A model of the complex response of Staphylococcus aureus to methicillin

It is widely accepted that β-lactam antimicrobials cause cell death through a mechanism that interferes with cell wall synthesis. Later studies have also revealed that β-lactams modify the autolysis function (the natural process of self-exfoliation of the cell wall) of cells. The dynamic equilibrium between growth and autolysis is perturbed by the presence of the antimicrobial. Studies with Staphylococcus aureus to determine the minimum inhibitory concentration (MIC) have revealed complex responses to methicillin exposure. The organism exhibits four qualitatively different responses: homogeneous sensitivity, homogeneous resistance, heterogeneous resistance and the so-called ‘Eagle-effect’. A mathematical model is presented that links antimicrobial action on the molecular level with the overall response of the cell population to antimicrobial exposure. The cell population is modeled as a probability density function F(x,t) that depends on cell wall thickness x and time t. The function F(x,t) is the solution to a Fokker-Planck equation. The fixed point solutions are perturbed by the antimicrobial load and the advection of F(x,t) depends on the rates of cell wall synthesis, autolysis and the antimicrobial concentration. Solutions of the Fokker-Planck model are presented for all four qualitative responses of S. aureus to methicillin exposure.     

Physics of Root Growth

THE elongation of a plant cell involves the yielding of the cell wall under the action of tensile stresses in the wall¹. The rate of elongation, R, can be expressed simply as R=mW, where m is the extensibility of thé cell wall material and W is the wall pressure. Changes in either m or W have been used to explain the effect of biochemical factors on the growth rate of plant cells^2,3. Cell growth is also affected by the physical environment and this becomes particularly important in the case of plant roots where soil water potential and the mechanical resistance of the soil to deformation can become rate controlling. Working with 3-5 day old radicles of Pisum sativum, growing in soil cores, we have obtained values of wall pressure in terms of these two properties and we find that the rate of root elongation can be described by a simple extensibility equation.     

An Increase in Mechanical Extensibility during the Period of Light-stimulated Growth

The sporangiophore of Phycomyces responds to a temporary increase in light intensity with a transient increase in growth rate that begins 2 to 3 minutes after the initiation of the stimulus and continues until approximately the 12th minute. Tensile tests conducted on the stage IVb sporangiophore demonstrate that an increase in mechanical extensibility of the cell wall occurs 2 minutes after the initiation of a light stimulus and continues until approximately the 15th minute. This finding supports the theory that light-stimulated plant cell expansion and rate of expansion is a function of the mechanical extensibility of the cell wall.     

Phytochrome-regulated Expansion of Mustard (Sinapis alba L.) Cotyledons

In developing mustard (Sinapis alba L.) seedlings continuousred light acting via the agency of phytochrome stimulates therate of expansion of cotyledons. Although phytochrome actionon cotyledon expansion is evident only after 36 h from sowing,the photoresponse escapes from reversibility at about 15 h fromsowing. The time lag of 21 h between loss of photoreversibilityand the onset of photoregulated cotyledon expansion indicatesthe existence of long-lived components in the phytochrome-triggeredsignal chain. Phytochrome-regulated cotyledon expansion doesnot require the involvement of photosynthesis, as applicationof SAN 9789, an inhibitor of chloroplast biogenesis, did notaffect cotyledon expansion. The role of turgor pressure-relatedcellular parameters such as osmotic potential () cell wall extensibility(m), hydraulic conductivity (L) and yield threshold (Y) forcell expansion were examined during photoregulated cotyledonexpansion. Using the general equation of cell growth dv/dt =[Lm/(L+m)]( - Y), where dv/dt is the rate of volumetric growth,it was demonstrated that the light-mediated cotyledon expansionresults from an increase in cell wall extensibility (m). Theseresults are discussed in relation to the photoregulation ofcotyledon expansion. Key words: Cell wall extensibility, growth, Sinapis alba L., phytochrome     

