Properly changing the time scale of a parameter in a growth model

Properly changing the time scale of a parameter in a growth model

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I have a two-population model, where one of the populations is a bacterial culture with its growth described as a logistic growth, with a reported growth rate of $2 loghbox{bacteria hour}^{-1}$ and a death rate of $1.5 loghbox{bacteria hour}^{-1}$ and a carrying capacity of $7 loghbox{bacteria}$. However, my model studies the interaction between bacteria and fungi in a time scale of days, but if I try to directly rescale these parameters, I get a growth rate of $48 loghbox{bacteria day}^{-1}$ and a death rate of $36 loghbox{bacteria day}^{-1}$, which is absurd, given the carrying capacity of bacteria.

Is there a standard method for properly rescaling these parameters for bacterial growth, without having to change the model? What alternatives do I have?

These values are not absurd: it's just a different unit, it does not change the reality of th phenomenon.

The fact that your population can grow by a $48$ or $36~mathrm{log}$ factor in one day, while the carrying capacity is $10^7$ bacteria, only means that the growth happens in less than one day (once the latency phase is over). Many bacterial population can grow in less than one day, it is not shocking.

In my opinion, it is not absurd to use the growth rates in $mathrm{log}~ extrm{bacteria} cdot extrm{day}^{-1}$. There is not need to rescale values if you just want to convert hours into days.

Age and Growth Determination of Fish (With Diagram)

Determination of age and growth of fishes are one of the important aspects in the development of fisheries. Both age and growth are closely related with one other. As the fish ages, it grows, but after attaining a particular size, growth stops. Age gives an idea about sexual maturity, spawning time, catchable size, growth rate and longevity. Knowledge of all these parameters is essential in fisheries production.

Growth of fish or any organism is the change in length and weight with increase of age as a result of metabolism of nutrition. Hence growth is an index of healthy food and oxygen supply in the water-body. Proper growth of fish also indicates that the water body is devoid of any pollution.

Knowledge of age and growth of fish has many applications:

1. We can calculate the time of sexual maturity of different species.

2. Further, we can know their spawning time.

3. Growth rate of fish also indicates the suitability of particular species for a particular type of the water-body.

4. Growth rate and age of fish also indicate the size of fish at different stages, e.g., fry, fingerling, and adult of different species.

5. The study of age and growth is helpful in catching fishes by using nets of desirable mesh size.

Methods of Age and Growth Determination:

Various methods employed for determination of age of fish are:

This method is most commonly used for determination of age of osteichthyes (bony fish), which are provided with cycloid and ctenoid scales. The structure of scale and its development is useful in the interpretation of growth zones. The structure of scale can be seen very easily under the microscope after washing with dilute solution of caustic soda followed by staining with borax carmine.

A well-developed scale has the following structures:

It is a clear area in the centre, but may be shifted from the center due to irregular growth of anterior or posterior parts of scale caused by unusual overlapping of scales.

These are concentric rings present around the focus, they run parallel at regular intervals or distances. They appear as ridges.

The grooves are found between the ridges of circuli and they are responsible for maintaining the regular space between them.

These are grooves found radially, viz., they run from focus to margin of scale. Radii cut the circuli present in their path.

These are wide circular troughs found in aged fish over one year. Each trough contains a few incomplete and narrow circuli different from the circuli outside it, which are complete and more widely spaced. The number of annuli represents age of a fish in years (Fig. 14.1).

At the time of development of scale, focus is established first and represents the original size of the scale. As scale grows older, other structures are added and perform their functions. The grooves and circuli represent growth activity. They also indicate osteoblastic activity as a result of which secreted material is deposited around the focus. In this way every year many such circuli and grooves are formed.

A characteristic bone material, ichthylepidin, is deposited in circuli and thus their height increases which depends on the calcification. Annuli show slow growth in a year but in many fishes, during winters, annuli grow remarkably and are added yearly as fish grows.

Thus annuli are very useful in counting the age of fish and serves as year-marks on the scale for age determination. The annuli are best seen at anterior part of the scale.

Types of Annuli:

The true annuli has following characteristic features:

(i) In cycloid scales true annuli is represented by a closely situated circuli, which is covered by widely spaced circuli.

(ii) Two complete circuli surround the trough, which is wider on the anterolateral and posterior side.

