Most conspicuous organisms are multicellular and most multicellular organisms develop somatic cells to perform specific, nonreproductive tasks. The ubiquity of this division of labor suggests that it is highly advantageous. In this article I present a model to study the evolution of specialized cells. The model allows for unicellular and multicellular organisms that may contain somatic (terminally differentiated) cells. Cells contribute additively to a quantitative trait. The fitness of the organism depends on this quantitative trait (via a benefit function), the size of the organism, and the number of somatic cells. The model allows one to determine when somatic cells are advantageous and to calculate the optimum number (or fraction) of reproductive cells. I show that the fraction of reproductive cells is always surprisingly high. If somatic cells are very small, they can outnumber reproductive cells but their biomass is still less than the biomass of reproductive cells. I discuss the biology of primitive multicellular organisms with respect to the model predictions. I find a good agreement and outline how this work can be used to guide further quantitative studies of multicellularity.
Every organism is exposed to mutations that cause variation in inherited traits. Competition between slightly different organisms leads to the proliferation of variants that increase fitness. Most adaptations will fine-tune existing systems but some adaptations lead to new features. The evolution of multicellularity was clearly such an adaptation. It opened a door to a whole new world of possibilities (Bonner 1965; Buss 1988; Maynard-Smith and Szathmary 1997; Bonner 2001; Knoll 2003; Nowak 2006).
In their simplest form multicellular organisms are just clusters of identical cells. Such undifferentiated multicellular organisms can evolve fairly quickly from unicellular ancestors through mutations of surface proteins (Boraas et al. 1998; Rainey and Travisano 1998; Velicer and Yu 2003). Cells in such clonal aggregates do not have to compete against each other for reproduction because they are genetically identical (Buss 1988). This alleviation of reproductive competition has a profound effect. It allows for a division of labor. Cells can specialize on nonreproductive (somatic) tasks and peacefully die because their genes are passed on by genetically identical reproductive cells that benefited from the somatic function. This division of labor turns multicellular organisms into more than just lumps of cells. They contain cells that are different in function and appearance. Today a plethora of differentiated organisms exist, demonstrating the evolutionary success of division of labor.
Most theoretical studies of multicellularity analyze the change in level of selection and the consequences for reproductive competition (Buss 1988; Maynard-Smith and Szathmary 1997; Michod 1997; Michod and Roze 1997, 2001). Michod et al. (2006) analyzed which trade-offs between the life-history components viability and fecundity favor the evolution of somatic cells. Recently, I analyzed how the preexistence of undifferentiated multicellularity accelerates the evolution of specialized cells (Willensdorfer 2008). In this article I study which conditions make differentiated multicellularity desirable. When has a differentiated multicellular organism a higher fitness than an undifferentiated or unicellular organism? I present a model in which the fitness of an organism depends on a quantitative trait. The quantitative trait is determined by the number and type of cells that the organism is composed of. The mathematical model allows one to study which kind of benefits multicellularity must convey to compensate for its disadvantages. I calculate how much (compared to a reproductive cell) a somatic cell has to contribute to the quantitative trait to make division of labor advantageous and determine the optimum number/fraction of somatic cells.
The following section describes the model in detail. In the Results, I will first consider the evolution of undifferentiated multicellularity. To study the evolution of differentiated multicellularity I analyze the fitness of organisms of constant size. Thereafter, I study multicellularity in organisms where the size of the organism and the fraction of somatic cells are determined by the same evolutionary force. In the Discussion, I use the insights from my analysis to discuss a broad spectrum of primitive multicellular organisms.
Biological fitness is usually measured in number of offspring per organism per generation. This can get very confusing during the derivation of a model that compares organisms of different sizes and generation times. It is conceptually much easier to compare at which rate one unit of parental biomass produces one unit of offspring biomass. This biomass production captures an organism's ability to grow and reproduce. It should be clear that the organism with the highest offspring biomass production will outcompete all other organisms because its fraction of biomass in the environment will increase by a constant factor for each unit of time that passes. It should also be clear that the rate of biomass production is proportional to fitness if organism size and generation time are constant. Hence, one can consider offspring biomass production as a generalization of fitness that simplifies fitness comparisons between organisms of different sizes and generation times.
In this section I derive how somatic cells, body size, and benefits of multicellularity affect the rate of biomass production. I distinguish between reproductive and somatic cells but allow for only one kind of somatic cell. Somatic cells are different from reproductive cells in that they are terminally differentiated. Their biomass does not contribute to the next generation. Reproductive cells, on the other hand, contribute to the next generation. In the context of this article I limit myself to asexually reproductive cells.
To derive how somatic cells affect fitness, let us first assume that multicellularity and organism size have no effect on the rate of biomass production. This case is illustrated in the first two rows of Figure 1. A unicellular organism has the same fitness as a four-cell organism. Each organism produces four offspring after two rounds of cell division. Indeed, if size and multicellularity have no effect on biomass production, then all organisms that are entirely composed of reproductive cells will have the same fitness.
Let us use a four-cell organism to derive the cost of somatic cells. This cost stems from the inability of somatic cells to contribute biomass to the next generation. To quantify this cost, we can compare the rate of offspring biomass production of a four-cell organism that has two somatic cells with the rate of offspring biomass production of a four-cell organism without somatic cells. Row three in Figure 1 shows that if the organism has two somatic cells that are as large as the reproductive cells, then the differentiated organism is able to produce only two new four-cell organisms (eight cells) whereas the undifferentiated organism produces four new organisms (16 cells). Hence the rate of production of the differentiated organism is half the rate of production of the undifferentiated organism. Row four in Figure 1 illustrates how this effect depends on the size of the somatic cells. Let us assume that the two somatic cells are negligibly small. Because organism growth in natural populations is mostly limited by the speed of cell growth and not by the speed of cell division, the production of two negligible small somatic cells is essentially free, and the rate of biomass production remains unchanged. Row four in Figure 1 illustrates this. For negligibly small somatic cells, we need to start with two four-celled organism to get the same initial biomass as in the other three rows. As one can see, after two generations we have the same biomass as in row one and two. In general, the rate of offspring biomass production is reduced by the fraction of biomass that does not contribute to the next generation.
In the following we assign variables to the number and biomass of somatic and reproductive cells to quantify this fitness reduction. I use the Greek letters α and β to denote the biomass of a reproductive and somatic cell in the adult organism. As we will see, it is sufficient to consider the size of a somatic cell relative to the size of a reproductive cell. Let B=β/α denote this size ratio. Let Nr and Ns denote the number of reproductive and somatic cells. An adult organism is composed of N=Nr+Ns cells and has a biomass of αNr+βNs. Of that biomass βNs rests in somatic (sterile) cells and αNr in reproductive cells. Hence, βNs/(αNr+βNs) of the organism's biomass is lost in each generation and reflects the cost of somatic cells. The fitness of an organism with somatic cells relative to the fitness of an organism without somatic cells is given by
for B=β/α as defined above.