On the control of tooth replacement in reptiles and its relationship to growth

The teeth of nearly all non-mammalian vertebrates are replaced in waves which sweep through alternate tooth positions. It is argued that tooth replacement in these animals represents growth of the dentition. It is shown that the pattern of tooth replacement could be described by the exponential equation t_(n)r, = k e^ar+bn when t(n)r is the time at which the rth replacement erupts in the nth position and k, a and b are constants. The length of a replacement wave (w) which is visible in the mouth, can be calculated from the equation w = 2(a?b)/a?2b for forward travelling waves. The effect of different ratios, $\text{a}\text{b}$, on wavelength is described. The model can be interpreted as describing the effect of a zone of inhibition which (it is argued) temporarily surrounds any newly initiated tooth. The increasing time required to dissipate the inhibition around successive replacement teeth is related to the age of the animal. This increasing time permits successive teeth to grow for longer periods than their predecessors and can account for a gradual increase in the size of successive teeth. A similar mechanism could account for the phasic nature of bone growth. It is indicated that the model could be difficult to test.     

Growth, in vivo extensibility, and tissue tension in developing pea internodes

The relationship between growth, in vivo extensibility, and tissue tension in the first 3 internodes of 5, 6, and 7 day-old pea plants (Pisum sativum L. cv Alaska), grown under continuous red light was investigated. The upper 15 millimeters of each internode was marked with ink and its elongation growth measured over the next subsequent 8 hours. In vivo extensibility was measured by stretching living tissue at constant force (creep test) in a custom-built extensiometer. Tissue tension was determined by (a) measuring the rate of expansion of the isolated cortical cylinder after adding water and the amount of contraction of the epidermis after peeling, and (b) by use of the `split section test.' A good correlation between rate of elongation growth, in vivo extensibility, and tissue tension was established. The epidermis peeled from the growing third internode of 7 day-old plants and measured immediately showed a plastic extensibility (E<sub>pl</sub> twice that of peels from nongrowing excised sections. This high E<sub>pl</sub>-value was lost on incubation of the sections in distilled water, and was subsequently restored by incubating the sections in auxin (indole-3-acetic acid). We conclude that the in situ growth of the internodes is a function of tissue-tension, which provides the driving force of organ growth, and the extensibility (E<sub>pl</sub> of the outer epidermal wall, which is in the growing plant in a `loosened' state. We furthermore suggest that in the intact plant auxin is causally involved in the wall loosening process in the epidermis.     

How could the Gompertz-Makeham law evolve

In line with the origin of life from the chemical world, biological mortality kinetics is suggested to originate from chemical decomposition kinetics described by the Arrhenius equation k=A*exp(−E/RT). Another chemical legacy of living bodies is that, by using the appropriate properties of their constituent molecules, they incorporate all their potencies, including adverse ones. In early evolution, acquiring an ability to use new molecules to increase disintegration barrier E might be associated with new adverse interactions, yielding products that might accumulate in organisms and compromise their viability. Thus, the main variable of the Arrhenius equation changed from T in chemistry to E in biology; mortality turned to rise exponentially as E declined with increasing age; and survivorship patterns turned to feature slow initial and fast late descent making the bulk of each finite cohort to expire within a short final period of its lifespan. Numerical modelling shows that such acquisition of new functions associated with faster functional decline may increase the efficiency of investing resources into progeny, in line with the antagonistic pleiotropy theory of ageing. Any evolved time trajectories of functional changes were translated into changes in mortality through exponent according to the generalised Gompertz-Makeham law μ=C(t)+Λ*exp[−E(t)], which is reduced to the conventional form when E(t)=E₀−γt and C is constant. The proposed model explains the origin of the linear mid-age functional decline followed by its deceleration at later ages and the positive correlation between the initial vitality and the rate of ageing.     