(iii) The wide part of the trough contains incomplete circuli that do not grow completely around the scale.

(iv) The trough remains narrow at the anterolateral side. In the ctenoid scales specially, the outer circulus cuts across or crosses over the incomplete circuli lying in the anterior part of the trough.

Annuli are considered as year-marks in the age determination of fish in the case of the following facts.

a. When there is a correlation between the calculated age from the scale and the size of the fish.

b. The length frequency distribution should coincide with the calculated age from the scales.

c. The calculated age should be in agreement with the age determined by other methods.

Sometimes false annuli appear on the scales of fishes due to undesirable factors like retarded growth due to paucity of food, starvation, injury, disease, and fluctuation in temperature. These false annuli resemble the true annuli but they take position on the scale closer to the true annulus of the preceding year than the normal annulus for the next year which appear in case of normal growth (Fig. 14.2).

3. Overlapping Annuli:

The position of these annuli in posterior field coincide with the annuli of the preceding year while, in anterior part, it is separated from the preceding year’s annuli by 4 or 5 circuli. The overlapping of scales can occur due to a slow growth during the growing period, which is represented by an increase in the length but not in the weight of body.

4. Skipped Annulus:

This type of annulus by position coincides with the annulus of the preceding year, with no normal circuli forming in between. This abnormal function is due to the fact that the fish has not grown during one growing season (one summer) either in length or in weight.

Applications of Scale Method:

1. Fish of temperate regions shows clear rings, which are true marks. This is because there is a sharp difference between the temperatures of two seasons—summer—the period of faster growth, and winter—the period of slow growth or no growth. Therefore, the calculation of the age of fish by annuli is most reliable in temperate fish.

2. This method is more reliably applicable in case of salmons, carps, cod and herrings, established a method of estimating age of fish based on scales, which is as given on next page:

Limitations of Scale Method:

1. More than one annulus are added in the extreme conditions of life, e.g., extreme cold (causes cessation of feeding), change in food quality or starvation at the time of spawning. These additional rings are called supplementary rings, which cause problem in age determination by the scale method.

2. This method cannot be applied to those fishes which live in the water with more or less uniform temperatures (tropics). This may be because in these places fishes spawn more than once and, due to fluctuations in food and chemical compositions of water due to rains and floods, formation of annulus may not be an annual feature. Thus, in the fishes of tropical regions, the growth rings do not actually represent year-marks.

Annuli are also present on some bones. The important bones such as operculum, vertebra, supra occipital and scapula are provided with annulations. These annuli are increased in number with the age of fish. The growth rate is different in different seasons. Number of annual rings are helpful in calculating the age of fish.

Similarly, the centrum of fish is also helpful in calculating the age of fish. The centrum of fish vertebrae possess rings, which are used in age determina­tion. For counting the rings on the centrum, it is exposed by removing the tissues attached to it using solution of 0.7% pepsin in 0.2% of hydrochloric acid. The rings on the centrum are counted under the microscope.

The otolith or ear-stone is present in the internal ear of the fishes and helps in balancing the body. The otolith has annulations formed by the regular deposition of calcium salts. The formation of annulations varies with the growth of fish, and the growth of fish varies with the season. The growth rate of fish is faster in the summer as compared to the winters, thus annulations formation is faster in the summer.

To get the otolith, fish is killed and dissected and otolith is taken out and annulations are counted. For this, first of all the otolith is broken and cut in transverse plane followed by polishing with liquid of high refractive index, such as immersion oils, creosote, etc. The otolith thus prepared is examined under the microscope to count the annulations for age determination and the growth rate analysis Graeme, (2007).

The determination of growth rate through this method requires the knowledge of age, which can be known from time of breeding. The hatchlings are kept in a tank in appropriate conditions for two or three seasons. The growth is measured periodically. The fishes are marked with tags and reintroduced in the water. The fishes are recaptured at regular intervals and idea of growth rate in relation of the time is taken.

6. Length Frequency Distribution Method:

Peterson introduced this method in the nineteenth century. In this method, in the sample of fishes of a particular fish, frequency analysis shows length of individuals of one age varying around the mean length according to normal distribution.

Data of the sample is plotted and peaks are counted which represent grouping of fishes of successive lengths. Age groups are thus separated and the age of a given species is determined.

