Questions & answers 2007

Go to the collection 2001 , 2003 . 2004 . 2005 .

The following questions have been put foreward by the participants of the 2007 course on DEB. The questions were formulated by the participants, after discussions in weekly meetings they also summarized the answers. The questions refer to the given section numbers of the DEB book 2000.

Go to chapters 0 1, 2, 3, 4, 5, 7, 8, 9, 10

Chapter 1: Energetics and models

Quest 1.1.3: At {5}, it is mentionned that "a typical bacteria cell has about 5x106 ATP molecules". In the quizz of chapter 1, question 5 concerns the small role of ATP in DEB theory. The second item is: "Because ATP stores little energy only." I understood that the main reason of the small interest of ATP in DEB is because of the large turnover, but I thought that ATP can contain a large amount of energy, thus I would say that the previous item is false. However, the given "answer" is: True, the number of ATP molecules per cell is small. Can we consider that 5x106 ATP molecules is a small number?

Answ: All these quantifiers are relative. The total amount energy in ATP (energy per molecule times the number of molecules) is small because it fuels only 2 seconds of synthesis, while it typically takes a full day of synthesis to double the mass of a cell. So at the organisation level of the individual ATP cannot play an important role. At lower organisation levels, ATP is obviously very important.

Quest 1.2.1: In ecology, we can also define qualitative criteria for model validation if the theory behind the model is consistent. Bifurcation analysis, for instance, can give some qualitative ideas that can be tested experimentally in some cases. See for instance: Lutz Becks, Frank M. Hilker, Horst Malchow, Klaus Jurgens & Hartmut Arndt, 2005, 'Experimental demonstration of chaos in a microbial food web', Vol 435|30 June 2005 |doi:10.1038/nature03627

Answ: As long as validation does not literally mean that a model is "true" after validation, this seems to be correct. A consistent model can still give a good fit to experimental data for the wrong reasons, which includes qualitative properties. Apart from the goodnoess of fit, we can look at the parameter values that we obtain. Do they have a sound meaning? If the fit is good for the wrong reasons, some parameter values might be unrealistic. The term "giving support" , rather than "validation", better reflects the meaning of a good fit with reasonable parameter values.

Quest 1.2.3: As far as I understand, dimensions are when we can give the units. In the paragraph on the allometries, it is written that: "this function has three rather than two parameters [which are they?]. It can be reduced to two parameters for dimensionless variables only [why is that?]". I do not understand the next three sentences.

Answ:The parameters in the function y(x) = y(x1) (x/ x1)beta are x1, y(x1) and beta; x is the independent variable and y is the dependent variable. Variables typically vary in value, but parameters typically have fixed values.

There is a difference between dimension and unit. A variable can have dimension "time", but units "second", "day", "year", etc. You can choose a unit, but a dimension you cannot choose, it is given by the physical interpretation of the variable. The course document on methods in theoretical biology has more info on this; in the chapter on statistics you will find info on units and strategies to choose units, and how this interacts with numerical errors in computations.

Back to the allometric function: x and x1 must have the same dimension to avoid dimension problems, so we are able to define a new variable z = x/x1, which is dimensionless. Now we can rewrite the y as function of z as y(z) = y(1) zbeta, and this function has only two parameters, namely y(1) and beta.

When we would plot log(y) against log(z), a straight line results. If we have a set of data points, we can fit a line, and extract two parameters. No problem. When we plot log(y) against log(x), again a straight line results, and from fitting a line to a set of data 2 parameters can be extracted, but our function has 3 parameters: so we have a problem in the case that no other arguments are available to fix the third parameter (namely x1). The data contain no information about the value of this parameter. We can give x1 an arbitrary value, but the value of another parameter (namely y(x1)) depends on the arbitrary choice. So this value has no scientific meaning.

Chapter 2: Basic concepts

Quest 2.1.1: At {17} and throughout the book the author is referred to "supply" and "demand" systems. Is there an explicit definition of those systems? If yes, this is related to feeding or/and to allocation rules?

Answ: Since the DEB theory is supposed to be applicable to all organisms, both supply and demand systems follow the same rules within the DEB theory. An explanation might be that demand systems evolved from supply systems, froze the existing metabolic rules, lost metabolic flexibility (to deal with extreme starvation conditions), but increased in behavioural flexibility. Demand organism are often mobile and move to there were the food is and this food typically consists of other organisms. Hence they encounter less frequently extreme starvation conditions; they typically cannot shrink during starvation, but die. The increased behavioural flexibility gives them the possibility to specialize on one type of food species and translates in a small value for the half saturation coefficient for demand systems. They also have a relatively large difference between the peak and the standard metabolic rate, and have typically closed circulation systems (efficient transport under extreme metabolic performance), some developed endothermy (birds & mammals) and many have higly developed sensors. Supply organisms typically move less and find their food via a kind of (activated) diffusion process. They can better deal with starvation (shrinking). They have less developed sensors and are metabolically more flexible Especially those that do not live of other organisms typically have a number of reserves equal to the number of complementary resources. By far the majority of species are supply systems, but the few demand systems got relatively more research attention (because they resemble humans better and humans like humans; we are a social demand species).

Quest 2.2.2: How does DEB theory deal with diatom reproduction/growth? Diatoms are among the most abundant primary producers on earth and they are at the base of the major oceanic food webs. So, they don't necessarily have a "most unique life cycle". Odd, perhaps, but very common, considering the thousands of diatoms species.

Usually, diatoms reproduce vegetatively by cell division. The cell is made of two asymmetric valves in size (see figure). During the life cycle of the diatom the cell size gradually shrinks as each valve produces a smaller complementary valve. When the smaller valves have completely formed the two cells split. But because one side of the diatom is smaller when they split one is the same size as the original, while one is smaller. When they shrink to a certain size they have to produce sexually. To do this they create a auxospore which then will become a diatom. Does this poses a problem to DEB model or any kind of constraint?

Answ: DEB theory aims to be a fundamental theory that applies to all life on earth, so the fact that any particular taxon is ecologically important or not should not matter. The way DEB theory handles these "details" of diatoms is via the shape correction function. Since diatoms do not increase the surface area that is involved in resource/nutrient uptake from the environment (outer membrane, chloroplast) during the cell cycle, diatoms classify as V0-morphs. The constant surface area becomes a parameter and parameter values are individual-specific in DEB theory; we need differences in parameter values among individuals to allow for evolutionary change. Over the generations the (constant) surface area decreases in one of the two daughter cells. The switch to sexual reproduction counts as a change in stage; suicide reproduction is discussed in chapter 7. Maturity is typically reset at division, but when the cell size steps down, maturity is possibly not fully reset; a careful analysis of how resource levels affect size at division and at the switch to sexual reproduction should tell this. The valves are made of silica, which makes silica an essential nutrient for diatoms. Diatoms are also special in having a huge vacuole (for storage of nutrients); its contents does not require maintenance, which gives them a large surface area for the mass that they have to maintain. This allows them to grow fast, and explains why they are typically the first to bloom each season. Chapter 9 discusses why many population dynamical aspects of organisms that divide into two parts hold approximately if they are treated as V1-morphs, despite deviations from this pattern at the level of individuals.

Quest 2.3.1: Is the difference between weak and strong homeostasts that

  1. in strong homeostasis we consider the lack of changes in composition of structural mass AND reserves INDEPENDENTLY and in weak homeostasis they BOTH do not change, or
  2. in strong homeostasis we consider the structural mass and reserves only while in weak homeostasis we take the whole organism into account?
If 2 is true, what is MORE in an organism than the structure and reserves? I do not understand WHY strong homeostasis has nothing to do with reserve dynamics while the weak homeostasis does.

Answ: Stong homeostasis applies to stuctural mass and reserves separetely while weak homeostasis applies to the whole organism (stuctural mass + reserves), but only if the environment remains constant long enough. In other words, in strong homeostasis stuctural mass does not change in composition and reserves do not change in composition. In weak homeostasis, stuctural mass + reserves do not change in composition, while their amounts can change; reserve and structure grow in "harmony". This is why weak homeostasis imposes constraints on how reserve can vary relative to structure; strong homeostasis has no such implications. If an organism is exposed to a constant environment, it might take a time to adapt its reserve to this (new) environment.

At {83} the reserve dynamics are partly derived from the assumption of weak homeostasis. SousDomi2007 shows a stronger result and simpler result (see comments), namely that reserve dynamics follows from weak homeostasis directly (without further assumptions). In KooyTroo2007 you can find new ideas about the evolutionary origins and some mechanisms behind various forms of homeostasis.

Apart from the biological realism it is entertaining to think about the problem of model identification (how are we going to use measurements to test model predictions). Since it is not possible to tell which molecule in an organism belongs to which pool (reserve or structure), we need indirect methods to access the amount of reserve and structure and their chemical composition. In chap 4 you will see how the concept of weak homestasis is used to solve this problem. Suppose now that weak homeostasis does not apply. How are we going to quantify reserve and structure and their composition? KooySous2007 deals with this problem in more detail, but it is better to delay reading this paper till after chap 4.

Quest 2.3.2: In Fig. 2.5 it would have been more convincing if weights and lengths of the same animals was shown. In this way it is difficult (or even impossible) to compare. Furthermore, it would have been nice if the author also showed that the parameters resulting from both fits (scaled size at birth and rB) are the same (or very similar), so that it is clear that it does not matter what size measure you take.

Answ: If I remember the paper correctly, the data refer to the same group of individuals. The difference in scatter is convincing for me, at least. This figure, like almost all other figures has an m-file in DEBtool, where the parameter values are given. When you look into this file you can exactly see how the result is produced. The software also allow you to fit two von Bertalanffy curves simultaneously with the same von Bertalanffy growth rate.