Growth Physics in Nitella: a Method for Continuous in Vivo Analysis of Extensibility Based on a Micro-manometer Technique for Turgor Pressure

The view that the plant cell grows by the yielding of the cell wall to turgor pressure can be expressed in the equation: rate = cell extensibility × turgor. All growth rate responses can in principle be resolved into changes in the 2 latter variables. Extensibility will relate primarily to the yielding properties of the cell wall, turgor primarily to solute uptake or production. Use of this simple relationship in vivo requires that at least 2 of the 3 variables be measured in a growing cell. Extensibility is not amenable to direct measurement. Data on rate and turgor for single Nitella cells can, however, be continuously gathered to permit calculation of extensibility (rate/turgor). Rate is accurately obtained from measurements on time-lapse film. Turgor is estimated in the same cell, to within 0.1 atm or less, by measurement of the ability of the cell to compress gas trapped in the closed end of a capillary the open end of which is in the cell vacuole. The method is independent of osmotic equilibrium. It operates continuously for several days, over a several fold increase in cell length, and has response time of less than one minute. Rapid changes in turgor brought on by changes in tonicity of the medium, show that extensibility, as defined above, is not constant but has a value of zero unless the cell has about 80% of normal turgor. Because elastic changes are small, extensibility relates to growth. Over long periods of treatment in a variety of osmotica the threshold value for extensibility and growth is seen to fall to lower values to permit resumption of growth at reduced turgor. A brief period of rapid growth (5× normal) follows the return to normal turgor. All variables then become normal and the cycle can be repeated. The cell remains essentially at osmotic equilibrium, even while growing at 5× the normal rate. The method has potential for detailed in vivo analyses of “wall softening.”     

The Arabidopsis thaliana hypocotyl,a model to identify and study control mechanisms of cellular expansion

Developmental biology studies in general benefit from model organisms that are well characterized. Arabidopsis thaliana fulfills this criterion and represents one of the best experimental systems to study developmental processes in higher plants. Light is a crucial factor that drives photosynthesis, but that also regulates plant morphogenesis. As the hypocotyl is completely embryonic of origin, its growth occurs solely by expansion of the cells and this process is strongly dependent on the light conditions. In this review, we provide evidence that the hypocotyl serves as ideal model object to study cell expansion mechanisms and its regulation. We focus on the regulation of hypocotyl development by light and highlight the key modulating proteins in this signaling cascade. Downstream of light-signaling, cellular expansion is greatly dependent on specific cell wall depositions, which is related to cortical microtubular (re)arrangements and on composition and/or extensibility of the cell wall. We discuss possible further experimental approaches to broaden our knowledge on hypocotyl development, which will give an outlook on the probable evolution of the field.     

Probabilistic Model of Microbial Cell Growth,Division, and Mortality

After a short time interval of length δt during microbial growth, an individual cell can be found to be divided with probability P_d(t)δt, dead with probability P_m(t)δt, or alive but undivided with the probability 1 − [P_d(t) + P_m(t)]δt, where t is time, P_d(t) expresses the probability of division for an individual cell per unit of time, and P_m(t) expresses the probability of mortality per unit of time. These probabilities may change with the state of the population and the habitat''s properties and are therefore functions of time. This scenario translates into a model that is presented in stochastic and deterministic versions. The first, a stochastic process model, monitors the fates of individual cells and determines cell numbers. It is particularly suitable for small populations such as those that may exist in the case of casual contamination of a food by a pathogen. The second, which can be regarded as a large-population limit of the stochastic model, is a continuous mathematical expression that describes the population''s size as a function of time. It is suitable for large microbial populations such as those present in unprocessed foods. Exponential or logistic growth with or without lag, inactivation with or without a “shoulder,” and transitions between growth and inactivation are all manifestations of the underlying probability structure of the model. With temperature-dependent parameters, the model can be used to simulate nonisothermal growth and inactivation patterns. The same concept applies to other factors that promote or inhibit microorganisms, such as pH and the presence of antimicrobials, etc. With P_d(t) and P_m(t) in the form of logistic functions, the model can simulate all commonly observed growth/mortality patterns. Estimates of the changing probability parameters can be obtained with both the stochastic and deterministic versions of the model, as demonstrated with simulated data.     