So far I have assumed that the rate of production is independent of organism size. An overwhelming amount of empirical data shows that the rate of production decreases with the body mass of the organism (Peters 1986). (Note that I use the term organism size and body mass interchangeably.) The empirical evidence shows that the annual rate of production per average biomass scales with W−γ, where W denotes the average body mass of an adult organism and γ is a scaling factor. This relationship holds from unicellular organisms to mammals, with body masses ranging from approximately 10−10 to 103 kg. As summarized by (p. 134 Peters 1986), the exponent γ might range from 0.23 to 0.37. For small organisms γ is close to 1/4, the typical allometric exponent of size. I will therefore use γ= 1/4 to discuss quantitative results. From above, we know that the adult body mass of an organism in this model is given by W=αNr+βNs. Hence, the rate of production decreases by the factor (αNr+βNs)−γ.
The benefit of multicellularity can be modeled as a function of the number of reproductive and somatic cells. Let f(Nr, Ns) denote this benefit function. All things considered, the fitness of a multicellular organism is given by
where B=β/α is the size of a somatic cell relative to the size of a reproductive cell.
This model confirms common sense. If multicellularity does not affect fitness, that is, f(Nr, Ns) is constant, then a unicellular organism has a higher fitness than a multicellular organism because F(Nr, Ns) < F(Nr, 0) < F(1, 0) for Nr > 1 and Ns > 0. For undifferentiated multicellularity the fitness of an organism is given by N−γrf(Nr, 0) and multicellularity is only advantageous if it conveys benefits that compensate for the disadvantages caused by the size increase. In other words, f(Nr, 0) has to increase more steeply than N−γr decreases.
Central to my analysis of multicellularity is the function f(Nr, Ns), which captures the benefit of multicellularity. I will assume that somatic and reproductive cells contribute to a quantitative trait, x, and that the benefit of multicellularity is a function, f(x), of this trait. In the following I illustrate this approach by modeling predator evasion and flagellation in terms of this model.
For predator evasion, the quantitative trait is given by the geometric size of the organism. The geometric size of the organism depends on its extracellular matrix and on the number and size of somatic and reproductive cells. The geometric size determines to which extent the organism is able to evade predation. (Note that I use the term organism size and body mass interchangeably throughout the article. However, in the context of predator evasion one has to be more careful because geometric size conveys the benefit whereas body mass constitutes the cost. It might be possible to increase geometric size substantially without increasing body mass by a lot.) If the organism is big enough, the predator is unable to ingest it and the benefit of multicellularity, f(x), is close to 1. For small organisms predation might be severe and f(x) close to 0. One can expect a steep increase of f(x) as the organism size surpasses the maximum particle size the predator can ingest. Figure 2A shows a benefit function that could be used to model predator evasion.
For flagellation, the quantitative trait is given by the flagellar drive that the cells of the organism provide. The more cells, the more flagellar drive, which improves the organisms ability to maintain its position in a favorable environment. For this example the benefit function can be expected to be concave. An initial increase in flagellar drive might be very beneficial by allowing the organism to maintain its position. At some point, however, the organism has enough flagellar drive to maintain its position for most of the time and a further increase in flagellar drive does not yield a substantial benefit. Figure 2B shows a benefit function that could be used to model benefits from flagellation.
As we will see, my analysis of this model does not require an exact specification of the benefit function. But it is necessary to make assumptions about the contribution of individual cells to the quantitative trait. I will mainly consider quantitative traits to which individual cells contribute additively. I use the letters a and b to denote the contribution of a reproductive and a somatic cell to the quantitative trait. In an organism with Nr reproductive and Ns somatic cells, the quantitative trait, x, is given by
Hence, we have
Let A=a/b denote the contribution of a reproductive cell relative to the contribution of a somatic cell. To simplify the analysis, I scale the argument for f so that b= 1, that is, a somatic cell contributes one unit to the quantitative trait. I only consider organisms in which somatic cells are usually smaller than reproductive cells (β≤α) and, hence, B=β/α∈ (0, 1]. The term AB appears several times in my analysis of this model. Because AB= (a/α)/(b/β) it can be interpreted as the contribution of one unit of reproductive biomass to the quantitative trait relative to the contribution of one unit of somatic biomass. It is evident that there is no need for somatic biomass if reproductive biomass contributes more to the quantitative trait. Thus, AB < 1 is a necessary condition for differentiated multicellularity to evolve.
Using A=a/b and rescaling f's argument, we can rewrite (2) as
Note that if the contribution of reproductive cells to the quantitative trait is insignificant (i.e., A Nr+Ns≈Ns) then the benefit of multicellularity is simply a function of the number of somatic cells.
So far I have not made any assumptions about f(x). To simplify the analysis I restrict f(x) to monotone increasing and bound functions. Both constraints are reasonable. An increase of the quantitative trait should not lead to a decrease in fitness, and the fitness of an organism cannot be increased infinitely. I scale, f(x) such that f(x) = 1. To guarantee that the fitness of an organism is always nonnegative I only consider benefit functions with f(x) ≥ 0. Moreover, for the analysis of evolutionary transitions to differentiated multicellularity, it is necessary to assume that undifferentiated organism have positive fitness, that is, f(NA) > f(A) > 0.
In this article I will use the term linear benefit function to refer to truncated linear functions. They are given by f(x) =cx if x < 1/c and f(x) = 1 if x≥ 1/c where c > 0. Strictly speaking no truly linear function can exist in this model because every function is truncated to one. It is worth noting that the truncated part of a benefit function is evolutionary speaking not very interesting. Every increase in x comes to a cost because it requires either an increase in organism size or a loss of a reproductive cell. Hence, once the benefit function is truncated to one, fitness will always decrease with increasing x.
Throughout the manuscript the reader should keep in mind that the number of cells can only change in increments of one. As a consequence, x can also only change in discrete increments. To find the point of optimum fitness, I will, in most cases, treat x as a continuous variable. The “real” optimum will generally be very close to this continuous optimum. For linear benefit functions, for example, if the continuous optimum is at 1/c, then the real optimum value for x could be “a notch up” or “a notch down” from this value. Whether it is the larger or the smaller value depends on the specific parameters and is mostly irrelevant for the drawn conclusions.
Table 1 summarizes the variables and parameters used in this model.
Table 1. Notation summary
The number of reproductive cells of the organism; Nr≥1
The number of somatic cells of the organism; Ns≥0, an organism is undifferentiated if Ns=0
The total number of cells of the organism; N=Nr+Ns≥1
The fraction of reproductive cells; q≥1/N; q=1 for undifferentiated organisms
The size of a reproductive cell
The size of a somatic cell
The size of a somatic cell relative to the size of a reproductive cell; B is bound to [0, 1]
The contribution of a reproductive cell to the quantitative trait
The contribution of a somatic cell to the quantitative trait
The contribution of a reproductive cell to the quantitative trait relative to the contribution of a somatic cell
The contribution of one unit of reproductive biomass to the quantitative trait relative to the contribution of one unit of somatic biomass
The value of the quantitative trait. If contributions of somatic and reproductive cells to the trait are additive, then x=aNr+bNs
The benefit function. It captures the extend to which the organism benefits from the quantitative trait.
