification of the term, however. Furthermore, the water balance of most cities is not well known. Following prolonged drought, there has also been a growing need to make Australian cities more water self‐reliant: to source water from within. This article formalizes a systematic mass‐balance framework to quantify all anthropogenic and natural flows into and out of the urban environment. Quantitative performance indicators are derived, including (1) degree of system centralization; (2) overall balance; potential of (3) rainfall, (4) stormwater, and (5) wastewater to offset current demand; and (6) water cycle rate. Using the method, we evaluate Sydney, Melbourne, South East Queensland and Perth using reported and modeled data. The approach makes visible large flows of water that have previously been unaccounted and ignored. It also highlights significant intercity variation. In 2004–2005, the cities varied 54% to 100% in their supply centralization, 257% to 397% in the ratio of rainfall and water use, 47% to 104% in their potential stormwater recycling potential, and 26% to 86% in wastewater recycling potential. The approach provides a practical, water‐focused application of the urban metabolism framework. It demonstrates how the principles of mass balance can help foster robust water accounting, monitoring, and management. More important, it contributes to the design and quantitative assessment of water‐sensitive cities of the future.     

Modelling the effect of varying soil water on root growth dynamics of annual crops

We present a simple framework for modelling root growth and distribution with depth under varying soil water conditions. The framework considers the lateral growth of roots (proliferation) and the vertical extension of roots (root front velocity). The root front velocity is assumed to be constant when the roots descend into an initially wet soil profile. The lateral growth of roots is governed by two factors: (1) the current root mass or root length density at a given depth, and (2) soil water availability at that depth.Under non-limiting soil water conditions, the increase in root mass at any depth is governed by a logistic equation so that the root length density (R _v) cannot exceed the maximum value. The maximumR _v, is assumed to be the same for all depths. Additional dry matter partitioned to roots is initially distributed according to the current root mass at each depth. As the root mass approaches the maximum value, less dry matter is partitioned to that depth.When soil water is limiting, a water deficit factor is introduced to further modify the distribution of root dry matter. It is assumed that the plant is an energy minimiser so that more root mass is partitioned to the wetter regions of the soil where least energy will be expended for root growth. Hence, the model allows for enhanced root growth in areas where soil water is more easily available.Simulation results show that a variety of root distribution patterns can be reproduced due to varying soil water conditions. It has been demonstrated that broad patterns of root distribution reported in the literature can also be simulated by the model.     

Morphologically-structured models of growing budding yeast populations

It has been well recognized that many key aspects of cell cycle regulation are encoded into the size distributions of growing budding yeast populations due to the tight coupling between cell growth and cell division present in this organism. Several attempts have been made to model the cell size distribution of growing yeast populations in order to obtain insight on the underlying control mechanisms, but most were based on the age structure of asymmetrically dividing populations. Here we propose a new framework that couples a morphologically-structured representation of the population with population balance theory to formulate a dynamic model for the size distribution of growing yeast populations. An advantage of the presented framework is that it allows derivation of simpler models that are directly identifiable from experiments. We show how such models can be derived from the general framework and demonstrate their utility in analyzing yeast population data. Finally, by employing a recently proposed numerical scheme, we proceed to integrate numerically the full distributed model to provide predictions of dynamics of the cell size structure of growing yeast populations.     

Managing the Flow of Construction Minerals in the North West Region of England

This article, focusing on the flow of bulk construction minerals, establishes a mass balance framework for the North West of England, a region that imports more aggregate material than any other in the United Kingdom. The problems associated with construction minerals are of a different nature than most other resource flow issues: Depletion of resources and contamination are not considered major problems; rather it is the environmental impact resulting from life-cycle stages from extraction, transport, processing, through to final disposal that is most important. A mass balance framework can promote a better understanding of the regional flow of materials, and the impact of human activity on surrounding ecosystems, and hence underpin informed decision making. This is of particular relevance at the current time because increasing political emphasis is placed on sustainable resource management and resource productivity at the United Kingdom and European Union levels. Using a mass balance framework to analyze the sustainability impacts of construction and mineral flows in the North West of England, this study finds that flows resulting from construction activity account for 34,075 terajoules (TJ) of energy resulting in 2,701 gigagrams (Gg) of carbon dioxide emissions related to energy use, and 387 Gg of carbon dioxide emissions related to the transportation of the minerals. Against these impacts, the flow of bulk construction mineral salso supports 147,000 jobs within the region.     

On the Structure-Bounded Growth Processes in Plant Populations

If growing cells in plants are considered to be composed of increments (ICs) an extended version of the law of mass action can be formulated. It evidences that growth of plants runs optimal if the reaction–entropy term (entropy times the absolute temperature) matches the contact energy of ICs. Since these energies are small, thermal molecular movements facilitate via relaxation the removal of structure disturbances. Stem diameter distributions exhibit extra fluctuations likely to be caused by permanent constraints. Since the signal–response system enables in principle perfect optimization only within finite-sized cell ensembles, plants comprising relatively large cell numbers form a network of size-limited subsystems. The maximal number of these constituents depends both on genetic and environmental factors. Accounting for logistical structure–dynamics interrelations, equations can be formulated to describe the bimodal growth curves of very different plants. The reproduction of the S-bended growth curves verifies that the relaxation modes with a broad structure-controlled distribution freeze successively until finally growth is fully blocked thus bringing about “continuous solidification”.     

Cholesterol homeostasis in the guinea pig. The importance of quantitating net tissue accumulation of cholesterol in sterol balance studies

Sterol balance measurements of whole body cholesterol synthesis were performed in guinea pigs to determine the relative quantitative importance of dietary cholesterol intake, endogenous cholesterol synthesis, fecal steroid excretion and net tissue accumulation in cholesterol homeostasis of a rapidly growing animal. Sterol inputs were from diet (33%) and endogenous synthesis (67%); sterol outputs, as fecal neutral and acidic steroids, accounted for 60% of the total input, the remainder being used for the demands of tissue growth. The data demonstrate that the measurement of total body cholesterol synthesis can be grossly underestimated in this rapidly growing animal if net tissue accumulation of cholesterol is not considered in the balance measurement.     

Microtubular configurations during the cellularization of coenocytic endosperm in Ranunculus sceleratus L.

Summary Endosperm cellularization in Ranunculus sceleratus was studied in terms of the initiation of cell-wall formation in the coenocytic endosperm. The first endosperm cell walls were in an anticlinal position relative to the cell wall of the embryo sac and originated from the cell plates and not from wall ingrowths from the embryo-sac wall itself. Alveolar endosperm was formed 3 days after pollination. Microtubules were associated with the freely growing wall ends of the anticlinal walls and were observed in various orientations that generally ranged from angles of 45 ° to 90 ° to the plane of the wall. They were absent in the regions where vesicles had already fused. These microtubules may function in maintaining the growth and the direction of growth of the anticlinal wall until cellularization is completed. At the site where three neighbouring alveoli share their freely growing wall ends, remarkable configurations of microtubules were observed: in each alveolus, microtubules ran predominantly parallel to the bisector of the angle formed by the common walls. These microtubules may form a physically stable framework and maintain the direction of growth of the wall edges. It is concluded that the growing edge of the anticlinal endosperm wall and its associated microtubules are a special continuum of the original phragmoplast that gave rise to the anticlinal wall.     

宁夏盐池苜蓿人工草地地上生物量季节动态研究

在干旱半干旱地区对草地进行水分生态学研究有重要意义,笔者曾对此提出过一些见解,并做过一些工作。本文旨在从人工草地苜蓿(Medicago sativa)种群地上生物量动态方面考察,为合理管理和利用苜蓿人工草地提供科学依据。