Cytokinin-induced wall extensibility in excised cotyledons of radish and cucumber

The mechanism of cytokinin-induced cell expansion in cotyledons excised from dark-grown seedlings of radish (Raphanus sativus L.) and cucumber (Cucumus sativus L.) was studied. Cotyledons were incubated in dim light with or without 17 micromolar zeatin for periods up to 3 days. Fresh weights and osmotic potentials were measured daily. Cell wall extensibility properties were measured before and after the growth period. Also, experiments in which radish cotyledons were grown in mannitol solutions of various concentrations were performed. Comparisons of growth rates and increases of tissue osmotic potentials (toward zero) during growth without mannitol indicate that wall extensibility increased during the growth period and that this extensibility was enhanced by zeatin.     

POLYGALACTURONASE INVOLVED IN EXPANSION1 Functions in Cell Elongation and Flower Development in Arabidopsis

Pectins are acidic carbohydrates that comprise a significant fraction of the primary walls of eudicotyledonous plant cells. They influence wall porosity and extensibility, thus controlling cell and organ growth during plant development. The regulated degradation of pectins is required for many cell separation events in plants, but the role of pectin degradation in cell expansion is poorly defined. Using an activation tag screen designed to isolate genes involved in wall expansion, we identified a gene encoding a putative polygalacturonase that, when overexpressed, resulted in enhanced hypocotyl elongation in etiolated Arabidopsis thaliana seedlings. We named this gene POLYGALACTURONASE INVOLVED IN EXPANSION1 (PGX1). Plants lacking PGX1 display reduced hypocotyl elongation that is complemented by transgenic PGX1 expression. PGX1 is expressed in expanding tissues throughout development, including seedlings, roots, leaves, and flowers. PGX1-GFP (green fluorescent protein) localizes to the apoplast, and heterologously expressed PGX1 displays in vitro polygalacturonase activity, supporting a function for this protein in apoplastic pectin degradation. Plants either overexpressing or lacking PGX1 display alterations in total polygalacturonase activity, pectin molecular mass, and wall composition and also display higher proportions of flowers with extra petals, suggesting PGX1’s involvement in floral organ patterning. These results reveal new roles for polygalacturonases in plant development.     

Generalized stable population theory

In generalizing stable population theory we give sufficient, then necessary conditions under which a population subject to time dependent vital rates reaches an asymptotic stable exponential equilibrium (as if mortality and fertility were constant). If x ₀(t) is the positive solution of the characteristic equation associated with the linear birth process at time t, then rapid convergence of x ₀(t) to x ₀ and convergence of mortality rates produce a stable exponential equilibrium with asymptotic growth rate x ₀–1. Convergence of x ₀(t) to x ₀ and convergence of mortality rates are necessary. Therefore the two sets of conditions are very close. Various implications of these results are discussed and a conjecture is made in the continuous case.     

Effect of temperature on plant elongation and cell wall extensibility

Lockhart equation was derived for explaining plant cell expansion where both cell wall extension and water uptake must occur concomitantly. Its fundamental contribution was to express turgor pressure explicitly in terms of osmosis and wall mechanics. Here we present a new equation in which pressure is determined by temperature. It also accounts for the role of osmosis and consequently the role of water uptake in growing cell. By adopting literature data, we also attempt to report theoretically the close relation between plant elongation and cell wall extensibility. This is accomplished by the modified equation of growth solved for various temperatures in case of two different species. The results enable to interpret empirical data in terms of our model and fully confirm its applicability to the investigation of the problem of plant cell extensibility in function of environmental temperature. Moreover, by separating elastic effects from growth process we specified the characteristic temperature common for both processes which corresponds to the resonance energy of biochemical reactions as well as to the rapid softening of the elastic modes toward the high temperature end where we encountered viscoelastic and/or plastic behavior as dominating. By introducing analytical formulae connected with growth and elastic properties of the cell wall, we conclude with the statement how these both processes contribute quantitatively to the resonance-like shape of the elongation curve. In addition, the tension versus temperature "phase diagram" for a living plant cell is presented.     