The benefit function is scaled such that lim x→∞f(x)=1.
The allometric exponent which is approximately 1/4
I am interested in the kind of benefit functions, f(x), that promote the evolution of multicellularity. I am also interested in how much a somatic cell has to contribute to the quantitative trait (relative to a reproductive cell) to compensate for the loss in reproductive biomass and what the optimum fraction of somatic biomass would be.
As a first step, I analyze conditions for the evolution of undifferentiated multicellularity. Thereafter I study the evolution of somatic cells in organisms of constant size. I study the unconstrained model last. In the unconstrained model the size of the organism and the fraction of somatic cells can change freely. I show that it is possible to calculate the optimum fraction of reproductive cells and that this fraction is independent of the benefit function f(x).
The evolution of undifferentiated multicellularity corresponds to an evolutionary transition from organisms composed of one (reproductive) cell to organisms composed of several identical (also reproductive) cells. Undifferentiated multicellular organisms have, per definition, no somatic cells (Ns= 0). For Ns= 0, the fitness (2) simplifies to
Multicellularity (Nr > 1) is only advantageous if N−γrf(Nr, 0) > f(1, 0). In other words, the benefit function f(Nr, 0) has to increase faster than N−γr decreases.
I employ this simple case to illustrate the method that will be used to analyze the more complex cases. I would like to know for which functions f(x) multicellularity is advantageous and what the optimum number of reproductive cells is. To get a general idea of how f(x) affects fitness, we can determine the functions f(x) for which the fitness is constant with respect to Nr. Let fiso(x) denote these functions. I refer to them as isolines since they join points of equal fitness. They are analogous to the lines on topographic maps that join points of equal altitude. These isolines can be used to capture the fitness landscape in (x, f(x)) space and determine local and global optima for any benefit function. The fitness of an undifferentiated organism is given by
and is constant if f(aNr) ∝Nγr. Substituting x=aNr, we get
The gray curves in Figure 3 show these isolines. With the knowledge of these isolines it is easy to determine which functions, f(x), promote the evolution of multicellularity. It is also simple to determine the optimum number of reproductive cells. An organism with Nr reproductive cells has x=aNr as value for the quantitative trait. We have an optimum xopt=a Nr,opt if f(x) ≤fiso (x) for the isoline with f(xopt) =fiso (xopt). In a continuous setting the optimum satisfies for the isoline with f(xopt) =fiso(xopt).
To interpret isoline plots, it might be useful to keep the analogy with topographic maps in mind. One can think of x as the distance traveled along a particular trail, f(x), in a mountainous region. The highest point, in our case the optimum, xopt, is reached if the trail “brushes” the highest contour line along the trail, f(xopt) =fiso(xopt). None of the points along the trail will be above this contour line, f(x) ≤fiso(x).
Figure 3 shows two linear and one concave benefit function. The black bullets indicate the optimum for each benefit function. As one can see, a linear benefit function will always promote the evolution of multicellularity and will increase f(x) until it reaches one (or a value just below one due to the discreteness of x). A concave benefit function reaches the optimum earlier and results in smaller organisms. Undifferentiated multicellularity would not evolve if f(x) increases slower than the isolines. In particular, because the maximum fitness of an organism of size two is given by 2−γ, multicellularity would not evolve if 2−γ < f(a). For γ= 1/4, we have 2−γ= 0.84 and multicellularity would not evolve if unicellular organisms are able to benefit from the quantitative trait more than 84% of its full potential.
DIFFERENTIATED MULTICELLULARITY IN ORGANISMS OF CONSTANT SIZE
In the previous section I have shown that undifferentiated multicellularity is advantageous for many benefit functions. In the following I will analyze the evolution of somatic cells (differentiated multicellularity). For simplicity, I will first analyze the evolution of somatic cells in organisms of constant size.
From a biological perspective it is relevant to consider organisms of constant size, because many benefits of undifferentiated multicellularity imply constraints on the size of the organism. Predator evasion, for example, is known to promote the evolution of undifferentiated multicellularity (Boraas et al. 1998). An organism that uses multicellularity to evade predation is obviously constrained with respect to size. It has to be larger than the largest particle the predator can feed on. Replacing nine large reproductive cells with nine small somatic cells could decrease its size to dangerous levels.
For simplicity, I will first assume that somatic and reproductive cells have the same size (B= 1) and explore the more general case of B < 1 later.
Somatic cells are as large as reproductive cells (B = 1)
The size of an organism is given by S=αNr+βNs. If somatic and reproductive cells have the same size (B=β/α= 1), then the size of the organism can only be held constant if the number of cells that compose this organism is constant, that is, N=Nr+Ns= constant. In this case the fitness depends on one variable instead of two. Using Nr as this variable and keeping in mind that N−(1+γ) is constant, we can rewrite (2) as
because the trait value of an organism with N cells is given by x=N− (1 −A) Nr. For B= 1 it is evident that somatic cells will not evolve if A≥ 1. For the following we can therefore assume A < 1. For A < 1, the quantitative trait of an organism of size N has a value of at least AN (if Nr=N) and at most N− 1 +A (if Nr= 1). The fitness of an organism is constant if f[N− (1 −A) Nr]∝N−1r. Expressing the number of reproductive cells, Nr, in terms of the quantitative trait, x, we get
Because we rescaled f(x) so that a somatic cell contributes one unit to the quantitative trait, we can interpret N in (10) as the value of the quantitative trait of an organism entirely composed of somatic cells. Hence, N−x could be interpreted as the value by which the quantitative trait is reduced due to the existence of reproductive cells. It reflects the cost of reproductive cells in terms of the quantitative trait.
Let us calculate the optimum number of reproductive cells for a linear benefit function, f(x) =cx. Let us first only consider linear benefit functions with 0 ≤f(x) < 1 for all organisms composed of N cells, i.e., . As mentioned above, the optimum, xopt, satisfies for the isoline with fiso(xopt) =f(xopt). Hence, we have to solve the two equations and fiso(xopt) =f(xopt) for xopt and the (irrelevant) constant k in fiso(x) =k (N−x)−1. Solving these equations for the benefit function f(x) =cx results in
which corresponds to an optimum number of reproductive cells of
As we can see, the optimum value for the quantitative trait, xopt, is independent of A. The parameter A does, however, determine how many somatic cells are necessary to reach xopt, and hence Nr,opt. Equation (12) also shows that the number of reproductive cells is usually greater than N/2 and only equal to N/2 if reproductive cells do not contribute to the quantitative trait (A= 0).