This method is suitable for determining the age of younger fishes of 2 to 4 years. This method is not reliable for calculating the age of older fishes because of overlapping of length frequencies in individuals of different ages.

7. Pectoral Spine or Fin Ray Method:

Spines of the pectoral fin is also useful in age determination. For this 3μ to 4μ thick sections of spines are cut and mounted in glycerin and then observed in the microscope. 1% to 2% of indistinct annuli or fin ray consistency are compared to 15% to 20% of annuli on the scales and, thus age is calculated.

Properly changing the time scale of a parameter in a growth model - Biology

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The PUN approach

Given a set of experimental data, the first thing that one may want to do, in order to extract from it as much information as possible, is to find a suitable fitting function. As a next step it may be desirable to construct a model out of it. For this purpose one should restrict one’s attention to the raw data and analyse them independently of the field of application. Such an unbiased procedure may be provided by the Phenomenological Universalities (PUN) approach, recently proposed by P.P. Delsanto and collaborators [33, 34] and applied to a wide range of topics (auxology [35], tumor growth [36, 37], nonlinear elasticity [38], and others [39–41]).

In order to describe the PUN methodology from an applicative point of view, let us start with the first order nonlinear growth differential equation

where y(t) represents the variable of interest (e.g. the height of the children) and a(y) the (unknown) growth rate. Equivalently, Eq.(1) may be written as

where z = 1n(y). To integrate Eq.(1), or Eq.(2), it is necessary to make some assumption about the rate a, e.g. assuming that the time derivative of the growth rate or acceleration b is given by an expansion in a, i.e.

E.g. for N = 2 we have, in addition to the linear term βa, also the quadratic term γa 2 .

We call UN the class generated by the solution of the coupled differential equations (2) or (1) and (3), when in the latter only the first N terms are considered. The functions y(t) that one obtains for the first UN classes (N = 0, 1, 2) have a very wide range of applications. In fact:

for N = 0, i.e. U0, b = 0 y(t) represents a self-catalytic growth function. By integrating over the two ODEs, Eq. (3) and Eq. (1), we obtain

where a0 = a(t = 0) is the initial growth rate. Here and in the following we normalize the variable y(t), so that y(0) = 1.

for N = 1, i.e. U1, b = βa and

Equation (5) represents the Gompertz law [25], which has been extensively used in all kinds of growth problems for almost two centuries. The parameter β represents a retardation factor, which may be defined as β = − a 0 / z ∞ , where z ∞ is the asymptotic value of z(t).

for N = 2, i.e. U2, b = β a + γ a 2 and

which yields a generalization of West’s law [8, 9, 42]. Here γ = p − 1, where p is the growth exponent [33].


Our models illustrate features of biochemical pathways that allow for the discrimination of signals that only differ in their duration. It is important to note that many important, nonlinear effects, at the expense of analytical tractability, were excised in making the linear, weakly activated cascade approximation. For example, nonlinear positive feedback is known to give rise to bistability. Also nonlinear negative feedback can lead to oscillatory behavior. These effects, however, correspond to long time, steady state behavior and the present analysis focused on transient signals of different duration and it is therefore expected that such nonlinear effects are not expected to influence the qualitative behavior of the results in this study.

In summary, we computed the frequency dependent internal gain for two classes of biochemical pathways involving multiple stages of regulation. The first model consisted of a cascade of steps and showed how changes in the number of steps as well as the amplification/attention of the signal changed the networks' ability to filter high frequency (short duration) components of a signal. Another network consisted of a sequence of steps in the form of biochemical intermediates in which the output is connected to a downstream feedback loop or an interacting product. The gain in this network can have non monotonic behavior in which the low frequency components of the signal are also filtered at time scales commensurate with the induction of the regulatory loop. This behavior enables the network to filter out signals of long duration. The minimal topological features of these biochemical networks provide distinct and robust mechanisms for integrating signals that persist with different characteristic time scales. As different temporally regulated signals often lead to different transcriptional programs such as in NF-κB signaling [26, 27], it is tempting to speculate on the role that such filtering mechanisms may have in regulating gene expression.


