Thus far we have seen that strong trends in N:P with particle size could be an indicator of life, but that the total stoichiometric ratio of all biomass (filtered particles) does not have a single reliable value as a biosignature because this depends on the distribution of cell sizes. This approach also considers the entire set of particulate matter in isolation without considerations of environmental conditions. We need measurements that assess the “livingness” of a particular sample in the context of its environment, and one possibility is to simultaneously measure both the particulate and environmental (fluid) stoichiometries. It is also important to consider that macromolecular and elemental abundances in cells change as cells acclimate to environmental constraints, where there are known physiological optima based on environmental conditions (Burmaster 1979; Legović and Cruzado 1997; Klausmeier et al. 2004a, b, 2007, 2008), and which is the topic of the following chemostat models for microbial life living in aquatic environments.
A variety of efforts have shown how steady-state elemental ratios can be derived from physiological models coupled to flow rates in an environment (Legović and Cruzado 1997; Klausmeier et al. 2004a, b, 2007, 2008). These chemostat models contain the simplest components of a biogeochemical model: the influx of inorganic nutrients, consumption and transformation of nutrients into cellular materials as cells grow, and the loss of both biomass and inorganic nutrients from the system. Such models are typically written as
$$\begin{aligned} \frac{dR_{i}}{dt}= & {} a\left( R_{i,0}-R_{i}\right) -f_{i}\left( R_{i}\right) {\mathcal {N}} \end{aligned}$$
(14)
$$\begin{aligned} \frac{dQ_{i}}{dt}= & {} f_{i}\left( R_{i}\right) -\mu (\vec {Q})Q_{i} \end{aligned}$$
(15)
$$\begin{aligned} \frac{d{\mathcal {N}}}{dt}= & {} \mu (\vec {Q}){\mathcal {N}}-m{\mathcal {N}} \end{aligned}$$
(16)
where \(\mu (\vec {Q})\) is the growth rate as a function of all of the existing elemental quotas (cellular quantities), and is typically given by
$$\begin{aligned} \mu (\vec {Q})=\mu _{\infty }\min \left( 1-\frac{Q_{1,min}}{Q_{1}},1-\frac{Q_{2,min}}{Q_{2}},...,1-\frac{Q_{q,min}}{Q_{q}}\right) \end{aligned}$$
(17)
where q is the total number of limiting elements (Legović and Cruzado 1997; Klausmeier et al. 2004a, b, 2007, 2008). The function \(f_{i}\left( R_{i}\right) \) is the uptake rate for a given nutrient. The terms \(Q_{i}\), \(\mu _{\infty }\), and \(f_{i}\left( R_{i}\right) \) are all known to systematically change with cell size (see Box 2), where commonly the uptake function is given by
$$\begin{aligned} f_{i}=U_{max} \frac{R_{i}}{K_{i}+R_{i}} \end{aligned}$$
(18)
given the half-saturation constant \(K_{i}\) and the maximum uptake rate \(U_{max}\) (Burmaster 1979). In this model a is the flow rate of the system, which affects both the inflow of nutrients from outside the system where \(R_{i,0}\) is the concentration outside the system, and the loss of the nutrients from the system. Similarly, m is the mortality rate of the cells and is often taken to be equal to the flow rate a (Klausmeier et al. 2004a, b, 2007, 2008). In this system one nutrient is typically limiting because of the minimum taken in Eq. 17, and thus the equilibria of the system are typically dictated by the exhaustion and limitation of one nutrient. Previous work has shown that growth can be maximized in this framework by considering the allocation of resources to different cellular machinery, and that this leads to two optimum physiologies, one where maximum growth rate is optimized, and another where all of the resource equilibrium values are simultaneously minimized leading to resource colimitation and neutral competitiveness with all other species (Klausmeier et al. 2004a, b).