Remarkably, the optimum number of reproductive cells is independent of the slope, c, of the linear benefit function. We constrained the slope so that the linear function does not exceed one (f(x) < 1). We can loosen this condition because our derivation only requires f(xopt) =f(N/2) < 1 or c < 2/N. For linear benefit functions with steeper inclines, the optimum is reached at the point of truncation. All things considered, we have xopt=N/2 if c < 2/N and xopt= 1/c if c≥ 2/N. Later we show that if N is determined by the benefit of multicellularity, i.e., allowed to change freely, then the optimum number of cells is given by 5/c. Hence, a size of N < 2/c (for which xopt=N/2) is, as far as linear benefit functions are concerned, actually “unnaturally” small.
Somatic cells are advantageous if the optimum number of reproductive cells, Nr,opt, is less than N. From (12) we see that this is only the case if A < 1/2 which means that somatic cells have to contribute twice as much as reproductive cells to the quantitative trait to justify their existence.
We can summarize our results for organisms of constant size, uniform cell sizes, and (not too steep) linear benefit functions: (1) such organisms contain many reproductive cells, and (2) somatic cells in such organisms have to contribute substantially more to the quantitative trait than reproductive cells. In the following I analyze isoline plots to illustrate that this result holds for many nonlinear benefit functions.
Figure 4 shows four benefit functions and isolines fiso(x) ∝ (N−x)−1 for N= 32. The solid line represents a linear benefit function that satisfies the requirements from above (c < 2/N). It is evident that the isolines and the benefit function have the same slope at xopt=N/2 = 16. The Figure illustrates that xopt does not depend on A (the relative contribution of a reproductive cell to the quantitative trait). Even though xopt is independent of A, this parameter determines how many somatic cells, if any, are required to reach this xopt. The top axis shows how x corresponds to Nr for four different values of A. As A increases to one, the range of possible values for x, AN to N− (1 −A), shrinks (to the right). This is no surprise. If the contribution of reproductive and somatic cells to the quantitative trait is about the same, then the total value of the quantitative trait changes little if one substitutes a reproductive cell for a somatic cell. One can see that if A is larger than 1/2, then the quantitative trait of an undifferentiated organism already exceeds N/2, the optimum value for linear benefit functions. There is no need for somatic cells.
Figure 4 contains a concave benefit function (dash-dotted curve). It is easy to localize xopt for this function and one can see that the xopt is smaller than the xopt for the linear benefit function. This is generally true. Let us assume there is a concave benefit function with an . We can draw a linear benefit function, f, through and get . Because is concave and nonnegative, it is larger than the linear benefit function for any . Thus, the fitness of the organism with the concave benefit function is always larger than the fitness of the organism with the linear benefit function, for . From Equation (11) we know that the linear benefit function has its optimum at xopt=N/2, that is, is not optimal for the linear benefit function and, because , it is also not optimal for the concave benefit function. This shows that for concave benefit functions the optimum value for x is always smaller or equal to N/2.
For a concave benefit function organisms need less of the quantitative trait to optimize fitness. As a consequence, the functional demand on somatic cells to justify their existence increases (A has to be even smaller, see upper axis in Fig. 4). We see that the results from above, (1) organisms have few somatic cells and (2) somatic cells have to contribute substantially more to the quantitative trait, do also hold for concave functions.
Figure 4 shows that only initially convex function can lead to organisms with predominantly somatic cells. Such functions might describe situations in which the quantitative trait has to exceed a minimum threshold value to benefit the organism. In such situations the functional demand on somatic cells is relaxed and organisms might need many somatic cells to optimize fitness.
Somatic cells that are smaller than reproductive cells (B < 1)
If somatic cells are smaller than reproductive cells, then the total number of cells, N, can change even if the size, S, of the organism remains constant. If, for example, reproductive cells are twice as large as somatic cells (B= 1/2), then one reproductive cell can be replaced by two somatic cells, which increases the total number of cells by one but keeps S constant.
If somatic cells are half the size of reproductive cells, we can simply apply the results from above by changing the parameter A. Because contributions are additive in this model, somatic cells of size β that contribute b to the quantitative trait are as advantageous as somatic cells of size β/2 that contribute b/2 to the quantitative trait. In other words, if somatic cells that are as large as reproductive cells are beneficial, then somatic cells that are half as large and contribute half as much to the trait have to be beneficial as well. Hence, to get the results for B < 1, we only have to consider the results from above and replace A with AB and N with S. For example, for a linear benefit function the optimum number of reproductive cells is given by
and somatic cells are advantageous if AB < 1/2. For concave benefit functions, we can conclude that more than half of the biomass of the organism will rest in reproductive cells and that somatic cells need to satisfy AB < 1/2 to justify their existence. Note that AB quantifies the contribution of one unit of reproductive biomass to the quantitative trait relative to the contribution of one unit of somatic biomass. To be advantageous, one unit of somatic biomass has to contribute at least twice as much to the quantitative trait as one unit of reproductive biomass.
Appendix A contains a more technical and detailed analysis of the case B < 1. For completeness, I analyze a model in which the number of cells is held constant (as opposed to the size of the organism) in Appendix B.
THE COMPLETE (UNCONSTRAINED) MODEL
Let us now study the unconstrained model. By not restricting the number of cells or the size of the organism, I assume that its size and the optimum fraction of reproductive cells are governed by one evolutionary force. In this case the quantitative trait, x, can no longer be expressed as a function of Nr. It depends on Nr and Ns. This makes the calculation and visualization of isolines unwieldy. Instead, we can directly analyze the fitness landscape and draw conclusions about differentiated multicellular organisms. In the following we will realize that if the size and the composition of the organism can change freely, then the optimum fraction of reproductive cells in differentiated organisms is independent of f(x) and can be calculated.
To simplify our analysis, we treat q and N as continuous variables. We assume that f(x) is differentiable and monotone increasing . The fitness of an undifferentiated organism is given by N−γf(NA) and has to be positive whenever we discuss the transition from undifferentiated to differentiated multicellularity.
If the fitness function has an optimum in the interior of its domain, Nopt∈ (1, ∞) and qopt∈ (0, 1), then the partial derivatives and have to be zero at that optimum. As we will see, these conditions allow us to calculate qopt, which turns out to be independent of the specific benefit function (chosen from the set of benefit functions that lead to a maximum in the domain's interior). If the fitness function does not have an optimum in the interior of its domain it has it at its boundaries. The optimum cannot be at the boundary N=∞ because the benefit of multicellularity is bound whereas the cost of organism size is unbound (Nγ→∞ for N→∞). It can also not be at the boundary q= 0 because F= 0 for q= 0. Hence, the optimum is either at Nopt= 1 (unicellular organism) or at q= 1 (undifferentiated organism). As we will see, the following procedure requires and can therefore not be applied if Nopt= 1. Unsurprisingly, we cannot calculate the optimum fraction of reproductive cells in unicellular organisms.
In the following we calculate the optimum fraction of reproductive cells for benefit functions with Nopt > 1, that is, for benefit functions that favor multicellular organism. Because the optimum is not at the boundary N= 1, we know that the partial derivative with respect to N is zero at the optimum.