A general remark that applies to almost all tests against data from the literature: the people who produced the data had very different ideas in their mind. What we really need is people who setup experiments with the aim of testing DEB theory.

Quest 2.3.2: One of the state variables of the DEB model is the structural volume, which is linked to the length via the shape parameter. A way to know the dry flesh weight of an animal is then to measure the structural dry weight percentage and apply it to the structural volume (with the assumption that the density is equal to 1 g/cm^3). But, some animals (like molluscs) seem to have an oscillation in their dry weight percentage according to the season. How does the DEB model could explain this?

Answ: The oscillation you see in the (total) dry weight percentage is, according to DEB, due to the relative contributions of V, E and ER to the total (dry) weight. Especially the reproduction buffer ER typically follows a seasonal cycle, linked to species-specific buffer handling rules.

Quest 2.3.4: At {35} the author states that "the chemical composition of biomass becomes increasingly flexible with the number of delineated reserves, and depends on the nutritional conditions of the environment". We assume that this referrs to variable environments where weak homeostasis cannot be applied. If the environment were constant the reserve density would be constant and we would have constant chemical composition of the total biomass, so no flexibility. It is not clear to us why all restrictions (which are they?) in the composition of total biomass disappear if the number of reserves exceeds the number of chemical elements included minus one.

Answ: Imagine the following exercise. Suppose a real organism is composed of n chemical compounds. We can setup an n-dimensional vector-space for the chemical composition of biomass, with one axis for each chemical compound in mol/C-mol or C-mol/C-mol depending on the chemical compound containing carbon or not. The number of dimensions is very large indeed, depending on details on how we define a chemical species, but suppose it is some finite number.

When we delineate 0 reserves and 1 structure, the theoretical organism can be represented as a point in this vector-space, and its position depends on a set of n composition-parameters for structure. If we now delineate 1 reserve, this reserve can also be represented by a single point in this vector space. By manipulating food levels and plotting the resulting composition of biomass in our vector space, all compositions are on a segment of a line through the two points for structure and reserve. The two points are outside the segment, because by manipulating food densities it is impossible to create a (theoretical) organism that fully consists of reserve or of structure. In chap 3 we will see, however, that a starting embryo consists of almost pure reserve; and in chap 7 that we can approximate a "pure structure" situation under extreme starvation conditions. So the two points are almost at the edges of our line-segment. If we delineate 2 reserves and one structure, and manipulate the resource levels, we can fill an area on a surface. We will see in chap 5 that by damming up, we can built up huge amounts of each reserve. So even for unicellulars (for which multiple reserves are realistic, but which do not have an embryo-stage) we can approximate an almost "pure reserve" situation. If we continue to increase the number of reserves, to 3, to 4 etc to n, and manipulate the resource levels, we increase the volume of possible biomass compositions and with n-1 reserves we can fill part of an n-dimimensional composition-space. The final step in our exercise is to choose the compositional parameters far apart so that we can fill up the whole space, within realistic boundaries.

Quest 2.7: It is said at {62} that energy budgets in males and females are different (which is obviously true !), and that this is accounted for in DEB models via differing parameters values for males and females (so that differences among sexes are treated the same way that differences among species). In this context, how can we handle the problem of alternate sexes in crossed-fertilized hermaphroditic species? For example, the snail Lymnaea stagnalis plays either the role of the male or the role of the female, depending on the relative density of "male-motivated" snails in the population. Therefore, it regularly switches from male to female behaviour during its life-cycle, probably inducing changes in energy balances (since males are generally believed to allocate less energy in reproduction then females, see p 62). Some authors have claimed that the number of reproduction as a male tended to be equal to the number of reproduction as a female, so that every individual had invested a similar total amount of energy in reproduction at the end of its life. Is this sufficient to skip the problem of fluctuating sex-role in Lynmaea stagnalis? If not, could we imagine using fluctuating parameter values in one species or should we consider two species (as well we would in seperate-sexes species)?

Deriving from the same idea, how can we conceptually deal with selfing? I guess this can easily be solved by using similar parameter values for every individuals that compose the population. But what about alternate selfing ? Would we consider selfing and non selfing organisms as differents species as well ?

Answ: For species with well-defined sexes, and size dimorphy between the sexes, it is clear that the parameters need to be different. For the examples that you raise, this is less clear. One of the simplest first steps would be to investigate the growth curves: do you see changes related to the type of sexual behaviour? Changes in allocation parameters often show up in the growth curve, so that would a good place to start looking.

The nematode C. elegans is a selfing hermaphrodite that can also reproduce sexually. It turned out that the resource allocation is very similar when the organisms are selfing or sexually reproducing, but the reproduction behaviour is different (sexually reproducing worms start reproducing at a smaller size, because they do not have to produce sperm cells themselves). Experiments by Andries ter Maat and co-workers, who blocked the female or male functions in the pond snail, indicated that both functions are about equally costly from an energetic point of view. We expected that the female function would have been much more costly (because of the eggs), but this was not confirmed by experimental results. Concerning, mollusks, and especially oysters (protandrous hermaphrodite - no dimorphism), Stephane Pouvreau and co-workers never saw any differences between males and females among oysters when measuring ingestion rate, respiration rate, growth rate... A female or male oyster seems to have the same energy fluxes. Moreover, the volume of their gonad is the same. The only difference is in the number of gametes: in million for oocytes / in billion for spermatozoa..... but a spermatozoa has a diameter around 2-3 µm whereas it is around 50 µm for an oocyte... So, on a bioenergetic point of view, we conclude that a female = a male (for oysters...), and the standard DEB model for oysters has the same parameters for male and female (J. Sea Res. (2006) 56: issue 2).

Chapter 3: Energy acquisition and use

Quest 3: I fully agree on the fact that maintenance costs have to be paid first, it seems quite "natural". Then why not to substract the maintenance costs to the total amount of energy available before partitioning what's left between growth and reproduction (i.e. the maintenance costs are exluded from the kappa-rule), which is the "net production budget" proposed by Lika and Nisbet, 2000?

Answ: "Net production" models subtract maintenance from assimilation; Chap 10 compares this construct with the DEB formulation. You seem to suggest, however, to subtract maintenance from the catabolic flux (i.e. mobilized reserve). It would be easier to include the embryo stage this way; embryo's have zero assimilation. Simple rules for allocation to growth versus development or reproduction (such as a constant fraction allocation), directly links growth to reproduction; a pattern that is not widely observed. In many species reproduction is maximal when growth is caesed, which lead many workers to the conclusion that growth competes with reproduction.

Any hierarchic partition rule has far-reaching and frequently complex implications. The book is about the allocation strategy that survived my own critisism; a great many alternatives went into the waste box. Somatic maintenance does not have priority over everything, only over somatic growth (Chap 7 discusses extreme starvation conditions, and especially how maintenance is paid in such situations). This allocation rule gives you size control (= the reason why there is a maximum size). If you analyse alternatives, size control is certainly one of the many aspects that you have to keep in mind. It is the basis of the co-variation of parameter values across species in DEB theory (which is discussed in Chap 8); if you choose for an alternative allocation, this is another aspect you have to deal with and explain, for instance, why respiration is more or less proportional to weight to the power 3/4.

These remarks are just meant to illustrate some of the implications that require consideration. It would be nice to have a problem-free alternative for the DEB theory and to test it against data. All alternatives that I know suffer from substantial problems. Having considered a large number of alternatives, it would surprise me very much if, after all, a problem-free alternative would exist with a similar level of simplicity. Problem-free alternatives of substantially higher complexity in terms of required number of variables and parameters might exist. Such models are much more difficult to test against experimental data, however.

Quest 3.1: The feeding rate depends on the body mass, food availability and temperature in DEB theory. But apart from them, there is also an effect of the food, namely the food nutritional quality. Does it make sense and, if does, would it be possible to include this factor into the DEB models?

Answ: Chap 3 discusses food uptake given the food quality of a single type of food; it then depends on the structural surface area (not body mass in general because that includes reserve, and DEB theory assumes that food uptake is independent of reserve), food availability and temperature. Food quality is taken into account via the parameter values. The reason why a cow eats such a large amount of grass (in terms of energy consumption) is because of lack of proteins (especially if the cow is milked). Chap 5 discusses the effects of food quality, and several type of food.

Quest 3.1: We can read that "selection is rarely found [...] in mussels" {77} but what about the size-selectivity (between 2 and 50 µm for Mytilus edulis) and a quality selectivity with the pseudo-faeces production (which is discussed in a recent article? If food selectivity seems not to present a real cost (evaluated at 0.92% of the net energy intake for Mytilus trossulus; Arifin et al., 2001) it can affect the functional response? A lot of papers exposed an important variation of the uptake rate as a function of the food quality (e.g. for M. edulis; according to the PIM/POM ratio: Thompson & Bayne, 1972-1974, Bayne & Warrall, 1980 [...], or the diet : Kreeger et al., 2001 [...]). How does it affect the {Pam}/{Pxm} ratio?

Answ: Selection for food particles (e.g. on the basis of size and quality) affects the half-saturation coefficient, so a parameter. This is worked out in more detail in Kooy2006. Food quality affects the specific max uptake rate {JXAmEX, so two other parameters. Chap 4 discusses mass fluxes, and chap 5 extensions to more types of food. The remark in the book that selection is rarely found is mussels was meant to apply given that the various types enters the gut (so living bacteria versus dead POM and "algae"); the relative abundance of the various food types in the first section of the gut resembles that in the water. This remark should have been formulated more carefully.