In this model the steady-state biomass, \({\mathcal {N}}^{*}\), limiting resource, \(R^{*}\), and quota of the limiting resource, \(Q^{*}\), are given by
$$\begin{aligned} {\mathcal {N}}^{*}= & {} \frac{a\left( R_{in}-R^{*}\right) \left( \mu _{\infty }-m\right) }{Q_{min}\mu _{\infty }m} \end{aligned}$$
(19)
$$\begin{aligned} R^{*}= & {} \frac{Q_{min}m\mu _{\infty }K}{U_{max}\left( \mu _{\infty }-m\right) -Q_{min}\mu _{\infty }m} \end{aligned}$$
(20)
$$\begin{aligned} Q^{*}= & {} Q_{min}\frac{\mu _{\infty }}{\mu _{\infty }-m}, \end{aligned}$$
(21)
(Legović and Cruzado 1997; Klausmeier et al. 2004a, b, 2007, 2008) where, for extant life, the physiological features are known to depend on size according to
$$\begin{aligned} U_{max}= & {} U_{0}V_{c}^{\zeta } \end{aligned}$$
(22)
$$\begin{aligned} K= & {} K_{0}V_{c}^{\beta } \end{aligned}$$
(23)
$$\begin{aligned} Q_{min}= & {} Q_{0}V_{c}^{\gamma } \end{aligned}$$
(24)
$$\begin{aligned} \mu _{\infty }= & {} \mu _{0}V_{c}^{\eta }. \end{aligned}$$
(25)
where the empirical values for the exponents and normalization constants are provided in Box 2. Given these general physiological scaling relationships the steady states are
$$\begin{aligned} {\mathcal {N}}^{*}\left( V_{c}\right)= & {} \frac{a\left( R_{in}-R^{*}\right) \left( \mu _{0}V^{\eta }-m\right) }{m Q_{0}\mu _{0}V^{\gamma +\eta }} \end{aligned}$$
(26)
$$\begin{aligned} R^{*}\left( V_{c}\right)= & {} \frac{m Q_{0}\mu _{0}K_{0}V^{\gamma +\eta +\beta }}{U_{0}V^{\zeta }\left( \mu _{0}V^{\eta }-m\right) -mQ_{0}\mu _{0}V^{\gamma +\eta }} \end{aligned}$$
(27)
$$\begin{aligned} Q^{*}\left( V_{c}\right)= & {} \frac{Q_{0}\mu _{0}V^{\gamma +\eta }}{\mu _{0}V^{\eta }-m} \end{aligned}$$
(28)
where it is important to note that these equations provide results for a single cell size considered in isolation. Below we first consider how these functions change due to cell size using known physiological scaling and then general exponents, and then we derive an ecosystem-level perspective from these results and discuss potential biosignatures under a range of exponent values.
Single-species Biogeochemistry for Extant Life
The above model provides a simple but general biogeochemical system where cellular physiology is coupled to an environment, and can be deployed to address ecosystems of various ecological complexity. First we consider the case where an environment is dominated by a single species, which would correspond to the measurement of a consistent particle size in our framework. Taking the known physiological scaling relationships for extant life (Box 2) we find that the size of the organism has a strong effect on the stoichiometric ratios of both the particles and fluid. Figure 4 gives the steady state N:P of cells as a function of steady state environmental N:P and cell size. The variation in the steady-state environmental and cellular N:P was achieved by varying the inflow concentrations \(R_{i,0}\).
We find that the largest cells will show the greatest deviation from the environmental concentration for most environmental ratios. Differences between the fluid and particle stoichiometry may define a biosignature, and these will be most noticeable for environments dominated by the largest cells. It should be noted that these results depend on the specific scaling relationships of the physiological features given in Box 2, which could greatly vary for life beyond Earth and are even known to vary across taxa for extant Terran life (DeLong et al. 2010; Kempes et al. 2012).
Generalized ecosystem biogeochemistry
The above coupling of cells to an environment considers a biogeochemical dynamic with only a single species that is characterized by a given cell size. This is the most rudimentary possibility for an ecosystem and is generally unlikely, but considering the full range of possibilities for life in the universe, could be of relevance to particular astrobiological contexts such as environments with low energy flux and characterized by a single resource limitation. However, we would like to expand this perspective to more complicated ecosystems with greater diversity as represented by a variety of cell sizes.