As a first step, we calculate
Because 1 − (1 −A)q > 0, we can express in terms of ,
Applying the product rule of differentiation to (15) we get
If the number of cells, N, is optimal, then and hence which can be substituted into (18) to give
Differentiated multicellularity is advantageous only if at q= 1. The fraction of reproductive cells is optimal if . Because f > 0, the optimum fraction of reproductive cells can be determined by calculating for which q the factor in (22) equals zero. Multiplying this equation by 1 − (1 −A)q shows that terms quadratic in q, i.e., (1 −A)(1 −B)q2 cancel. Hence, this equation is linear in q and can be solved to give
Thus, we are able to calculate the optimum fraction of reproductive cells, qopt. We also see that differentiated multicellularity is advantageous ( at q= 1) only if qopt < 1. Remarkably, qopt is independent of the benefit function f(x).
The benefit function will, however, determine the size of the organism. We can replace q in (15) with the now constant qopt and get
This is of course very similar to (7). Again, the isolines fiso (x) ∝xγ (see Fig. 3) can be used to find xopt. Knowing xopt, we can calculate
For example, for linear benefit functions f(x) =cx (if x < 1/c and 1 otherwise) we have xopt= 1/c and, hence, Nopt= 1/[c(1 − (1 −A)qopt)]. For B= 1 this simplifies to (1 +γ−1)/c which equals 5/c for γ= 1/4.
Figure 5 shows an implicit plot of qopt as a function of the parameters A and B. The curve for qopt= 1 gives the threshold for parameter values that allow for the evolution of differentiated multicellularity. If qopt≥ 1, then no benefit function exists that can promote the evolution of differentiated multicellularity. The curve qopt= 1 is given by AB=γ/(1 +γ) (=1/5 for γ= 1/4). As for constant size, the term AB determines if somatic cells are advantageous or not. This time somatic cells are advantageous if one unit of somatic biomass contributes at least five times as much to the quantitative trait as one unit of reproductive biomass. If somatic cells are as large as reproductive cells (B= 1), then they can be beneficial only if A < γ/(1 +γ). Similarly, if somatic cells contribute as much to the quantitative trait as reproductive cells (A= 1), then, to be advantageous, their size has to be a fraction (<(1 +γ)/γ) of the size of reproductive cells.
Figure 5 shows that for small somatic cells (small B), the ability of reproductive cells to contribute to the quantitative trait (A) has little effect on qopt. For small B, the denominator in (23) is approximately 1 and qopt≈γ−1B. From (15) we also see that A appears in the equation for F only in the term 1 − (1 −A)q. If q is small (because of small B), then 1 − (1 −A)q≈ 1 and A has little effect on the fitness of the organism (unless A is very large).
That the effect of A (the ability of reproductive cells to contribute to the quantitative trait) on F and qr,opt depends on B (the size of somatic cells) is an important result for our understanding of the evolution of division of labor. Many reproductive cells have to grow to a minimum size before they can initiate cell division. Newly evolved somatic cells are presumably as large as reproductive cells but are instantaneously relieved of reproductive size constraints. The minimum size at which a somatic cell can still function might be much smaller than the minimum size of a reproductive cell. Organisms will have the tendency to evolve somatic cells that are as small as their function allows (decrease B). Equation (23) shows that a decrease of B increases the optimum fraction of somatic cells. This will further increase the selective pressure to reduce the size of somatic cells because the organism has now more somatic cells that should not be unnecessarily large. This feedback loop might continue until there are many, small somatic cells. It is worth noting that at this point (small q) the contribution of reproductive cells to the quantitative trait has no major effect on fitness (see eq. 15) and reproductive cells can cease to contribute to the somatic function (the quantitative trait in my model) with little effect on fitness. They are free to dedicate their existence fully to reproductive duties. This evolutionary feedback loop leads to the evolution of organisms with a strict division of labor between many, small somatic cells and few, large reproductive cells.
It is important to emphasize that we treated q and N as continuous variables. Especially for small multicellular organisms they are, however, discrete. For example, a bicellular organism can have a q of 1/2 or 1. From Figure 5 and (23) we know that somatic cells in such a bicellular organism are (even for A= 0) only optimal if they are much smaller than reproductive cells (B < γ). Also, for B= 1 the optimum fraction of reproductive cells is always larger than 1/(1 +γ) (=4/5 for γ= 1/4). Hence, a multicellular organism has to be composed of at least six cells to be able to attain qopt.
I presented a model to study the evolution of undifferentiated and differentiated multicellular organisms. In my model three factors determine an organism's fitness: (1) its biomass (or, loosely speaking, size), (2) its investment in somatic (terminally differentiated) cells, and (3) a quantitative trait that is determined by the number and kind of cells that the organism is composed of. The quantitative trait, x, affects the fitness of the organism via a benefit function, f(x) (see Fig. 1). For simplicity I assume that the cells of a multicellular organism contribute additively to the quantitative trait.
I analyze under which conditions (benefit function, contribution to the quantitative trait, size of somatic cells, etc.) the evolution of undifferentiated and differentiated organisms is favored, and calculate the optimum fraction of somatic cells. My analysis shows that the transition from unicellular to undifferentiated multicellular organisms is favored by many benefit functions. The evolution of undifferentiated multicellularity is, however, unlikely if the unicellular organism is already able to receive large benefits from the quantitative trait. In particular, multicellularity will not evolve if the unicellular organism benefits from the quantitative trait more than 84% (=2−γ for γ= 1/4) of its full potential.
The model suggests that primitive differentiated organisms will generally have a small fraction of somatic biomass. If somatic cells are as large as reproductive cells and the benefit function linear or concave, then the fraction of somatic cells is always less than or equal to 1/2. If somatic cells are smaller than reproductive cells, they might occur in large numbers, but their biomass will still be at most 1/2 of the total biomass. Somatic cells compose more than 1/2 of the organism only if the benefit function has convex parts. A possible source for such convexity could be a minimum threshold that the quantitative trait has to surpass to be beneficial for the organism.
In the following I discuss the biology of primitive multicellular organisms. First I discuss undifferentiated, then differentiated organism. I use experimental data from volvocine algae to demonstrate how experimental observations can be compared with model predictions from this work. At the end I point out the limitations of my model.
Most benefits of undifferentiated multicellularity relate to an organism's ability to evade predators or its ability to secure a favorable position in the environment. Predator evasion is commonly recognized as a driving force for the evolution of undifferentiated multicellularity (Buss 1988; King 2004). Boraas et al. (1998) showed that unicellular algae can evolve multicellularity within few generations after exposure to a phagotrophic predator. Phagotrophic and many other predators face an upper size limit for the particles they can ingest. A simple “sticking together” of cells provides protection by exceeding these size limits. In this case the quantitative trait is the size of the organism. If it exceeds a certain value, the organism benefits substantially from it (see Fig. 2A).