Three models were fitted to weight and height growth data of 961 children from birth to age 6 years separately for boys and girls (table 1). Children had between two and 11 weight and height measurements (table 1), and 75% of girls had at least four measurements (five in boys). Some descriptive characteristics of the study population are presented in online supplementary table 2. The overall mean maternal age at age 1 year of the study child was 25.8 years (SD 5.6). Almost three-fourth of all mothers could not read or write at the start of the cohort. Almost 10% of mothers were underweight before their pregnancy. Almost 10% of boys and 14% of girls had a lower birth weight (<2.5 kg) and less than 7% of children were born preterm. Almost 10% of children had malaria at 1 year of age and almost 74% of boys and 67% of girls had anaemia at 1 year of age.

Number of anthropometric measures per child and sex

The distribution of the residuals over age for weight and height of both boys and girls showed that, overall, residuals were centred around zero for all ages and there was no strong dependence of residual distribution over ages for all the models (figures 1 and 2).

Residuals of weight models for boys and girls (data of the first two deciles was gathered, as there were more than 10% of the data at birth).

Residuals of height models for boys and girls (data of the first two deciles was gathered, as there were more than 10% of the data at birth).

For weight models, the Jenss-Bayley model had the lowest AIC and BIC values, both in boys and girls, indicating a better fit than the two other candidate models (table 2). Both the Reed and the adapted Gompertz model had high delta AIC values for both boys (136 and 117, respectively) and girls (129 and 135, respectively), indicating essentially no support for these two candidate model to be better than the best-fitting model (Jenss-Bayley).

Comparison of the goodness-of-fit of the three candidate models

For height models, the Jenss-Bayley model and the adapted Gompertz model had the lowest AIC and BIC values for the height model in boys while the Jenss-Bayley model had lower AIC values in girls than the Reed and adapted Gompertz model. Concerning delta AIC values, both the Reed and the adapted Gompertz model had values <10 for boys (8 and 6, respectively), indicating considerably less support for these two models to be the best-fitting model. While in girls, these two candidate models had delta AIC values >10 (28 and 32, respectively), which does not indicate any support for these two models to have a better fit than the Jenss-Bayley model.

Using the Jenss-Bayley model, weight and height were calculated for all the children at age points, particularly when they were not measured, as well as instantaneous growth velocities at different age points, and growth trajectories were calculated (tables 3 and 4, and figure 3, respectively).

Estimated weight and weight growth velocity (SD) of girls and boys aged 0 to 6 years, from the Jenss-Bayley model

Estimated length/height and length/height growth velocity (SD) of girls and boys aged 0 to 6 years, from the Jenss-Bayley model

Weight and height mean growth trajectories from the Jenss-Bayley model. (A) Weight growth curve. (B) Height growth curve. The solid (black) line implies boys’ curve and the dashed (red) line implies girls’ curve.

Means and SD of parameters estimates of the Jenss-Bayley model along with two other candidate models (the adapted Gompertz model and the Reed model) fitted to weight and height of boys and girls are reported in online supplementary table 3.

Nutritional status for all children was estimated at age 6 years. About one-third of the children were wasted and underweight (online supplementary table 4). While 8% of boys and 4% of girls were stunted (online supplementary table 4). In comparison, in the WHO growth reference population, the percentage of children under −2 z-scores is expected to be 2.3%.


Physiologists have long been interested in a particular class of phenotypic plasticity termed acclimation, which is typically defined to include reversible changes in physiological phenotypes as a result of environmental exposures in the time range of days to months. Note, however, that there is a fuzzy boundary between acute physiological responses, which can occur within minutes or hours (e.g., the heat-shock response), and more typical acclimation responses on a longer time scale. In addition, the degree of reversibility of all of these responses may vary, with prior thermal exposure having the potential to cause long-term effects on phenotype ( Whitman 2009). If we return to Fig. 2, it is apparent that the TPCs for the acute effects of temperature on mitochondrial oxygen consumption differ when compared between mitochondria from fish held at different acclimation temperatures ( Fig. 2A–C). Both the exponent (slope) and the intercept of the exponential curves fitted to the data differ, and there is also a slight movement of the curve along the x-axis mitochondria from killifish acclimated at 5°C could not be assayed at temperatures greater than 35°C, while those from killifish acclimated to warmer temperatures could be assayed at temperatures up to 37°C. In addition, the data from the 25°C acclimation group conform fairly well to an exponential curve, but there is increasing deviation from a simple exponential curve and an increase in obvious breakpoints with decreasing acclimation temperature. Overall, the data shown in Fig. 2 indicate that one of the important effects of acclimation on oxygen consumption by killifish mitochondria is to change the nature of their response to acute temperature challenge.