Quest 3.1: The beetle I investigate lives in wood for the whole larval stage. This environment is abundant in food all year round and does not demand high costs of locomotion. But, on the other hand, it is extremely poor in nutrients. Thus, the limiting factor of food intake in these beetles could be the feeding rate or (and) digestion. The beetle larva needs to eat much to provide itself with sufficient amounts of nutrients, e.g. nitrogen. Its chitinised mouthparts do not grow proportionally to the rest of the body within a larval stadium, therefore limiting the food intake.

Might the interspecies number of moults in insect larvae be inversely correlated with food quality. It is possible to construct a model to verify this?

Answ: All Ecdysozoa (including Arthropods and nematods) have the problem of moulting. The connection between the size of the teeth and the feeding or digestion rate is perhaps less direct. In terms of the surface area of the gut the situation is even more complex, e.g. by the movements of the gut inside the individual that interfere with the effective surface area of the gut. All these details go lost in the whole picture. The exoskeleton probably does not limit energetics; on the contrary: As soon as it no longer "works well" it is replaced by a new (and bigger) exoskeleton. The details are very much taxon-specific. Copepods have a fixed number of moults (like many insect); daphnids have an unlimited number of moults. A fixed number of moults does imply that the individual looses its ability to grow in physical dimensions in the last moult; it still can grow (somewhat) in mass (dry weight), which has more direct relationships with maintenance requirements.

It would be nice to compare the species of Ecdysozoa on the basis of moulting strategies and relate it to their budgets.

Quest 3.1: Another important factor that affects the feeding rate is interference between individuals. Why is this not discussed in 3.1? If a copepod catches an algal cell on which another one is hunting, the second one looses time, which affects its ingestion rate. Figure 2.1 shows this clearly for fish, with great consequences for energy budgets.

Answ: Correct, interference is an important factor that should be taken into account. The strategy followed in the book is that Chap 3 keeps matters as simple as possible to reveal the basic principles. Extensions (which complicates matters) are discussed later. Chap 5 generalizes to more types of food, of reserves and of structures, Chap 6 discusses changing environments, adaptations and other sources of deviations from simple situations, and Chap 9 discusses population-level phenomena. Interference classifies in this context as a population-level phenomenon. Several types of interactions are discussed there, one of them is competitive interactions. The comments for {17} present a quantitative analysis of the type of interaction you indicated.

Quest 3.1.2: Do feeding costs include the costs of digestion as well? If not, then it is difficult for me to imagine what actually are these costs associated with in the "real life". If yes, then these costs could be inversely proportional to the nutritional quality of the food, but I cannot imagine how could one accommodate it nicely within the DEB models. Both approaches proposed in the chapter, the proportionality to feeding rate or body volume seem unrealistic to me.

Answ: Fouraging is part of the movement costs, so part of the activity, and is included in somatic maintenance. The costs for the conversion from food to reserve, however, is paid from food. So food has a dual role in providing both the energy and the building blocks for this transformation. As a result there must be overhead costs associated with assimilation, that generally result in a carbon dioxide production and a dioxygen (or other election acceptor) consumption. We will discuss this in more detail in chap 4.

Notice that in the conversion from reserve to stucture we have a very similar situation, where reserve has a dual role in providing both the enrgy and the building blocks for the transformation. So there again we have overhead costs and products that are released into the invironment.

The typical way to quantify the costs in practice, is via the dynamic mass balance at the level of the individual. KooySous2007 discusses this in more detail.

Quest 3.1.2: On {73}, the 3rd paragraph starts with the sentence "The energy required for walking or running is found to be proportional to velocity". From this we cannot generally conclude that energy costs of walking or running a certain distance are independent of speed and just proportional to distance. From physics we know that even in the simplest case where resistance is proportional to velocity, the energy costs to cover a certain distance is proportional to the product of the velocity and the distance covered. We cannot generally conclude that energy costs of walking or running a certain distance are independent of speed and just proportional to distance.

Answ: It takes double the amount of energy (joules/time) to double your speed while walking (ref[311] {73}). That halves your time to do that specific distance. Thus time*joules/time = joules remains constant. Thus travelling cost are proportional to the distance travelled. The detailed physics behind movements is not simple at all; many references deal with this problem, sometimes in great detail. The situation for mosquito's, dinosaurs, fish and bacteria are rather different. I directly admit that the DEB book is very simplistic about this, possibly too simplistic. On the other hand I am sure that you don't want to have models with some 50 parameters for movement costs that take very different values if the subject walks uphill or downhill. Resistance is obviously important generally, but would it be important in the limited range of typical speeds? A cheeta running at 120 km/h will doubtlessly experience effects of resistenance, but it runs this fast only during very few minutes each week. Most of the time it has speed zero in the shade. Many applications of DEB theory require more detail in particular aspects. The first purpose of DEB theory is to find the simplest formulation that covers all species and all basic aspects of energy budgets, and start from there to add modules for particular applications.

Quest 3.2: Why is the energy taken to convert food to reserves not paid from reserves, but from food?

Answ: A detailed answer is long; this is one of the many possibilities I tried in the past. It gives al sorts of problem when food intake occurs after long starvation. You also need to assign a priority. I tried between maintenance and growth, but got very complex and unrealistic results.

Quest 3.3: Why does urea not appear in the general DEB scheme, whereas faeces appear? The only difference I know between urine and faeces from what I have red in chapter 3 is that faeces is part of the environment, whereas urine is part of the body. Ok with that but I hardly see the subsequent implications for energetics. Two example of questions on this topic:

It is said {127} that "nitrogen is excreted in the transformation of food to reserve". It is thus part of the side products of assimilation. But it is said p 81 that "urine is not directly derived from food". So where does the side products of assimilation "go" (urine or faeces or both ?) and where does urine comes from?

At {81} we read assimilated energy = energy contained in food- digestion costs- energy contained into faeces. It is explicitly said that the energy in urine is included in assimilated energy (where ?) "because it does not directly derive from foom and it is excreted " (what is it the opposite of faeces). I do not understand why this arguments justifies including urine in assimilated enrgy.

Answ: The reason why faeces as a product is discussed in chap 3 is because is has a very simple relationship with the budget, it is namely only linked to assimilation, and in chap 3 I tried to keep things as simple as possible. The situation for urea (and dioxygen and other products) is more complex because only part of the urea production is linked to assimilation; other parts are linked to maintenance (protein turnover) and growth (overhead costs for growth where part of the protein is used as fuel to extract energy). Chapter 4 discusses these matters in depth.

Chapter 10 compares DEB with static approaches, where urea is subtracted as losses from the assimilation. Very misleading in a DEB-context, because part of the urea-production is linked to production processes.

Quest 3.4: What does "reserves are at equilibrium" biologically mean and when (or at what conditions) does it happen? I can see what is a chemical equilibrium, but I hardly figure what is reserve equilibrium (equilibrium between what and what ?)

Answ: "Reserves in equilibrium" simply means the following. If you expose an individual to a constant environment is takes some time (both in reality, and in the simple abstract DEB world) to adapt the amount of reserve to the new situation. The weak homeostasis principle tells that the reserve density will settle to some constant value (despite possible continued growth), but it takes (or better: can take) some time to arrive at this value (depending on the species and the body size of the individual). To illustrate the concept in practice. Appetite (in humans) is (indirectly) linked to reserve density. Between meals we typically starve and start a new meal at a certain level of appetite. In adults this typically occurs some 3 times a day, but much more frequent in babies. Here you see the effect of body size on the response time of reserve. Babies equilibrate their reserve density faster than adults. This basic pattern is obviously a bit modified by cultural habits, but not to the extend that the pattern cannot be recognized.

Quest 3.4: Looking at the slide collection for chapter 3, I found something about polymer reserves. As far as I remember from chapter 2, polymer reserves are usullay bounded to cell membranes, so that they are not instantly available for the cell: only monomers are available for the catabolic power. Am I right? Then how does polymer/monomers dynamics work? What is this polymerization inhibition thing?

Answ: Good question! It took me some 25 years to come up with an answer that I liked. I wrote is out in the appendix of KooyTroo2007 on a possible mechanism behind the reserve dynamics of the standard DEB model, which includes ideas on monomerization and inhibition of this process.

Quest 3.4: I find this part (3.4) a difficult one so I just want to try to clearify some things in my mind. The second requirement for the reserve dynamics is that "the reserve density at steady state, [E]*, should not depend on structural body mass, V" and it's motivated by the weak homeostasis assumption. The weak homeostasis assumption is applied at equilibrium conditions. I can understand therefore that at equilibrium conditions the last term of the not-numbered equation at {83} vanishes and thus there is no dependence of V. However I cannot see clearly what happens when we have non-equilibrium conditions. I understood that in order to fill also the 3rd requirement, in non-equilibrium conditions this last term is somehow proven to be zero, so it vanishes again and there is no dependence of V. To find out how it is proven to be zero I also looked at the comments but when I arrive at the solution G([E]*,[E],V) = G*([E],V)/([E]*-[E]) I cannot see why this is independent of [E]* only if G*([E],V)=0. If there is a way to explain this last point I would be grateful. Because this way it seems to me as if we have a term that we don't want so we just put it zero to remove it.

Answ: Correct, this is probably the most difficult part of the book (but possibly also the most important one, because the use of reserves drives metabolism). SousDomi2007 found an improvement of the derivation, the is also simpler (see comments). The use of reserves is only a function of the amount of reserve and of structure. It has no "knowledge" about assimilation, and, therefore, this function must be the same for equilibrium as well as non-equilibrium situations. If we can show that a certain term in the use of reserve must disappear in equilibrium situations, it must always disappear.