Multicellularity is also known to improve an organism's ability to obtain a favorable position in the environment. In particular, the flagellation constraint dilemma is believed to play an important role in the evolution of multicellularity (Margulis 1981). Many eukaryotic cells face the dilemma that they are unable to maintain flagellation during cell division and, hence, lose motility (Bonner 1965; Margulis 1981; Buss 1988; Koufopanou 1994; Kirk 1997). In an undifferentiated multicellular organism, motility can be maintained. Multicellularity can also increase the speed of an organism. Many cells can provide more drive than a single one (Sommer and Gliwicz 1986). For this example, the quantitative trait, flagellar drive, determines the organism's ability to reach a favorable position in the environment, which constitutes a benefit (see Fig. 2B).
Multicellularity can also improve an organism's ability to float. Many algae lack flagella. They regulate buoyancy through the production of carbohydrate ballast and/or gas inclusions (Graham and Wilcox 1999). Filamentous growth in combination with the secretion of extracellular polymeric substances allows the formation of mats that provide a stable structure that can be used to regulate buoyancy by trapping bubbles (Phillips 1958; Graham and Wilcox 1999). In this case the quantitative trait might be given by the tightness of the mat. Tight mats allow to trap many bubbles and allow the cells in that mat to stay close to the surface water where they receive more light.
Let us now consider differentiated multicellularity. My analysis predicts that primitive differentiated multicellular organisms will generally have many reproductive cells. More precisely, in most cases I would expect more than half of an organism's biomass to rest in reproductive cells. In the following I discuss algae and slime molds, two groups of organisms for which quantitative data exist.
Volvocine algae are an excellent group of organisms to study differentiated multicellularity. Their multicellular complexity ranges from undifferentiated to highly differentiated organisms (Kirk 1997). The most primitive differentiated forms have somatic cells that maintain flagellation during cell division of reproductive cells. The flagellar beating is also important to provide a constant nutrient supply. It stirs the medium and prevents a nutrient depletion of the organism's boundary layers that would occur due to the nutrient uptake by the organism itself (Solari et al. 2006a). According to the source-and-sink hypothesis (Bell 1985) somatic cells can also increase the uptake rate of nutrients, but experiments by Solari et al. (2006a) suggest that the stirring of the medium plays a more important role in nutrient supply.
The smallest differentiated colonies in volvocine algae have 32 cells and 4, 8, or 16 are somatic (Goldstein 1967; Bonner 2003b). This is in agreement with the model prediction. Allometric data about soma and germ in volvocine algae show that there is more germ tissue than somatic tissue in all species (Koufopanou 1994, Fig. A1). Furthermore, we can use the data collected by Koufopanou (1994) to calculate q and B. In Figure 6, I compare the data with the model predictions (23). The good agreement between the model and the data suggests that the unconstrained version of the model is a good approximation of the evolutionary forces that govern multicellularity in volvocine algae.
To compare the model predictions with the experimental data, I assumed that reproductive cells do not contribute to the quantitative trait (A= 0). In this case the benefit of multicellularity is modeled as a (largely arbitrary) function of the number of somatic cells. To calculate the optimum fraction of somatic cells, it was not necessary to specify this function. For A= 0, my model is surprisingly general and should be applicable to a wide range of primitive differentiated multicellular organisms.
Figure 6 suggests that the allometric exponent for volvocine algae, 1/5, is smaller than the typical allometric exponent of 1/4. It should be possible to determine experimentally the allometric exponent of volvocine algae and compare my prediction with experimental observations. From experiments similar to the ones conducted by Solari et al. (2006a,b), one could also learn something about the shape of the benefit function by manipulating the number of (functional) somatic cells.
The model predicts the optimum fraction of reproductive cells over several orders of magnitude. It is unreasonable to assume that the ecological advantages of division of labor are the same for these organism. For example, small organism might benefit more from the increased flagellar drive whereas large organism benefit more from the stirring of boundary layers (Solari et al. 2006a,b; Kirk 1997). But, as mentioned above, the same model can be applied even if the kind of ecological benefit that somatic cells provide are different for different species. The only requirement is that the benefit of multicellularity is determined by a quantitative trait and that cells contribute (approximately) additively to this trait.
Another organism group that is commonly used to study primitive differentiated multicellularity is the slime molds. Slime molds such as Dictyostelium discoideum feed as individual cells until food becomes scarce, at which point they form a multicellular mass that migrates to a suitable spot and differentiates into a fruiting body. The somatic stalk of the fruiting body lifts the spores above the ground to facilitate more efficient dispersal (Bonner 1967, 2003a). In this case, the quantitative trait that conveys the benefit of multicellularity is the height of the stalk. The higher the stalk the more efficient is the dispersal. According to my model, we would expect the biomass of the stalk to be less then 50% of the total biomass of the fruiting body. Farnsworth (1975) measured the percentage (dry weight) of stalk as a function of the temperature during culmination. This percentage changes from about 20% at 18°C to 13% at 27°C. Hence, most of the fruiting body is indeed composed of reproductive cells. The data suggest that somatic cells are slightly more advantageous at lower temperatures because the fraction of somatic cells increases. Similarly, in Myxococcus xanthus, a fruiting body forming bacterium, more than 61% of the cells in a fruiting body are spores (O'Connor and Zusman 1991, Table 2).
It would be interesting to collect data about soma and germ in slime molds that is analogous to the data collected for volvocine algae. A comparison of such data with the presented model would be of particular interest because the quantitative trait is most likely the size of the stalk. This trait is easy to measure and would allow conclusions about the benefit function. The model presented in this article should guide the researcher in their data collection and presentation. For example, Koufopanou (1994) reported the average and standard deviation of the number and size of somatic and reproductive cells. In the light of my analysis it seems to be of greater biological importance to report the average and standard deviation of the fraction of reproductive cells, q, and the size of somatic cells relative to reproductive cells, B.
For most conspicuous organisms such as plants and animals the number and biomass of somatic cells vastly outnumbers that of reproductive cells. Notably, all of these organisms are much more complex than the primitive multicellular organisms that are the focus of this study. They contain many somatic cell types that form organs and interact with each other in complex ways. It is important to keep in mind the kind of organisms that the model is able to describe. I make two key assumptions: (1) The benefit of multicellularity can be modeled as a function of a quantitative trait. In particular, for a given value of the quantitative trait, the benefit does not depend on the number of reproductive cells, and (2) that cells contribute additively to that trait. If one of these two assumptions is not satisfied, the results can be quite different.
In Appendix C I analyze a model in which the fitness of an organism depends on how much of a limiting resource (e.g., Nitrate) each reproductive cell obtains. This resource, after it has been acquired by the (maybe multicellular) organism, has to be divided between reproductive cells. This is in disagreement with assumption (1) because the benefit depends on the number of reproductive cells (the less reproductive cells, the more nutrients each reproductive cell receives). As shown in Appendix C, such a situation does not favor the evolution of (undifferentiated) multicellularity, and differentiated organisms will tend to be small and contain a large somatic biomass.