The idea that thermal acclimation of biological rate processes (which are affected by temperature on an instantaneous basis) is likely to be due to changes in the position and shape of the TPC on an acute time scale has long been appreciated ( Precht 1949 Prosser 1958). However, many studies on biological rate processes fail to consider this shorter temporal scale, and instead simply examine metabolic rate at the acclimation temperature across various treatments. In fact, it has been argued that this is the most biologically relevant way to assess performance if you wish to estimate selection on TPCs, as animals are most likely to be exposed to temperatures close to the temperature to which they are acclimatized in nature. For example, one of the most convincing empirical examinations of the hypothesis of oxygen and capacity limited thermal tolerance is an examination of variation in the TPCs for aerobic scope in different stocks of sockeye salmon ( Eliason et al. 2011), which found a strong correlation between the Topt for aerobic scope, a variety of underlying physiological traits, and the historical mean river temperature at the time of migration. This study utilized fish acclimated for 1 week to the temperature at which they were collected, and physiological rate processes were tested only at this acclimation temperature.

The potential challenges associated with examining rate processes at the temperature of acclimation can be seen if we replot the data on the oxygen consumption of killifish mitochondria from Fig. 2 to show oxygen consumption at the acclimation temperature ( Fig. 5). In this case, oxygen consumption increases linearly with acclimation temperature, unlike the exponential increases seen in the acute TPCs ( Fig. 2). Thus, in cases where an organism has the capacity to alter the acute TPC by acclimation, considering only the TPC on an acclimated time scale (as in Fig. 5) will conceal the underlying mechanistic complexity. This observation highlights a critical property of TPCs. As pointed out by David et al. (2003), most studies of phenotypic plasticity assume a null hypothesis of no passive plasticity at the acute time scale, but for TPCs, the null hypothesis is the presence of passive plasticity (sensitivity to temperature) at the acute time scale, making the interpretation of plasticity from TPCs on longer time scales of temperature exposure more complicated than for many other traits.

Maximal in vitro (state III) consumption of oxygen by killifish mitochondria at the acclimation temperature. Liver mitochondria were isolated from killifish (F. heteroclitus macrolepidotus) acclimated to three different temperatures and were tested in vitro at their acclimation temperature. Mitochondria were provided with saturating substrate (pyruvate + malate), oxygen, and ADP to stimulate maximum oxygen consumption. Oxygen consumption is expressed per mg mitochondrial protein. Data are from Fangue et al. (2009a).

Maximal in vitro (state III) consumption of oxygen by killifish mitochondria at the acclimation temperature. Liver mitochondria were isolated from killifish (F. heteroclitus macrolepidotus) acclimated to three different temperatures and were tested in vitro at their acclimation temperature. Mitochondria were provided with saturating substrate (pyruvate + malate), oxygen, and ADP to stimulate maximum oxygen consumption. Oxygen consumption is expressed per mg mitochondrial protein. Data are from Fangue et al. (2009a).

The interactions between passive plasticity and active plasticity have prompted a certain degree of confusion with respect to terminology. The extent of phenotypic plasticity for a trait is usually visualized using a reaction norm ( Woltereck 1909), which is simply a graph of the values of a trait of interest against values of an environmental variable. The slope of a reaction norm summarizes the extent of plasticity in the trait. When the slope of the reaction norm is horizontal, the trait does not vary with the environment (i.e., it lacks plasticity), whereas when the slope of the reaction norm is steep, the trait varies greatly as the environment changes (i.e., it has high plasticity). From this description, the similarities between TPCs and reaction norms are clear TPCs are simply reaction norms for a particular class of trait (biological rate processes). The complex nonlinear shapes of TPCs make interpretation of the slope somewhat more difficult, but the real difficulties arise because of the presence of passive plasticity at an acute time scale for TPCs.