The third term in the eq at {83} without a number must be zero because [E]* depends on assimilation, and the use of reserve should not depend on assimilation. This third term can only be equal to zero if G([E},V) = 0.

Quest 3.5: It is said at {87} that cells dispose of two types of information on their environment (size and energy density in the blood). On the other hand, it is well known that hormones acts as messengers that may inform cells on the state of others cells, and thus regulate energy use. Excluding hormones from the set of available informations seems not biologically realistic to me.

It is also said, {20} that the physiological role of hormones can be understood only by looking at other variables and compounds. I understand that dealing explicitly with hormones might be too complicated (even impossible....) and perharps not worth (?, depending on the context of the study), but I wonder how we can deal with it implicitely?

Anyway, I am quite surprised to see that hormonal regulations are accounted for neither in reserve dynamics, nor in the allocation rules: everything runs without "retrocontrol" from one physiological function on the other ones except energy partitioning? Is that realistic from a biological point of view, since hormonal control loops are important stabilizing mechanisms? A cell is allowed to divide when it senses a certain concentration of inhibitors and activators. These regulators are often secreted by a specialized clusters of cells. So there is a point source. Often inhibitors are secreted by the cells themselves. Thus only those cells near a growth signal excreting cells receive enough positive reinforcement to divide.

If the signal/hormone secreting cells can "sense" the energy/reserve densities in the organism then global information is available locally. These local cells can secrete information containing hormones, which are then available in the direct vicinity and on an intermediate level between local and global.

Programmed cell death removes the webbing between your fingers in a late embryonic stage. To me it seems that the use of hormones makes "where to grow" more controlled but not the "when"; with "where" being a limb or vertebrae or muscles-tissue.

Answ: Hormones do play an important, but implicit, role in DEB theory. They might stimulate the individual to grow, for instance, but if there is no material available for growth, the system does have a problem. If it grows too slowly, on the contrary, mobilized reserve (frequently consisting of monomers) build up, which also can give all sorts of problems. The organism must, therefore, grow at exactly the correct rate that just consumes all mobilized reserves that is allocated to growth; a demanding task for hormones. The problem of "where to grow" is only roughly dealt with in the surface area to volume relationships (e.g. isomorphy). DEB theory does not deal with the lower levels of organisation explicitly, but can be (in principle) be extended with modules that do.

All species on earth suffer from this problem (and many related problems), but comparing different species (animals, plants, etc) it is clear that they use very different homones for solving very similar problems. So there is hardly any generality in chemically explicit models. Since DEB theory aims at generality (successful or not), it needs to be chemically (and biologically) implicit.

Quest 3.5: Near the end of p. 87, the book says "Once in a somatic cell, energy is first used for maintenance, the rest is used for growth. This makes maintenance and growth compete directly, while development and reproduction compete with growth plus maintenance at a higher level." I don't really get where the competition takes place between maintenance and growth. To me, it reads "Whenever energy is available, first maintenance needs are fulfilled. Then, whatever energy is left can be used for growth."

Answ: Multicellular organisms, and especially animals, typically have a specialization of tissues and organs, and only few of them are involved in reproduction. The allocation to these two types of tissues/organs must be made at a high level in the allocation decision tree while each somatic cell needs to allocate to somatic maintenance and growth. This allocation is, therefore, at a lower point in the decision tree. The term "competition" is used here to indicate that the sum of the reserve fluxes allocated to somatic maintenance and growth is given at any point in time, so all what is going to somatic maintenance is not going to growth; the outcome of this competition does not affect the size of the total flux.

Quest 3.5: What is the mechanism behind the kappa-rule? Does it only apply for organisms that use gametes for reproduction? Section 3.9.2 discusses dividing organisms; they divide when maturity reaches threshold. For a special choice of maturity maintenance, this means division at a fixed size. If division is at a fixed size, and there is no reproduction through gametes, it seems that there is really no need for the kappa rule; the organism may follow that rule, but it is of little interest to the investigator.

Answ: Section 3.5 discusses a mechanism behind the kappa-rule. Please note that the reserve dynamics (3.4) excludes that kappa can be a function of reserve; it can still be a function of structure. LikaKooy2003 discusses bang-bang strategy (first grow, then reproduce; this strategy is still possible within the DEB theory) and compares it with indeterminate growth. The kappa-rule is supposed to apply to all organisms, and for a lot of work with organisms that only have juvenile stages, the value of kappa, and so in fact the kappa-rule itself, is hardly relevant within the context of the DEB theory. Notice, however, that this does not necessarily apply for alternatives for DEB allocations rules; it is an attractive property of DEB theory.

Quest 3.5: At {87}, I read that under poor conditions "allocation to reproduction is blocked". Is this really in agreement with the kappa-rule? Bivalves use energy that is allocated to reproduction to cover maintenance during poor conditions. "Reproductive storage" (e.g. glycogen in bivalves) can be considered like an another "reserve" with a "smooth dynamics", able to reduce environment fluctuations at a larger temporal scale than reserve can. Gametes and/or gonads can be resorbed (either because of "poor conditions" (e.g. in bivalves), lack of mating (e.g. in insects) or because gonad constrution/resorption is cyclic in adults (e.g. in oligochaetes).

Answ: Chap 7 discusses starvation conditions. This chapter explains why, under extreme starvation we should expect deviations from the kappa-rule, when structure is used for maintenance (shrinking) and second maintenance parameter plays a role. The maintenance process has an energy (work) and material (for the recycling of protein e.g.) aspect. Since the composition of reserve and structure can differ, these maintenance parameters must be different. Caroline Tolla made a detailed study on this (Toll2006). In animals, where we typically have a reproduction buffer, the preference for paying maintenance costs is (according to DEB theory) from reserve, then from the reproduction buffer, and finally from structure. Yes, the reproduction buffer can be used to replace the reserve, but we don't have rules for the use of this buffer, other than "if necessary".

Quest 3.6: When reading the section about maintenance, I realised that lots of processes are responsible of maintenance costs. In the beginnig I thought that maintenance could be related to something like respiration but I realised I was wrong. We have to distinguish somatic maintenance from maintenance linked to growth for example. The model that I use for bacteria, for instance, is really easy and comprises only one maintenance term. My question is, how can we separate the respiration due to the somatic maintenance (it's something like a basal respiration) and the respiration due to the growth for example, which also is a maintenance cost?

Answ: Respiration is a term that is typically used in a sloppy way in the literature. It can stand for the use of dioxygen, the production of carbon dioxide and the heat production. These fluxes are not all proportional to each other. DEB theory has no concept "maintenance for growth", but is has overhead costs for growth, and growth does contribute to all fluxes that are indentified in the three definitions of respiration. How the contribution from assimilation, maintenance and growth to respiration can be quantified and estimated from data is discussed in chap 4.

Quest 3.6.2: What could be the advantage of endothermy in the DEB context? Could it be associated with an increased growth rate? Keeping your body temperature up keeps the growth rate up, since endotherms are demand systems and if they don't grow at first they cannot make up for this lack of growth later. Or are endotherms demand systems BECAUSE they keep their growth rate steady?

Answ: Endothermy represents a further step in homeostasis, where now not only the chemical composition is constant, but also the temperature. Given a constant local environment for enzymes, their properties might be further optimised (on an evolutionary time scale) for that environment, which reduces maintenance costs. By having a constant high body temperature, the (max) searching and feeding rates are high which helps to find ectothermic prey that are less active under cool conditions.

Quest 3.7: The investment ratio given on {94}, shouldn't it involve the overhead costs also, or is that incorporated in the "costs of the new biovolume"?

Answ: The energy inverment ratio is a compound parameter. Depending on the physicial frame that you use, it contains the primary parameter [E_G] (energy cost for structure in the time-energy-length frame) or y_VE (mass cost for structure in the time-mass-length frame). These primary parameters include the overhead costs of structure.

Quest 3.7: Does the heating volume stands for the reduction in volume endotherms experience due to the energy costs of heating. Why does the individual reduce in volume because of heating, but not for payment of maintenance, e.g.? Is it associated with the loss of water? And what emperical data supports this statement?

Answ: The heating costs are in fact part of the somatic maintenance costs; the only difference with more typical maintenance costs is that it is proportional to structural surface area, rather than to structural volume. The term "reduction" does not imply shrinking, but just refers to the ultimate size being less than that of an ectothermic individual with otherwise the same parameter values.

Quest 3.7: I wonder how to estimate growth efficiency with a DEB model. When we use an empirical model, such as the Monod one (Michaelis-Menten kinetics), growth efficiency is simply the constant dV/dX where V is the structural volume and X the food density, as the model considers that all the assimilated food is used for the growth. There is no maintenance and no reserve. But when we work with a DEB model, do we have to include reserve in the growth efficiency estimate? I think yes if we consider that the reserve are comprised in the total volume of the organism, but I'm not sure. This will give something like: growth efficiency = (dV+ p.d e)/dX where p is the proportionality factor between the reserve density e and the total volume of the organism. And what happen if we have several substrates and thus several reserves compartments?

Answ: Growth efficieny is more complex in the DEB context, compared to the standard literature, and several different choices are possible; chap 9 discusses some choices. One such a choice is C-mol of biomass ( = structure + reserve) per C-mol of substrate at steady state. Such an efficiency is a function of the growth rate. If reserve would nut be considered, it is still a function of the growth rate, because of maintenance. If you exclude maintenance as well you arrive at a rather strange efficiency measure that microbiologists call "true yield" (strange enough, because this is not the yield that you acturally get). Another choice of an efficiency measure is C-mol of structure per C-mol of substrate. This choice makes the result perhaps better comparable with the literature. It would, however, be infinitely large for embryos because they grow, but assimilation is zero. They represent a transient state, however, not a steady state. This further illustrates that efficiency measures are less useful at the level of the individual.