One might also wonder how my model can be applied to organisms for which the distinction between somatic and reproductive cells is not so clear cut. For primitive differentiated multicellular organisms, “somatic” cells can be characterized by a delayed cell division or a reduced probability of reproduction, rather than no cell division or no reproduction at all. It is straight forward to incorporate this developmental plasticity into my model by modifying the term that captures the cost of somatic cells. This cost is given by the biomass that is lost due to the existence of somatic cells (or more generally the resources that are allocated to somatic cells). For terminally differentiated cells this is just given by the biomass of the somatic cells but can be modified to reflect any developmental plasticity. If, for example, “somatic” cells have approximately a 50% chance of reproduction, then the average evolutionary cost of somatic cells is given by 50% of the somatic biomass.
Throughout this article I optimized fitness for constant parameters A (relative contribution of a reproductive cell to the quantitative trait) and B (relative size of a somatic cell). What would happen if these parameters where allowed to evolve as well? This question is easily answered by looking at equation 5. It shows that fitness is monotone increasing with decreasing B and/or increasing A. This is intuitively clear because (everything else being the same) smaller somatic cells are less costly, and reproductive cells that can contribute more to the quantitative trait are advantageous. However, there is certainly a minimum threshold size below which somatic cells are no longer able to perform their somatic function, and there is an upper limit to how much a reproductive cell can contribute to the quantitative trait. The two parameters might also depend on each other. In Volvocine algae, for example, an increase in the number of cells requires bigger reproductive cells (B decreases) which, in turn, contribute less to flagellation (A decreases) due to the flagellation constraint dilemma (Koufopanou 1994; Kirk 1997; Solari et al. 2006a; Michod 2007). Possible interactions between A and B are probably only relevant for organism with few somatic cells. If the organism contains many somatic cells, the relative contribution, A, of reproductive cells to the quantitative trait becomes irrelevant (see The Complete Model). In this case the benefit of multicellularity is just a function of the number of somatic cells and their size will be reduced as much as possible.
Michod and coworkers derived and analyzed models to study the transition from undifferentiated to differentiated life (Michod et al. 2006; Michod 2006, 2007). In their models the product of fecundity, , and viability, determines fitness, . They study which trade-offs between bi and vi make differentiated multicellularity advantageous. They show that differentiated multicellularity is only advantageous if this trade-off is convex (vi decreases as a convex function of bi).
In this article fitness is a product of three components. The cost of organism size, 1/(αNr+βNs)γ, the cost of somatic cells, (αNr)/(αNr+βNs), and the benefit of multicellularity f (aNr+bNs). For constant size and linear benefit function (ignoring truncation of the function), fitness is proportional to αNr (aNr+bNs). This is analogous to Michod's model if bi=α for reproductive cells and bi= 0 for somatic cells and vi=a for reproductive cells and vi=b for somatic cells. Hence, Michod's model is similar to a special case of the model presented in this article. The main difference is that Michod et al. (2006) allow somatic cells to contribute biomass to the next generation (bi can be positive for somatic cells) and that they are mainly interested in the consequences of interactions between the different parameters.
In this article, on the other hand, I study a more general model in which the size and composition of organisms play an important role. I limit myself to somatic cells that do not contribute biomass to the next generation and consider under which conditions (differentiated) multicellularity evolves and what the optimum composition of differentiated organisms is. Furthermore, the last factor in my model can describe any kind of benefit of multicellularity. The benefit is not limited to an increase in viability. Multicellularity (including contributions from somatic cells) can increase fecundity as well.
It is important to differentiate between convexity/concavity of the functions that Michod et al. (2006) study and the convexity/concavity of benefit functions. Although in Michod's model convexity (of the functions that describes the relationship between bi and vi) is a necessary condition for the evolution of differentiated multicellularity, benefit functions are not required to be convex. Linear and concave benefit functions can lead to the evolution of differentiated multicellularity. In the presented model, benefit functions with convex parts are only special for organisms of constant size, because only they can lead to organisms with a large fraction of somatic biomass. In the unconstrained model, the shape of the benefit function has no affect on the composition of a differentiated organism, it affects only its size.
In this work I mathematically described the cost and benefit of differentiated and undifferentiated multicellularity. I showed that multicellularity can evolve readily if cells of a multicellular organism contribute additively to a quantitative trait that benefits the organism in a manner that is independent of the number of reproductive cells. Multicellularity is especially beneficial if a single-cell organism alone cannot benefit from the quantitative trait substantially. If the single-cell organism is able to exploit the quantitative trait to 84% (=2−γ for γ= 1/4) of the quantitative traits full potential then the shape of the benefit function is irrelevant and multicellularity will not evolve. I showed that evolutionary forces that are based on such additive quantitative traits will generally lead to multicellular organism with few somatic cells even if somatic cells contribute much more to the quantitative trait than reproductive cells.
In particular, for the complete model (organism size and fraction of somatic cells is determined by the benefit of multicellularity) and for somatic cells that are as large as reproductive cells, the optimum fraction of somatic cells is always less than γ/(1 +γ) = 1/5. As a consequence, under such conditions multicellular organisms can only reach this optimum if they are composed of at least six cells. Somatic cells can be numerous if they are very small compared to reproductive cells but their biomass will still be less than that of the reproductive cells. In the presence of many, small somatic cells, the contribution of reproductive cells to the quantitative trait has little effect on the fitness of the organism. This allows reproductive cells to specialize on the reproductive function and paves the way for a strict division of labor between reproductive and somatic cells.
Associate Editor: J. Wolf
I thank A Knoll and D Hewitt for their biological perspective. I am grateful to R Bürger and M Nowak for comments on the manuscript. I thank an anonymous reviewer for many helpful and detailed comments. I was supported by a Merck-Wiley fellowship. Support from the NSF/NIH joint program in mathematical biology (NIH grant r01gm078986) is gratefully acknowledged. The Program for Evolutionary Dynamics at Harvard University is sponsored by J. Epstein.
OPTIMUM NUMBER OF REPRODUCTIVE CELLS IN ORGANISMS OF CONSTANT SIZE WITH SMALL SOMATIC CELLS
In this section I derive the results for organisms of constant size S and somatic cells that are smaller than reproductive cells (B < 1). The unit for S is chosen so that an undifferentiated organism is composed of S reproductive cells. The quantitative trait of such an undifferentiated organism totals AS. A differentiated organism of constant size with one reproductive cell has (S− 1)/B somatic cells and its quantitative trait equals (S− 1)/B+A. Notably, an organisms quantitative trait can range from AS to (S− 1)/B+A and depends on A and B.
and are independent of A. The term S/B can be interpreted as value of the quantitative trait that an organism entirely composed of somatic cells would have.