The finite element implementation of a K.P.P. equation for the simulation of tsetse control measures in the vicinity of a game reserve

An equation, strongly reminiscent of Fisher’s equation, is used to model the response of tsetse populations to proposed control measures in the vicinity of a game reserve. The model assumes movement is by diffusion and that growth is logistic. This logistic growth is dependent on an historical population, in contrast to Fisher’s equation which bases it on the present population. The model therefore takes into account the fact that new additions to the adult fly population are, in actual fact, the descendents of a population which existed one puparial duration ago, furthermore, that this puparial duration is temperature dependent. Artificially imposed mortality is modelled as a proportion at a constant rate. Fisher’s equation is also solved as a formality.The temporary imposition of a 2 % day⁻¹ mortality everywhere outside the reserve for a period of 2 years will have no lasting effect on the influence of the reserve on either the Glossina austeni or the G. brevipalpis populations, although it certainly will eradicate tsetse from poor habitat, outside the reserve. A 5 km-wide barrier with a minimum mortality of 4 % day⁻¹, throughout, will succeed in isolating a worst-case, G. austeni population and its associated trypanosomiasis from the surrounding areas. A more optimistic estimate of its mobility suggests a mortality of 2 % day⁻¹ will suffice. For a given target-related mortality, more mobile species are found to be more vulnerable to eradication than more sedentary species, while the opposite is true for containment.     

Gibberellic Acid Regulates Cell Wall Extensibility in Wheat (Triticum aestivum L.)

Mutations (Rht genes) blocking sensitivity to gibberellic acid (GA) were used to examine phytohormone mediated cell wall expansion in wheat (Triticum aestivum L.). Irreversible extensibility of immature leaf segments, as determined by stress/strain (instron) measurements, declined with Rht gene dose. Exogenous GA₃ significantly increased wall extensibility in the nonmutant controls but had no effect on the near-isogenic GA-insensitive genotypes. Furthermore, ancymidol, an inhibitor of gibberellin biosynthesis, diminished wall extensibility in the nonmutant control. Extensibility of immature segments was highly correlated with mature leaf sheath length (R = +0.95). The results indicate that wall yielding properties of expanding wheat leaves are associated with leaf cell expansion potential and that GA is involved in the determination of those properties.     

Salinity Stress Inhibits Bean Leaf Expansion by Reducing Turgor, Not Wall Extensibility

Treatment of bean (Phaseolus vulgaris L.) seedlings with low levels of salinity (50 or 100 millimolar NaCl) decreased the rate of light-induced leaf cell expansion in the primary leaves over a 3 day period. This decrease could be due to a reduction in one or both of the primary cellular growth parameters: wall extensibility and cell turgor. Wall extensibility was assessed by the Instron technique. Salinity did not decrease extensibility and caused small increases relative to the controls after 72 hours. On the other hand, 50 millimolar NaCl caused a significant reduction in leaf bulk turgor at 24 hours; adaptive decreases in leaf osmotic potential (osmotic adjustment) were more than compensated by parallel decreases in the xylem tension potential and the leaf apoplastic solute potential, resulting in a decreased leaf water potential. It is concluded that in bean seedlings, mild salinity initially affects leaf growth rate by a decrease in turgor rather than by a reduction in wall extensibility. Moreover, longterm salinization (10 days) resulted in an apparent mechanical adjustment, i.e. an increase in wall extensibility, which may help counteract reductions in turgor and maintain leaf growth rates.     