Imagine, for example, a hypothetical reaction norm showing the effects of acclimation temperature on metabolic rate measured at the acclimation temperature (similar to the presentation shown in Fig. 5). In a case where this graph has zero slope, we would conclude that there is no plasticity in this trait. However, recall that there is substantial passive plasticity in metabolic rate at the acute time scale of temperature exposure. Thus, the only way to achieve a horizontal slope on the acclimated reaction norm is for active plasticity to have caused physiological changes that maintain the acclimated metabolic rate constant with increasing temperature. This leads to the apparently absurd conclusion that the only way to achieve a reaction norm demonstrating a lack of plasticity is for the organism to exhibit substantial plasticity. Of course, this logical knot can be untied by recalling that two different types of plasticity acting at two different time scales are being considered here. To avoid confusion, we would prefer that the term plasticity only be used to describe an active response by the organism, but we acknowledge that this is not the generally accepted usage.

We are not the first to point out the challenges of dealing with environmental effects acting at multiple time scales ( Fry 1971 Pigliucci 2001 Chown and Terblanche 2007). For example, Kingsolver et al. (2004) suggested that graphs showing effects of temperature at time scales relevant to acclimation should only be thought of as reaction norms when they take into account the effects of changes in the shape of the curves at acute time scales. Chown and Terblanche (2007) made a similar proposal, suggesting that “where variation in performance curves is being assessed, the use of the term reaction norm should be restricted to the response of the curves rather than being meant to imply the curves too”. In this context, we suggest that it might be useful to develop (or encourage the use of) alternatives to typical TPCs that help to keep this potential complexity at the forefront of thinking. At least for traits where it is possible to measure the acute TPC, it would almost certainly be useful to utilize three dimensional plots, with acclimation temperature on one axis and acutely experienced temperature on another. These distinctions can be important, because failing to properly distinguish between passive plasticity and active plasticity can have critical consequences for the development of mechanistically based models of the impacts of climate change ( Wythers et al. 2005).

Ecological scaling and biomass size spectra

The study of the distribution of biomass by size in the pelagic systems has been a significant step in the search for generalizations in aquatic ecology. Regularities in the size structure of pelagic communities have been observed in offshore systems (e.g. Sheldon et al.,1972 Beers et al.,1982 Platt et al.,1984 Rodriguez and Mullin, 1986a,b Witek and Krajewska-Soltys,1989 Quiñones et al.,2003) and lakes (e.g. Sprules et al., 1983, 1991 Sprules and Knoechel, 1984 Sprules and Munawar, 1986 Echevarría et al.,1990 Ahrens and Peters,1991 Gaedke,1993). In coastal pelagic ecosystems the biomass size distribution does not present patterns as regular as those observed in oligotrophic systems but biomass is not randomly distributed across body size (e.g. Jimenez et al., 1987, 1989 Rodriguez et al., 1987). A regular pattern in the biomass size distribution has also been found in salt marshes (Quintana et al.,2002) and benthic communities (e.g. Schwinghamer, 1981 Warwick 1984 Schwinghamer, 1985 Saiz-Salinas and Ramos, 1999 Quiroga et al., 2005).

On the other hand, aquatic food webs are strongly size-structured with larger predators eating smaller prey(Sheldon et al., 1972 Dickie et al., 1987). Many species grow in mass by five orders of magnitude cannibalism, cross-predation and transient predator-prey relationships are common(Cushing, 1975 Kerr and Dickie, 2001). However, mean body mass of species is only weakly correlated with body mass in the whole food web (Fry and Quiñones, 1994 France et al., 1998 Jennings et al., 2001, 2002). These observations provide compelling reasons to adopt size-based rather than species-based analyses of food web structure in pelagic ecosystems(Jennings and Mackinson,2003).

Example of a unnormalized biomass size spectrum from Sprules et al.(1991) and compiled from their sampling of Lake Michigan, showing the component trophic groups. (After Thiebaux and Dickie, 1993.)Biomass (g m -2 ) Mb (g).

Example of a unnormalized biomass size spectrum from Sprules et al.(1991) and compiled from their sampling of Lake Michigan, showing the component trophic groups. (After Thiebaux and Dickie, 1993.)Biomass (g m -2 ) Mb (g).

Normalized biomass size-spectra in carbon units from several stations in the New England Seamounts Area (Northwest Atlantic). Size range:1.6×10 -9 to 1.33×10 3 μg C individual -1 (from bacteria to meso-zooplankton). Depth range:0-400 m. (After Quiñones et al.,2003.)