For a linear benefit function f(x) =cx we can calculate the optimum as xopt=S/(2B), which corresponds to
Reproductive cells will constitute 1/[2(1 −AB)] > 1/2 of the biomass of the organism. As for B= 1, most of the organism's biomass will be reproductive cells. Somatic cells are only beneficial if AB < 1/2.
Figure A1 shows the isoline landscape and three benefit functions. Figure A1B illustrates how the isolines depend on the parameters S and B. Figure A1A shows the correspondence between x and Nr. For example, an increase of A would move the upper left end of the line that maps x onto Nr to the right and steepen its slope. Increasing B would move the lower right point of this line to the left and also steepen the slope. It would also change the isolines that approach S/B asymptotically. As for constant N most benefit functions and parameter combinations will lead to a fairly large number of reproductive cells or a large fraction of reproductive biomass. Only for benefit functions with convex parts would Nr,opt be small.
OPTIMUM NUMBER OF REPRODUCTIVE CELLS IN ORGANISMS WITH A CONSTANT NUMBER OF CELLS
In this section I analyze the evolution of small somatic cells (B < 1) in organisms that are composed of a constant number of cells (N= constant). The fitness (5) is given by
and constant if f[N− (1 −A) Nr]∝N−1r[1 +B (N/Nr− 1)]1+γ. Expressing Nr in terms of x, we get
We can interpret N−x as the decrease of the quantitative trait due to the existence of reproductive cells, and x−AN as the increase in the quantitative trait (compared to undifferentiated organisms) due to somatic cells.
Let us now study how a change in the size of somatic cells affects the optimum number of reproductive cells. The isolines are given by 1/(N−x) for B= 1. For B < 1, we can rewrite equation (A6) as
and note that fiso(x) approaches (N−x)γ for B→ 0. The shape of the isoline is entirely determined by the factor (1 −AB)/(1 −B) and different combinations of parameters A and B can result in the same isoline. Because A appears only in the term 1 −AB, it has less influence on the shape of fiso(x) if B is small. This can be explained intuitively. If somatic cells are very small, they are not very costly and how efficient they are (compared to reproductive cells) is less important.
Figure A2A shows isolines for different parameter combinations. I choose A= 0.1, 0.25, 0.5, and 0.75, and B= 0.02, 0.1, 0.2, 0.5, and 1. It illustrates the analytical results. Isolines change from (N−x)−1 (purple line) to (N−x)γ (gray line) and different parameter combinations can result in similar isolines. The parameter A has little affect on the isoline if somatic cells are small (small B). The top axes of Figure A2A show how x corresponds to Nr for different values of A. Interestingly, if B is small enough, then isolines can have a negative slope. In other words, even a constant benefit function would promote the evolution of somatic cells. We can calculate that the slope of fiso(x) is negative at if B < γ/(1 +γ) = 1/5. For somatic cells of that size the disadvantage of loosing a reproductive cell is compensated for by the size decrease (smaller organisms have higher rates of production). We encounter this threshold during our analysis of the complete (unconstrained) model.
Figure A2B shows the fitness, F, as a function of Nr. I plot F for A= 0.1 and indicate the maxima for the other A values. As expected, a decrease in B leads to an increase of F and a decrease of Nr,opt. Also, F and Nr,opt increase with A, but less so if B is small.
MULTICELLULARITY IN ORGANISMS IN WHICH THE BENEFIT OF MULTICELLULARITY DEPENDS ON THE NUMBER OF REPRODUCTIVE CELLS
In this section I analyze a small but significant variation of my model. To model the benefit of multicellularity, I assumed that this benefit is a function of a quantitative trait to which somatic and reproductive cells contribute additively. In this section I analyze a model in which the benefit of multicellularity (for a given value of the quantitative trait) does also depend on the number of reproductive cells. In this version reproductive and somatic cells contribute (additively) to the acquisition of a resource that is desperately needed by reproductive cells. The more a reproductive cells has of this resource, the faster it can grow and the larger is the probability of survival and, consequently, the fitness of the organism. Let f(x) denote the benefit from this resource if the organism manages to supply each reproductive cell with an amount x of the resource.
If one somatic (reproductive) cell acquires β (α) of the resource, then a total of αNr+βNs can be allocated between the reproductive cells and each cell would receive (αNr+βNs)/Nr. The benefit of multicellularity is then given by
and the fitness of the organism is given by
In the following, I analyze this fitness function. I show that such benefits of multicellularity are (1) an unlikely source for the evolution of undifferentiated multicellularity, (2) differentiated organisms of constant size would have few reproductive cells, and (3) if organism size and the fraction of somatic cells are governed by f(x) then the optimum number of reproductive cells is given by one.
It is easy to see that for Ns= 0, equation (A9) simplifies to F=N−γrf(a). Fitness, F, is strictly monotone decreasing with Nr and optimal for Nr= 1. Obviously, if all cells contributed linearly to the acquisition of a resource and divide that resource equally among each other, then each cell gets as much as it would get if it were on its own. In other words, if cells contribute linearly to the acquisition of a resource that is apportioned among them, then multicellularity conveys no advantages and a unicellular organism has the highest fitness. Limited resources that have to be divided between reproductive cells can only trigger the evolution of multicellularity if cells act synergistically. The multicellular organism has to be more than just the sum of its parts.
For constant size S=αNr+βNs, the fitness (A9) is given by
The amount of resources that each reproductive cell receives depends on A and B. It can range from to . The isolines are given by
As one can see, the isolines are linear. For linear benefit functions f(x) =cx with f(S/B) < 1 the optimum number of reproductive cells equals one if AB < 1. Somatic cells are advantageous if one unit of somatic biomass contributes more to the quantitative trait than one unit of reproductive biomass. More so, once differentiated multicellularity is advantageous the optimum number of reproductive cells is given by one.
Figure A3 shows isolines and the nonlinear relation between x and Nr. I plot three possible benefit functions. It is obvious that only very steeply increasing benefit functions would lead to organisms with many reproductive cells. Most benefit functions would result in few reproductive cells.
The complete model
Let us now analyze the unconstrained model. As in the main text, the optimum size of the organism as well as the optimum number of reproductive cells are governed by the benefit function f(x). As we will see, we can use this to our advantage and calculate the optimum number of reproductive cells analytically. We can even calculate the isolines with respect to size and conclude that benefits from resources that have to be allocated between reproductive cells are an unlikely cause of the evolution of multicellularity.
The fitness is given by
It is easy to show that for constant Ns, fitness increases when Nr decreases.
Because there can never be less than one reproductive cell the optimum number of reproductive cells is given by Nr,opt= 1. The fitness of the corresponding organism is given by
The isolines with respect to N are given by
Hence, multicellularity based on divisible resources is only advantageous for initially convex benefit functions that grow faster than ≈x1+γ. The resulting organism would contain one reproductive cell and would generally be fairly small (contain few cells). This suggests that benefits from resources that have to be allocated between reproductive cells are an unlikely cause of the evolution of multicellularity.