Morphological and physiological adaptive traits of Mediterranean narrow endemic plants: The case of Centaurea gymnocarpa (Capraia Island,Italy)

A case study on Centaurea gymnocarpa Moris & De Not., a narrow endemic species, was carried out by analyzing its morphological, anatomical, and physiological traits in response to natural habitat stress factors under Mediterranean climate conditions. The results underline that the species is particularly adapted to the environment where it naturally grows. At the plant level, the above-ground/below-ground dry mass (1.73 ± 0.60) shows its investment predominately in the above-ground structure with a resulting total leaf area per plant of 1399 ± 94 cm². The senescent attached leaves at the base of the plant contribute to limit leaf transpiration by shading soil around the plant. Moreover, the dense C. gymnocarpa leaf pubescence, leaf rolling, the relatively high leaf mass area (LMA = 12.3 ± 1.3 mg cm⁻²) and leaf tissue density (LTD = 427 ± 44 mg cm⁻³) contribute to limit leaf transpiration, also postponing leaf death under dry conditions. At the physiological level, a relatively low respiration/photosynthesis ratio (R/P_N) in spring results from high R [2.26 ± 0.59 μmol (CO₂) m⁻² s⁻¹] and P_N [12.3 ± 1.5 μmol (CO₂) m⁻² s⁻¹]. The high photosynthetic nitrogen use efficiency [PNUE = 15.5 ± 0.4 μmol (CO₂) g⁻¹ (N) s⁻¹] shows the large amount of nitrogen (N) invested in the photosynthetic machinery of new leaves, associated to a high chlorophyll content (Chl = 35 ± 5 SPAD units). On the contrary, the highest R/P_N ratio (1.75 ± 0.19) in summer is due to a significant P_N decrease and increase of R in response to drought. The low PNUE [1.5 ± 0.2 μmol (CO₂) g⁻¹ (N) s⁻¹] in this season is indicative of a greater N investment in leaf cell walls which may contribute to limit transpiration. On the contrary, the low R/P_N ratio (0.05 ± 0.02) in winter is resulting from the limited enzyme activity of the respiratory apparatus [R = 0.23 ± 0.08 μmol (CO₂) m⁻² s⁻¹] while the low PNUE [3.5 ± 0.2 μmol (CO₂) g⁻¹ (N) s⁻¹] suggests that low temperatures additionally limit plant production. The experiment of the imposed water stress confirms that the C. gymnocarpa growth capability is in conformity with the severe conditions of its natural habitat, likewise as it may be the case with others narrow endemic species that have occupied niches with similar extreme conditions.     

Diffusion Coefficients of Endogenous Cytosolic Proteins from Rabbit Skinned Muscle Fibers

Efflux time courses of endogenous cytosolic proteins were obtained from rabbit psoas muscle fibers skinned in oil and transferred to physiological salt solution. Proteins were separated by gel electrophoresis and compared to load-matched standards for quantitative analysis. A radial diffusion model incorporating the dissociation and dissipation of supramolecular complexes accounts for an initial lag and subsequent efflux of glycolytic and glycogenolytic enzymes. The model includes terms representing protein crowding, myofilament lattice hindrance, and binding to the cytomatrix. Optimization algorithms returned estimates of the apparent diffusion coefficients, D(r,t), that were very low at the onset of diffusion (∼10⁻¹⁰ cm² s⁻¹) but increased with time as cytosolic protein density, which was initially high, decreased. D(r,t) at later times ranged from 2.11 × 10⁻⁷ cm² s⁻¹ (parvalbumin) to 0.20 × 10⁻⁷ cm² s⁻¹ (phosphofructose kinase), values that are 3.6- to 12.3-fold lower than those predicted in bulk water. The low initial values are consistent with the presence of complexes in situ; the higher later values are consistent with molecular sieving and transient binding of dissociated proteins. Channeling of metabolic intermediates via enzyme complexes may enhance production of adenosine triphosphate at rates beyond that possible with randomly and/or sparsely distributed enzymes, thereby matching supply with demand.