Normalized biomass size-spectra in carbon units from several stations in the New England Seamounts Area (Northwest Atlantic). Size range:1.6×10 -9 to 1.33×10 3 μg C individual -1 (from bacteria to meso-zooplankton). Depth range:0-400 m. (After Quiñones et al.,2003.)

A detailed analysis about constructing normalized (NBS) and un-normalized size spectra can be found in Blanco et al.(1994 1998).

On the other hand, Vidondo et al.(1997) have argued in favor of using the Pareto type II distribution, which is widely used in many disciplines to describe size distributions, for representing and modeling size spectra. To apply such an approach adequately, each particle should contribute one point to the Pareto plot and, therefore, all the information contained in the observations is used. The Pareto approximation is ideal for automatic sizing instruments, such as flow cytometers and electronic or laser particle counters. Although in theory it is possible to estimate the parameters of the underlying Pareto distribution from the NBS-spectra, this procedure is not recommended from a statistical standpoint(Vidondo et al., 1997). It is important to note that, in systems that are far from equilibrium, there may be size distributions that cannot be appropriately described by the Pareto nor the normalized biomass-size-spectrum model, such as multimodal distributions(Gasol et al., 1991 Havlicek and Carpenter,2001).

The size-spectrum approach is rooted in the well-accepted concepts of the pyramids of biomass and numbers (Cousins, 1980, 1985 Platt, 1985) and research in this field can be traced back to the first half of the century (e.g. Elton, 1927 Ghilarov, 1944). However, it is the work of Sheldon et al.(1972, 1973) that provided new impetus to the field by publishing a set of particle-size spectra from oceanic areas (for a historical perspective see Platt, 1985). Sheldon et al.(1972), based on his field observations, proposed the `linear biomass hypothesis', which states that in the pelagic system there is roughly the same biomass at all size classes. The regularities in pelagic size structure observed by Sheldon et al.(1972, 1973) and the fact that most aspects of energy and material flow of an organism are size dependent(Peters, 1983 Calder, 1984 Schmidt-Nielsen, 1984) led to the development of theoretical models to explain and quantify the regularities(e.g. Kerr, 1974 Sheldon et al., 1977 Platt and Denman, 1977, 1978 Silvert and Platt, 1978, 1980 Borgmann, 1982, 1983, 1987 Dickie et al., 1987 Boudreau and Dickie, 1989 Boudreau et al., 1991). The first theoretical models about the size structure of the pelagic ecosystem were proposed by Kerr (1974),Sheldon et al. (1977) and Platt and Denman (1977, 1978). Whereas the two first models were based on the trophic-level concept, the last stands on the consideration of a continuous flow of energy from small to large organisms. Kerr and Sheldon's models propose that biomass is constant when organisms are organized in logarithmic size classes. On the other side, Platt and Denman's model predicts a slight decrease of biomass with organism size with a slope of-0.22 and proposes an allometric structure for the pelagic ecosystem (Platt and Denman, 1977, 1978). Until now the most comprehensive biomass size spectra constructed in close to steady state systems (i.e. North pacific Central Gyre, Rodriguez and Mullin, 1986boligotrophic areas of the Northwest Atlantic, Quiñones et al., 2003)support the Platt and Denman's model. It is important to note that in Platt and Denman's model the exponent (-0.22) represents a balance between catabolism and anabolism and, consequently, from a scaling standpoint it is coherent with the recently proposed `metabolic theory of ecology'(Brown et al., 2004).


Libraries of S. cerevisiae and E. coli promoter reporters measured under different conditions reveal scaling relationships between expression profiles across conditions and suggest that most changes in activity are due to global effects.

  • Between any two conditions, the activity of most promoters changes by a constant global scaling factor that depends only on the conditions and not on the promoter's identity.
  • The value of the global scaling factor between any two conditions corresponds to the change in growth rate and magnitude of the condition-specific response.
  • When specific groups of genes are activated, they also tend to change according to scaling factors, changing the degree to which the entire group is activated, while preserving the ratios between genes within the group.
  • Altogether, a handful of scaling factors are sufficient for quantitatively describing genome-wide expression profiles across conditions.