The number of reserves that should be delineated should equal the number of independently varying complementary recourses, which is typically much less than the number of recourses. Growth efficiencies are typically substrate-specific, and if several substrates vary in abundance, their usefulness reduce.

Quest 3.7: On {96}, it is said that the max surf spec assimilation rate {pAm} relates to food-energy conversion. And the book gives the example of chickens that eat animals (high protein level) in the early juvenile period before becoming more herbivorous (same example with milk and mammals) to grow faster. Does this mean that the value of {pAm} can change along the life cycle or that {Jxm} changes according to food-energy conversion to keep {pAm} constant? Because if {pAm} is 'authorized' to change along the life cycle, that means for example, that the maximum length (Lm = kappa * {pm}/ [pM]) is not a constant for a same species?

Answ: Chapter 3 treats food in a symplistic way: just one type, constant quality, single size of particles, no size-dependent selection etc. In chapter 5 adds a bit more realism, and we will meet the concept substitutable compounds. Some changes in types of food don't translate in changes in budget. {pAm} is basic and typically not very dependent on food quality, because it quantifies the maximum processing capacity of the individual, and so the level of internal transport etc. The max spec feeding rate {JXAm} depends much more on food quality, and variations in the digestion efficiency {pAm}/{JXAm} are primarily caused by changes in {JXAm}. In fact this digestion efficiency is treated as a primary parameter in KooySous2007 (because it quantifies food quality best), from which {JXAm} follows, given {pAm}. The fact that Japanese people recently grow to larger sizes problably relates to changes in diet. The slide collection gives the example of house mice that switched to carnivory on an ocean island and grow to huges sizes (for mice). So diet related changes in {pAm} do occur.

Quest 3.7: Two elements seems to be distinguished in proteins turn over: synthesis (which costs belong to maintenance costs) and net synthesis (which belongs to growth costs). I do not catch the difference between synthesis and net synthesis, since it seems to me that both implie building new proteins from generalized compounds (so that the underlying biological mecanism is the same)?

Answ: Somatic maintenance has a catabolic and an anabolic aspect. The basic substrate for maintenance is structure, and reserves are just to compensate for the leaks (both the energy and the building block aspects). Growth also hase a catabolic and an anabolic aspects, but the substrate for growth is reserve (again both the energy and the building block aspects). The stoichiometric requirements for somatic maintenance and growth differ, even if we focus on the structure-recycling process, which is ony one of the somatic maintenance topics.

Quest 3.7: How can (compound) DEB parameters be obtained from feeding experiments with oyster larvae?

Answ: If you start from a small length, and asymptotic length is large, the von Bert curve is (initially) linear (in length versus time), with slope vB = rB Li. This can be seen from d/dt L = rB (Li - L) \simeq rB Li = vB. (I here introduce a new symbol vB with dimension length per time). This slope vB is measurable in larvae for various values of f so you can obtain rB Li = vB(f) = f Lm/(3/kM + 3 f Lm/v) = f/(3 g/v + 3 f/ v) = v f/(3 g + 3 f) see (3.22) at {95} as a function of f and can try to extract the compound parameters v and g that are in this expression. Notice that dim(v) = length/time, so it is sensitive to the shape of the larva. It is best to work with volumetric length.

Quest 3.7.1: I'm wondering how are calculated the 3 DEB parameters (conductance, maintenance rate and Eb/E0) in Table 3.1 ? Somebody has an idea ?

Answ: Fig 3.15 illustrates some cases of collected in Table 3.1 and DEBtool/fig/fig_3_15 exactly shows how you can obtain the values. If you have similar data: replace the data you find in this script-file by yours, run the script to get the results.

Quest 3.7.1: You would expect that the O2-consumption to go down in fig 3.15 for the soft-shelled turtle, since growth is obviously retarded. However, the curve and the data (!) seem to indicate that O2-consumption can drop to extremely low levels. I presume it will not become zero, or does it? There should be maintenance costs for the individual waiting in the egg to hatch? The peak in O2-consumption is after 60 days, but at that point, the left plot says that the turtle is still happily growing. For the crocodile at page 99, the timing seems to be better.

Answ: The curves on respiration, yolk and embryo weight are fitted simultationsly, see DEBtool/fig. This further illustrates that you need models to analyse data. Intuition is sometimes a very helpful first step, but then follows the analitical step.

Quest 3.8: Maturity does not involve mass and energy, since it is information. This information is "written" in the body volume. Stage transitions occur when the amount of energy that has been invested in maturity exceeds certain thresholds. The reproduction buffer starts to receive reserve after puberty (the stage transition fron juvenile to adult). This occurse at a fixed amount of stucture if [pJ] = [pM] (1 - kappa)/ kappa. If this condition does not apply, the amount of structure at puberty depends on food history within a species in variable environments. The maturity maintenance costs in adults are constant, because the maturation level is constant, even if the organism keeps growing. Is this correct?

Answ: Yes, this is basically correct. In retrospection I now think that, in my wish to simplify matters as much as possible, I was too hasty to introduce the special constraint on the maturity maintenance costs that gives stage transitions at fixed amounts of structure. With toxic compounds, parasites or light cycles (see 7.1.5 at {227}) we can change the maturity maintenance and deviate from this situation. Maturity maintenance should in general not be linked to size, but to the level of maturity. This theoretical elegance comes with the price of an explicit third state variable (namely maturity).

Quest 3.8: What is "level of maturity": a volume of gonad + a volume of gametes? What about sexual behaviour (e.g. building and maintenance of secondary sexual characters, movements to search and "seduce" a mate, food gifts, etc....). Since juveniles do not exhibit those costs, they should not be accounted for in the adult reproduction costs (since both are quantitatively equivalent due to the kappa-rule). Are they accounted for in the "general maintenace", as any other activity (e.g. feeding behaviour)? If yes, are they also proportional to body size?

Answ: The formal status of maturity (level) is information; it does not represent mass or energy. It costs work (which can be quantified as the cumulative investment of reserve) to build it up (and to maintain), but this effort cannot be recovered when the level of muturity decreases (e.g. as a result of parasitism, see comments).

Chapter 4 discusses the implications for mass flows. In 3.9.1 it is discussed why we need a buffer with reserve that is allocated to reproduction and a buffer handling rule that convert the content of the buffer into offspring. Since gonads have mass, they have no direct relationship with maturity (but they do have it with the reproduction buffer).

The increase of the level of maturity stops at puberty; this freezes the maturity maintenance costs. So during the adult state the maturity maintenance costs are no long proportional to their size; they typically remain constant. Behaviour, and so also sexual behaviour, is included in somatic maintenance, and typically comprosis a small fraction only. Somatic maintenance has a (leading) term that is proportional to structural volume and a term that is proportional to structural surface area. Remember that animals (for whom sexual behaviour is most eye-catching) are just one of the many groups of organisms that DEB theory aims to capture; this explains why it is not in the core theory, but in extension modules.

Quest 3.9.2: At {119}: "The interdivision time for Escherichia coli can be as sort as 20 minutes under optimal conditions, while it takes an hour to duplicate the DNA." I’m afraid that I cannot assimilate this sentence.

Answ: At a fixed threshold value of the maturity the signal "start duplication of DNA" is given, maturity is reset to zero, and an internal clock starts running. If DNA is duplicated, which takes a fixed amount of time, the cell is ready to separate into two daughter cells. Separation lags behind DNA duplication. Since bacteria typically have a single circular DNA molecule, the partially duplicated molecule is connected to the parent molecule at both ends; one end is running along the parent molecule (like a zipper), and is called the division fork. At high growth rates the parent molecule can show several division forks.

Quest 3.10: Chapter 3 is about lots of details, and only at the end in a summary it is shown how they fit together. Would a start with the summary not be easier to follow? Each of these hypothesis leads to a number of questions, that in my opinion, a "naive" reader who not think about "the same way" if beginning with section 3.10. I like the fact that underlying processes are explained and illustrated with biological examples BEFORE we get the summary of fundamentals. It gives us the approriate background to really think about the consequences about the hypothesis that are made.

Answ: In introduction of chapter 3, {65,66}, gives a broad overview of what is comming up, exactly to avoid the feeling of becomming submerged in "details". Tests against realism is by the "little" details. The purpose of these tests is not only to show that the assumptions make biological sense but also to illustrate how to translate abstract concepts to predictions for measurable quantities and some implications of these assumptions. Only later in the book you will see how much "details" are left out to reveal in chapter 3 a rather simple system that is shared by all organisms, while life is in fact quite complex.

If you like a formal approach to the same topic, with less details, SousDomi2007 is very nice reading.

Quest 3.10: What does disspating power at {124} stands for? Overhead costs of the other functions? Does it corresponds to the production costs of the excreted products (CO2, NH3...) which are generated from reserves due to the "work" of other catabolic powers? It is said that since the reserve composition is very similar in the mother and its eggs, kappaR is close to one. So the reproduction power hardly contributes to the dissipating power. This implies that reproduction does not have overhead costs

Answ: Dissipating power is the energy aspect of the collection of processes that use reserve as substrate and, after work has been done, completely ends up as products in the environment. It excludes overhead for growth, for instance, because part of the reserve that is allocated to growth is fixed in structure, so not all of this flux ends up as products in the environment. The term "dissipating power" is somewhat misleading because the power that dissipates is larger than the dissipating power (namely the overheads of assimilation and growth). The reason why it is handy to delineate dissipating power will become apparent in chapter 4. Because the reproduction process transforms reserve of the mother to reserve of the eggs, there is little chemical work, so problably also little use of dioxygen linked to this transformation, while it is (or can be) a substantial flux. This directly illustrates that the use of dioxygen (and so respiration) is not an ideal quantification for the reproduction process. Respiration serves a less central role in a DEB context, compared to the main literature on metabolism. Notice that reproduction uses reserve as substrate, and reserve has to be synthesized from food, so the conversion from food to offspring can have substantial overhead costs.