Properly changing the time scale of a parameter in a growth model - Biology

As an object moves through the atmosphere, the gas molecules of the atmosphere near the object are disturbed and move around the object. Aerodynamic forces are generated between the gas and the object. The magnitude of these forces depend on the shape of the object, the speed of the object, the mass of the gas going by the object and on two other important properties of the gas the viscosity , or stickiness, of the gas and the compressibility , or springiness, of the gas. To properly model these effects, aerodynamicists use similarity parameters which are ratios of these effects to other forces present in the problem. If two experiments have the same values for the similarity parameters, then the relative importance of the forces are being correctly modeled. Representative values for the properties of air are given on another page, but the actual value of the parameter depends on the state of the gas and on the altitude.

Aerodynamic forces depend in a complex way on the viscosity of the gas. As an object moves through a gas, the gas molecules stick to the surface. This creates a layer of air near the surface, called a boundary layer, which, in effect, changes the shape of the object. The flow of gas reacts to the edge of the boundary layer as if it was the physical surface of the object. To make things more confusing, the boundary layer may separate from the body and create an effective shape much different from the physical shape. And to make it even more confusing, the flow conditions in and near the boundary layer are often unsteady (changing in time). The boundary layer is very important in determining the drag of an object. To determine and predict these conditions, aerodynamicists rely on wind tunnel testing and very sophisticated computer analysis.

The important similarity parameter for viscosity is the Reynolds number . The Reynolds number expresses the ratio of inertial (resistant to change or motion) forces to viscous (heavy and gluey) forces. From a detailed analysis of the momentum conservation equation, the inertial forces are characterized by the product of the density r times the velocity V times the gradient of the velocity dV/dx. The viscous forces are characterized by the dynamic viscosity coefficient mu times the second gradient of the velocity d^2V/dx^2. The Reynolds number Re then becomes:

Re = (r * V * dV/dx) / (mu * d^2V/dx^2)

The gradient of the velocity is proportional to the velocity divided by a length scale L. Similarly, the second derivative of the velocity is proportional to the velocity divided by the square of the length scale. Then:

The Reynolds number is a dimensionless number. High values of the parameter (on the order of 10 million) indicate that viscous forces are small and the flow is essentially inviscid. The Euler equations can then be used to model the flow. Low values of the parameter (on the order of 1 hundred) indicate that viscous forces must be considered.

The Reynolds number can be further simplified if we use the kinematic viscosity nu that is euqal to the dynamic viscosity divided by the density:

Here's a Java program to calculate the coefficient of viscosity and the Reynolds number for different altitude, length, and speed.

To change input values, click on the input box (black on white), backspace over the input value, type in your new value, and hit the Enter key on the keyboard (this sends your new value to the program). You will see the output boxes (yellow on black) change value. You can use either Imperial or Metric units and you can input either the Mach number or the speed by using the menu buttons. Just click on the menu button and click on your selection. The non-dimensional Mach number and Reynolds number are displayed in white on blue boxes. If you are an experienced user of this calculator, you can use a sleek version of the program which loads faster on your computer and does not include these instructions. You can also download your own copy of the program to run off-line by clicking on this button:

The Similarity Parameter Calculator was modified in May, 2009, by Anthony Vila, a student at Vanderbilt University, during a summer intern session at NASA Glenn.

For some problems we can divide the Reynolds by the length scale to obtain the Reynolds number per foot Ref. This is given by:

The Reynolds number per foot (or per meter) is obviously not a non-dimensional number like the Reynolds number. You can determine the Reynolds number per foot using the calculator by specifycing the length scale to be 1 foot.

Multi-scale modelling and simulation in systems biology

The aim of systems biology is to describe and understand biology at a global scale where biological functions are recognised as a result of complex mechanisms that happen at several scales, from the molecular to the ecosystem. Modelling and simulation are computational tools that are invaluable for description, prediction and understanding these mechanisms in a quantitative and integrative way. Therefore the study of biological functions is greatly aided by multi-scale methods that enable the coupling and simulation of models spanning several spatial and temporal scales. Various methods have been developed for solving multi-scale problems in many scientific disciplines, and are applicable to continuum based modelling techniques, in which the relationship between system properties is expressed with continuous mathematical equations or discrete modelling techniques that are based on individual units to model the heterogeneous microscopic elements such as individuals or cells. In this review, we survey these multi-scale methods and explore their application in systems biology.