Chapter 4: Uptake and use of essential compounds

Quest 4.2: At {129} we can read : The overhead costs of the reproduction event are taken into account in the allocation to reproduction (I understand allocation as (1-kappa)pC-pJ), but on {114} we can read : (1-kappaR) dissipates and represents the overhead involved in the conversion from reserves energy of mother to the initial energy available to the embryo. Is it possible to distinguish reproduction reserves and gametes in the DEB context? Where does the cost kappaR need to be paid? Cardosa et al., 2006 based it on the gonado-somatic ratio, so on a mass ratio; larger is kappaR, larger kappa need to be to compensate the overhead costs!? How to measure kappaR? If reserves and reserves in the reproduction buffer are added together {129}, what about the gametes?

Answ: Gametes inside an organism are formally a type of tissue. In many species very small gamete-precursors are formed in an early stage, and later they ripe and become much larger and prepared for release. In other species, gametes are only formed just prior to release. It can also differ between sexes. This shows that, in a DEB context, gametes cannot simply be identified with the reproduction buffer. Material in this buffer can have its own distribution in the body. To what extend gametes are part of the reproduction buffer depends on the dynamics; in most species they can be considered as part of the reproduction buffer.

I am not quite sure where the most important overheads in reproduction are paid exactly, during the filling of the buffer or in the conversion of the buffer to the gametes. Probably both steps involve some overhead costs and in many situations the conversion of reserve to gametes will be the dominant source of overhead costs.

KappaR can be quantified (in principle) by measuring the carbon dioxide production in association with reproduction. To single out which part of the carbon dioxide production is linked to reproduction requires comparison is different individuals under controlled conditions. Duable, but not easy.

If kappaR is increased, so the overhead costs for reproduction is decreased, kappa so be increased as well to arrive at the same reproduction output. Kappa is in both cases used as the fraction of resources (here reserve) allocated to the system itself. In chapter 5 we will see a new fraction kappaE with the same notation-strategy. Reproduction is in this context seen as an allocation to other individuals (namely offspring); therefore reproduction is linked to 1 - kappa.

The reason for adding that reserve and the reserve buffer (gametes are in this combined pool) is to work out the mass balances. It is not a logical step for the dynamics of the systems "individual". Mass balances follow from the dynamics. Quest 4.3: What are reduction degrees, and why is it odd that the third row of the inverse of the matrix of chemical indices for the minerals has an interpretation in terms of reduction degrees?

Answ: The definition of the term reduction degrees is given in the glossary. The interpretation of that third row is remarkable, because neither the valences of the chemical elements, nor the electrical charges are used in DEB theory that leads to the result. Quest 4.3.4: The example at {134} gives an estimation of the composition of the reserves and of the structural mass. Is this possible only when working with chemostats (as all figures depend on the throughput rate)?

Answ: Just like for isomorphs at the individual level (see KooySous2007, for V1-morphs at the population level there are also several possibilities. They basically boil that you need info on composition changes under known changes in substrate availability (e.g. during starvation, or several throughput levels in steady-states of chemostats, or known resource cycles).

Quest 4.4.1: On {137}, it is shown that dioxygen consumption is proportionnal to 3 rB(Vi1/3V2/3-V)+ kMV

In van Haren and Kooijman (1993)-fig 8, it is shown that dioxygen consumption is proportional to L3 + L2v/kM. Are these two formulations equivalent?

Answ: Yes, substitution of Eq (3.23) learns that these two expressions are the same.

Quest 4.4.3: Page {141} says that the accumulated damage during the embryonic stage is negligibly small. Surely an embryo doesn't take a whole lot of damage, but whatever damage it does take is propagated throughout its life for lack of a backup copy of whatever was damaged. The high dilution rate through growth should make up for this, but basically any cell that gets hit early enough during development turns into a whole clump of "bad" cells later on. Maybe many embryos don't even make it to their birth this way, whereas their mothers do invest resources there. It just seems like a bit of a shortcut to just deny that damage is done to embryos...

Answ: The conclusion that damage is small during the juvenile stage is not an assumption. What is an assumption is that the transformation from the use of dioxygen to damage remains constant over the full life cycle. Wether or not a damaged cell can be replaced is very much dependent on the taxon. The assumption is also that cells that are hit by aging are taken out of the cell cycle, so aging does not amplify this way (according to DEB theory)

Quest 4.4.3: Organisms that propagate through reproduction have "resetted" embryo's, i.e. individuals that start without damage, and have thus not aged yet. Organisms that propagate through division suffer from the problem that both daughter cells get some of the damage of the parent cell. Are there any mechanisms that can reset individuals of these species? What damage-preventing or -repairing mechanisms exist in these species?

Answ: The idea is that at the cellular level cells are either hit by aging, and then taken out of the growth/division cycle, or they are not hit by aging. In a multicellular organism this translates into gradual aging (an increasing fraction of the cells is hit by aging), and the dynamics of damage production becomes relevant to describe the aging process.

Quest 4.4.3: Suppose that all damaged protiens are removed by the organism, which would increase the somatic maintenance cost, rather than the hazard rate. Would this yield similar results? As increasing the daily total O2 consumption increases again the rate of DNA damage occurrence. Which again leads to more proteins which needs to be removed, which costs reserve energy again. etc.

Answ: Although it is close to impossible to know for sure that any particular individual died from aging, quite a few well-fed individuals seem to die by aging. Your mechanism excludes this.

Quest 4.5.1: I don't understand why UQ and WQ depend on the dioxygen flux as for RQ. For RQ it is implicit for me because we link O2, which is taken up, to CO2, which is produced. So it is interesting to see if the flux of O2 consumed is equal to that of CO2 produced (in microbiology we often used RQ to go from O2 to CO2 fluxes if only one of this data set is available for example). But what is the interest to do this for the UQ and WQ?

Answ: I never found UQ and WQ in the literature and invented them myself. Like carbon dioxide, water and N-waste are products as well, and if the RQ for carbon dioxide is interesting, why are the UQ and WQ less interesting? My original motivation was in the sloppy use of the term "respiration" in the literature. It might mean "use of dioxygen", "production of carbon dioxide" or "heat production". Brown and co-workers, for instance, are never explicit in what they choose when they try to explain why "respiration" is propto weight3/4. They think of respiration as "the rate of living", whatever the exact meaning.

Indirect calorimetry states that heat production is a weighted sum of use of dioxygen, production of carbon dioxide and production of nitrogen waste (frequently ammonia in aquatic organisms). So here is a role for nitrogen waste. If all were proportional to each other, then things can be simplified, obviously, and e.g. Brown was is not need to be explicit in his choice, and people working with indirect calorimetry would be wasting their time.

My reason to introduce UQ and WQ is to show that simple ideas about these ratios, such as that they are constant, imply that the chemical composition of biomass cannot change in relation to food intake, and that indirect calorimetry can be simplified considerably. I think that the method of indirect calorimetry cannot be simplified, that complex changes in biomass do occur, with the implication that one should not assume that RQ is constant.

What you see frequently in animal physiology is that people are assuming that RQ is constant, but at the same time use free (= unconstrained) regression coefficients in indirect calorimetry and have no problems to demonstrate complex changes in chemical composition of biomass. You now know that inconsistencies can easily occur in data-driven research.

Chapter 5: Multivariate DEB models

Quest 5.1.1: Do the binding probabilities only account for the relative affinity of the substrates for the carrier (in which case the sum of the binding probabilities equals one) or does these probabilies also depend on criteria that are not intrinsic properties of the substrates?

Answ: Binding probabilities depend on the properties of the substrates as well as that of the SUs. These probabilities are absolute, not relative, so their sum can exceed one. In more advanced SU-formulations, that deal e.g. with inhibition and co-metabolism, the binding probabilities depend in the binding of other substrates to the SU (see KooyTroo2007).

Quest 5.1.1: The special case of equal handling times for two substrates is evoked at {161}. It seems to me that two componds exhibiting different composition or chemical structure could not have equal handling times. Could you please give me an exemple of two substrates that would have the same handling time? How can we identify such substrates ?

Answ: Although different substrates will generally have different binding probabilities, this is not necessarily so. Moreover SUs are used as generalized enzymes, with applications in e.g. animal feeding behaviour. One could think of cats feeding on mice and voles, for instance, or mussels feeding on diatoms and dinoflaggelates. In such practical applications the details of food composition are typically unknown. In any case, the special case served to illustrate model properties.

Quest 5.1.1: What is `a well fed prey'? A prey in which weak homeostasis would apply? Is weak homeostasis in prey assumed when deriving conversion equations (i.e. equations that account for the conversion of prey biomass into predator reserves)?

Answ: Weak homeostasis is a property of a model; it says that IF food density is constant long enough, reserve density remains constant despite of growth. A model has or don't has this property, and can also be applied to conditions in which food densities change (generally leading to changes in reserve density, also in models that have weak homeostasis). Well-fed prey means prey with a reserve density equal to its maximum.

Quest 5.1.1: Why is the predator ingestion rate related to prey structure rather than to prey reserves? It seems to me that the number of ingested preys depend on preys nutritive quality (satiety phenomenon), which directly depends on prey reserve rather than on prey structure (see the daphnia/algae example {161})

Answ: It is here assume that structure dominates size, especially length, and that prey selection is based on length, while reserve density is less easy to see from the outside. Satiation is not a state variable in the standard DEB model, but the model can obviously be extended to include such a state variable. The reserve density of the prey does affect the conversion of prey into reserve of the predator (digestion efficiency). Many extensions are possible, such as that prey activity is a function of its reserve, and prey selection is a function of the activity of the prey; such an extension requires also more detail in the specification of somatic maintenance requirements. The best choice for the level of detail depends on the application. Models can easely become too complex.

Quest 5.1.1: Which parameter need to be specified to describe the energetic status of a prey?

Answ: DEB theory applies to both predator and prey, with the implication that the prey has structure, reserve and (typically) a buffer with reserve allocated to reproduction. These are dynamically changing variables, not parameters. Digestion efficiency should deal with this complex situation, where prey's reserve(s) and structure can be substitutable and/or supplementory compounds for the predator.

Quest 5.1.1: It seems to me that the diet is typically composed both of substitutable AND supplementary substrates. Furthermore, a number of species feed on numerous substrates.

Therefore, I wonder how can we chose between substrates for those that should be accounted for in the model :

  1. focus on the "preferred substrates" = those which are consummed in largest quantities, on the hypothesis that the contribution of other substrates can be neglected?
  2. focus on the substrates that bring a particular nutrient which the organism is not able to synthesize from other substrates (and use this particular substrate as a tracer for the assimilation process)?
  3. Other strategies?
  4. in an ecotoxicological context, focus on the substrates that are potentially contaminated, thus allowing bioamplification throught the food chain?
I guess there is no general rule, but can we distinguish particular benefit and constraint when choosing between these strategies?

Furthermore, in case of supplementary substrates, could we focus on the more limiting substrate and neglect the other one (thus using a monovariate model based on the dynamic of the limiting substrate rather than a multivariate model based on both substrates?

Answ: DEB theory did not choose for the strategy of select just a few compounds and ignore the rest, but did choose for the concept of generalized compounds, which is linked to various forms of homeostasis. Reserve of animals, i.e. organisms that live of other organisms, typically consists of a rich mixture of proteins, carbohydrates and fats. Reserves of autotrophs typically consist of simple chemical compounds (e.g. nitrate, phospate, or starch). The dynamics of SUs that deal with complementary compounds shows that only in a very narrow window of concentrations of more than one compound matters; in most cases just a single compound dominates the transformation. This is not an assumption, but a model property. In many situations we are forced to rely on approximations.

The non-limiting nutrients, which are not taken into account explicitly, are still there, and some of them might represent the bulk of biomass; still important if you want to make mass balances. In terms of total mass the contributions of P and vit B12 are very small indeed, but if they limit transformations, the greatly affect the change in state, and then must be considered explicitly.

Quest 5.1.1: How does figure 5.1 indicates there are two uptake routes for glucose? Why is biomass density is zero for low values? they data don't indicate this!

Answ: Fig 4.8 in section 4.7 on fermentation shows the growth of yeasts on glucose under anearobic conditions, where only one glucose carrier is active. If you compare this with Fig 5.1 under aerobic conditions, you will see that biomass as a function of throughput rate is more complex. Biomass density is zero at throughput rate zero because of maintenance. If the half saturation constant is very small, the curve can become very steep. Although the data indeed do not indicate this, there is no reason to assume that yeasts have no maintenance. Experiments with chemostats at low throughput rates are very complex, and one has to wait a long time for equilibrium; small changes in (e.g.) throughput rate have a very large effect. All these problems directly reveal through theoretical analysis. Meanwhile molecular biology has confirmed the existence of more glucose cariers, and indeed one has a high affinity and a low capicity, and the other vice versa.

Quest 5.1.2: `These three supplementary nutrients’ are processed in parallel, {164}, which means that an increase in the abundance of one nutrient can increase the assimilation of other. Does this statement imply that there are cases where the increase in the abundance of one nutrient does not affect or decreases the assimilation of the other nutrients?

Answ: Parallel processing of supplementary compounds always has the property that the increase in arrival of one compound has a stimulating effect on the processing of the other compounds. We developed several variations on this (co-metabolism, adaptation, mixtures between supplementary and substitutable properties). A recent review is given in the appendix of KooyTroo2007, with examples.

Quest 5.1.3: The remark that oxygen has "counterproductive" effects is somewhat confusing. Apparently 2P-glycolate is not a desired product for the algae.

Answ: The remark relates to the minus-sign in the numerator of Eq (5.11) and to the existence of a compensation point (mentioned two lines later).

Quest 5.1.3: Are equations (5.7) and (5.8) valid for small values of the flux ratios z_L1 and z_L2?

Answ: Correct; this constraint can be removed within the general underlying theory, but involves more parameters. The functionality of these extra parameters is very limited, given the limitations imposed by carbon dioxide and nitrogen. Even with these simplifications, the result is already rather complex, depending on the aim of your research.

Quest 5.1.3: How the DEB model can offer quantitative explanations for specific effects of photosynthesis such as photoinhibition?

Answ: The general answer is that light can effect parameter values, just like all chemical compounds can. Below a compound-specific and DEB-parameter-specific no-effect concentration a particualr chemical compound does not have an effect on that parameter, but if the compound exceeds that level, the parameter changes. As long as we are interested in small changes, we can quantify the effects linearly. The same method can be applied to the photon flux. Cor Zonneveld (see Zonn98) published on photo-inhibition, but that was before the invention of the SU's. This work should be reformulated in the present formulation of the DEB theory.

Quest 5.2: In Fig.5.3 {169} I don't really understand the feed-back flux from growth-SU to reserves. It seems to be explained by a "stoichiometric constraints" (?) but this doesn't help me more!

Answ: In the (parallel) processing of two supplementary compounds, at least one, but generally both compounds are partly rejected. The question is what happens to these rejected fluxes? The DEB theory states: a fixed fraction of these rejected fluxes is fed back to the reserve from which the arrival flux originates, and the rest is (actively) excreted into the environment.

Quest 5.2.1: In fig 5.4 and associated exemple, `cell content on P and Vit B12 have been measured rather than reserves': that is P and Vit B12 were used as a proxy for the global reserves, right? I do not understood why we are allowed to do that, so that I begin to have doubts on `what are reserves'. Tell me if I am right when saying that reserves are composed of generalized compounds, from which we distinguish only C, H, O and N. For me, P and vit B12 are not generalized compounds, so that I do not see how they can be treated as "reserve 1" and "reserves 2". (maybe vit B12 is decomposed into C, H, O, N ?). Should we conclude from this example that it is possible to "add" generalised compounds like P when usefull?

Moreover, it is said in the following lines that P and vit B12 content in cells have very small values, so that they hardly contribute to the total biomass (OK for that), so that they can be neglected. This step seems wierd to me because then, our organism has no reserve anymore !

Answ: We are here dealing with a phototrophic haptophyte (which also can sport heterotrophy, of course). We need more than one reserve to understand its metabolism. Typically the number of reserves that should be delineated should equal the number of independently acquired substrates. The chemical composition of these reserves is typically simple; a pure chemical compound is just a special case of a generalized chemical compound. Vit B12 was here measured as labelled cobalt, and phospate as elemental phosphorous (method AAS). Both these elements also occur in structure, can the measurement just represents the combination of what is in the reserve(s) and the structure. This is just another example of application of auxiliary theory that links results of measurements to quatities in the model. The choice of following C, H, O, N only is just convenience to simplify the notation in the book. DEB theory can deal with all chemicals elements simultaneously; needless to say that this increases the number of required parameters.

Chapter 7: Case studies

Quest 7.7.3: How can DEB theory be applied to cold-water corals, which are colonies that live hundreds of meters deep in cold waters? A coral larvae settles and grows out to a polyp. Branches extend from the initial polyp on which new polyps grow and so on. These branches and polyps together form the coral colony. One of the questions with respect to DEB is how I should consider such a colony. My guess is that coral polyps grow as V1-morphs (see also {109}), which implies that feeding surface increases in proportion of volume. I have modeled (hopefully correct by multiplying the assimilation power with the shape correction function for a V1-morph {27}) a V1-morph and of course the growth dynamics are different from the isomorph. In particular, whereas the isomorph approaches a maximal structural volume, the volume of V1-morph grows in an exponential-like fashion (see {109}): Structural volume of a V1-morph as a function of time. Two problems/questions arise for me: 1) The ultimate volume/length of an isomorph is determined by the growing importance of maintenance costs (proportional to cubed length) as compared to feeding (proportional to squared length), which ultimately limits growth. In a V1-morph, feeding surface is proportional to volume and therefore growth does not vanish with increasing volume. So what does determine the ultimate size of a V1-morph? Does the resource need to be explicitly modeled to provide constraints? 2) The literature that exists on estimating DEB parameters is (understandably) heavily focused on isomorphs. However, an important and easily measurable parameter that is commonly used to determine primary parameters is ultimate volume/length. Moreover, KooySous2007 considers mostly length related parameters as primary DEB parameters from which other less easily observable parameters are estimated. Does the fact that volume or length has little meaning for a V1-morph (see also DEB book) imply that these parameter estimation methods cannot be used for a V1-morph?

Answ: V1-morphs have no size control, i.e. if food density remains constant their volume increases exponentially without any boundary. This situation never lasts long in nature due to depletion of resources. To described how such systems grow in the natural environment, resource dynamics must be included. Corals don't behave like V1-morphs, however, because when the polyps become more abundant, they start to compete for food, like leaves on a plant compete for light. Initially the competition is weak, but over time it becomes stronger. If coral colonies continue to grow, and make contact with their neighbours, the situation has strong similarities with that of plants in a vegetation. This sequence of events is described in 7.7.3 at {252} in a DEB context. Section 9.1.2/9.1.3 discusses typical mixotrophic surface-water corals; do these deepwater species have (baterial) symbionts that can fix dinitrogen?

Quest 7.1.4: The example in Fig 7.2 shows that five parameters only are needed to fully describe growth in case of constant food density and in case of one switch in food density. How far is this extendable ? I am not sure that this still holds if more than one switch occurs during the experiment ?

Answ: In case of constant food density only three (compound) parameters are required for a given scaled functional response: initial and asymptotic length and the von Bert growth rate. If you have several levels of functional responses, the von Bert growth rate has to be replaced by k_M and v, see Fig 3.14. This makes 4 parameters. The scaled functional response typically also needs to be quantified. The half-saturation constant and the maximum ingestion rate sometimes need to be known. If one of the food levels is large compared to the half saturation constant, the scaled functional response at some low food density can be estimated via the ratio of the asymptotic lengths. Knowledge about feeding parameters is then not required.

Quest 7.1.4: The scaled functional response can be reconstructed based on growth + temperature patterns in mussels. In Fig 7.5, four size classes of mussels were used, probably because it corresponded to the availble data from Kautsky. Would this reconstruction have work for the growth response of one single size class (I think yes but I am not sure)?

Furthermore, growth was followed during 500 d, which is longer than the duration of the test we usually use in ecotoxicology. Thus, my question is: what is the minimal growth data set (number of observations per classes and number of size classes) that is needed to reconstruct the scales functional response ? Two cases are distinguished: food intake changes slowly (and e = f is Ok in Fig 7.5) or it changes fast (and e(t) must be reconstructed in Fig 7.6). How is the boundary between these two cases defined?

Answ: Yes it also works with one size class, see the piguin example of Fig 7.6, and also for shorter periods. There must be enough info in the data to do the reconstruction. The reduction of growth due to the approaching of the ultime size must be enough. If you have DEB parameters for the mussel already, even that is not necessary. In principle it is even possible to come up with a confidence interval for reconstructed food density, but that is more work. Body size scaling relationships can be used to judge the need for explicit reconstruction of scaled reserve density. If in doubt you can also reconstruct food intake under the assumption that scaled functional response changes in speudo steady state, and compare the impact.

Quest 7.6: Given the dependence of many physiological processes upon the surface area of membrane (or cell) I wonder if the DEB model could be use a weak homeostasis assumption of {E} = constant instead of [E] = constant.

Answ: It would certainly be interesting to work out various alternatives and compare the implications. These alternatives will be substantially more complex for auxiliary theory that links model variables to measurements. Structural homeostasis here works out such that interface between reserve and structure scales with E/L, strange enough. Although the mechanism for reserve dynamics in section 7.6 was the best I could produce at in 1999-2000, I was never happy with it: complex and approximative. The newly discovered one, in KooyTroo2007, is very much better. It is much simpler and not approximative. It still uses structural homeostatis as discussed in 7.6, and a good next question is how this could have evolved. I don't have satisfying anwers for this (yet?).

Quest 7.9.5: The example given in {263} suggests that parameter values may greatly vary among generations in bacteria, possibly leading to permanant changes in their values when the population is exposed to a "selective factor" during several generations.

In the same way, can we imagine that parameter values may significantly fluctuate during a lifetime in species that experience a highly variable environment and/or in long-lived species ?

Second, could parameters be conceptualised as "life cycle characteristics" (and thus parameters values would be specific of a given species in a given habitat type)? Then, could the monitoring of their values permit:

At last, It is suggested in {264} that parameter values are partly genetically determined, but that they can be modulated phenotypically {265}. Can we consider that some of them could be "more plastic" than others? Which ones?

Answ: I think that some parameters are much more likely to vary across generations than others. At {265} we specifically look at the relative frequency of carriers for specific organic substrates in a bacterium that can potentially live on many different organic substrates. So this bacterium has the problem of what carrier it should produce. Producing all of them is not economic, because most of the carriers are doing nothing due to the absence of substrate it can bind. The details on this type of adaptation are discussed in KooyTrooy2006, and summerized in the comments.

Chapter 8: Comparison of species

Quest 8:I noticed that the investment ratio is often approximated by 1 in the examples. What is the hypothesis that enable us to do so: feeding density = constant or feeding density = ad libitum or both or anything else?

Answ: The motivation is simply convenience, in absence of better estimates. g is likely to vary inservely to maximum body size among species (see chap 8). Meanewhile we have more experience with g estimates, cf KooySous2007.

Chapter 9: Living together

Quest 9.1.2 At the early beginning of {304}, implications of the competitive exclusion principle for DEB model are described. I understand why strict forms of competition are rare if each species want to achieve weak homeostasis, that is have constant food conditions. But I don't see how syntrophic relationships can help to maintain weak homeostasis. Do you know if this can apply to phytoplankton? Does phytoplankton species produce waste that can be used by other species of phytoplankton ?

Answ: When a species needs the products of another one, and is limited by the availability of these products, the abundance of that species becomes coupled to that of the species that produces the products. This also occurs in phytoplankton. First of all the N-waste and the carbon dioxide that is produced by phytoplankton species directly serve as substrates for other species, even for themselves. As is discussed in detail in chap 4, phytoplankton should be modeled with multiple reserves, which comes with an active excretion of reserves, such as carbohydrates. The latter compounds are known to fuel heterotrophic bacterial growth: this is known as the microbial loop. Syntrophy is no exception among organisms, it is the standard.

Quest 9.1.2 Does direct transfer mean that all products can be handled from the recipient (whereas in indirect transfer not all product has to be handled)? The maximum flux that can be handled is j_P,Am2 * M_V2. What exactly does j_P,Am2 acount for? Does this mean that the product is only associated to assimilation for the recipient?

Answ: In the context of 9.1.2 direct transfer means that the recipient receives (all of) the flux that is produced by the other, not necessarily that all of it is actually processed. Indirect transfer means that a product is released into the environment, and another organism takes (some of) it out. Most of that material will then not be availeble (low concentration, transport in the environment). j_P,Am2 is the maximum mass-specific assimilation rate of species 2.

Quest 9.1.2 We can find the constrains for achieving weak homeostasis by syntrophy assuming that all product is handled by the recipient (direct transfer). That is, no product formation is associated with donor growth, the turnover rates are equal, and the maintenance rate coefficients are equal for the two species. Then, for indirect transfer (where part of the product is handled, and part of it is not available to the recipient) at steady state the ratio of the biovolume densities of species (X_V1/X_V2) is done by the non-numbered equation in {304}, where we want X_V1/X_V2 to be independent from the throughput rate (h) (in order to have weak homeostasis- to be independent of the substrate availability of the donor) and the above constraints for weak homeostasis (of direct transfer) apply here also. Does the above reasoning seem correct? If that is so.. I can't see how can the X_V1/X_V2 can be independent of throughput rate h since in the equation there are the f1 and f2 which are dependent of h?

Also, concerning the constrains for weak homeostasis (of direct transfer) I wonder how they can be applied in reality? For example, in the case of the syntrophic relations between heterotrophic bacteria and phytoplankton, do they have equal turnover rates (k_E) or equal maintenance rate coefficients? How real can this be?

Answ: The result of this analysis on when we can expect situations of weak homeostasis in syntrophyic interactions is that at indirect transfer the ratio of the participant' abundances still can vary within a limited range only. The partners can improve the quality of weak homeostasis using direct transfer, rather than indirect transfer. Weak homeostasis can become perfect if the partners tune their reserve turnover rates. The mechanism I have in mind is that turnover rates are individual-specific and can vary (somewhat) across generations. Selection towards tuning of the reserve turnover rates on an evolutionary time scale can be expected if, e.g. the accumulation of unused products has negative effects on growth. The details of such a selection mechanism should be workes out in the context of adaptive dynamics, see e.g. the thesis of Tineke Troost.

Quest 9.3.1: On the figure on {343}, the unit of X_K1 is said to be \mug.mL^-1 but the unit of X_0 is mg.mL^-1. So it seem not possible to add these two quantities in the functional response without a correction.

Answ: Dimensions are different from units. It is never possible to add two quantities with different dimensions in a meaningful way. If their dimensions are the same (as here), but their units are different, you obviously need to correct for differences in units. The units of the half saturation constants are here chosen differently from that of the concentration in the feed to reduce the numbers of significant digits that are required to represent the values.

Quest 9.3.1: In the DEBtool corresponding to the Fig 9.19, an additional parameter is used to create these graphics which didn't appear in the text: the contribution of reserve in volume. It does not seem to have unit. It is used to obtain the correspondance between mean cells volume and reserve with this formula (for E. coli): M_CV_coli = V_coli * (1 + w_coli * e_coli) Where : M_CV_coli is the Mean Cell Volume of E. coli V_coli is the mean cell volume of E. coli (estimated parameter) w_coli is the contribution of reserve to volume of E. coli (unestimated parameter) e_coli is the reserve density of E. coli I don't really understand how w_coli is obtained and how it could be constant. The aim of my work is to sum the reserve and the structure of this model in order to plot the trajectory in an 3 axes phase area.

Answ: Since e_coli is dimensionless, w_coli has to be dimensionless as well because the product is added to the number one (which is dimensionless). The value is obtained by minimasation of the sum of squared deviations between the calculated and the measured values.

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