Classic resource competition theory in equilibrium (e.g. Tilman 1982; Levin 1970; Hutchinson 1953, 1957; Volterra 1927, 1931) indicates that for multiple species, in our case multiple cell sizes, to coexist on a single limiting resource they must all share the same \(R^{*}\) value. This is not naturally the case given the physiological scaling relationships outlined above, or the unlikelihood that many species will have identical physiological parameter values. In general, at most x number of species can coexist in equilibrium if there are x independent limiting factors (Levin 1970), and in our framework we can adjust the mortality rate, m, to abstractly represent the combination of many factors and to obtain coexistence. This adjustment could be the consequence of a variety of other factors such as variable predation, sinking rates, phage susceptibility, or intrinsic death. For our purposes this approach allows us to obtain a spectrum of cell sizes in connection with our earlier focus.
To enforce coexistance we take \(R^{*}\left( V_{c}\right) =R_{c}\), where \(R_{c}\) is a constant, in which case the required mortality rate is given by
$$\begin{aligned} m= & {} \frac{R_{c}U_{max}\mu _{\infty }}{R_{c}U_{max}+Q_{min}\mu _{\infty }\left( K+R_{c}\right) } \end{aligned}$$
(29)
$$\begin{aligned}= & {} \frac{R_{c}U_{0}\mu _{0}v^{\zeta +\eta }}{R_{c}U_{0}v^{\zeta }+Q_{0}\mu _{0}v^{\gamma +\eta }\left( R_{c}+K_{0}v^{\beta }\right) }. \end{aligned}$$
(30)
This function for m should be seen as the consequence of the complicated evolutionary dynamics of many species living in a coupled ecosystem where prey and predator traits have evolved over time and new effective niches have emerged. It should also be noted m is now size dependent compared with being set to constant value which was the case for the earlier results.
Our mortality relationship can be incorporated into \({\mathcal {N}}^{*}\) to give the scaling of biomass concentration for each cell size:
$$\begin{aligned} {\mathcal {N}}^{*} \left( V_{c}\right) =\frac{a\left( R_{in}-R_{c}\right) V_{c}^{-\zeta }\left( R_{c}+K_{0}v^{\beta }\right) }{R_{c}U_{0}}. \end{aligned}$$
(31)
This result has two important limits, where either the half-saturation constant is much smaller than the equilibrium value of nutrient in the environment, \(K_{0}v^{\beta }\ll R_{c}\), or is much bigger than this environmental concentration, which leads to
$$\begin{aligned} {\mathcal {N}}^{*}\left( v\right) = {\left\{ \begin{array}{ll} \propto V_{c}^{-\zeta } &{} K_{0}V_{c}^{\beta }\ll R_{c} \\ \propto V_{c}^{\beta -\zeta } &{} K_{0}V_{c}^{\beta }\gg R_{c} \end{array}\right. } \end{aligned}$$
(32)
These two relationships provide nice bounds on the scaling of \({\mathcal {N}}\) given the underlying physiological dependencies.
Similarly, the quota is given by
$$\begin{aligned} Q^{*}=Q_{0}V_{c}^{\gamma }+\frac{R_{c}v^{\zeta -\eta }U_{0}}{\mu _{0}\left( R_{c}+K_{0}v^{\beta }\right) }. \end{aligned}$$
(33)
which implies that the ratio of particle to fluid elemental abundance for the limiting nutrient is the following function of cell size
$$\begin{aligned} \frac{{\mathcal {N}}^{*}Q^{*}}{R^{*}}= {\left\{ \begin{array}{ll} \frac{a (R_{in}-R_{c}) \left( Q_{0} V_{c}^{\gamma -\zeta }+\frac{U_{0} V_{c}^{-\eta }}{\mu _{0}}\right) }{R_{c} U_{0}} &{} K_{0}V_{c}^{\beta }\ll R_{c} \\ \frac{a (R_{in}-R_{c}) \left( K_{0} Q_{0} \mu _{0} V_{c}^{\beta +\gamma -\zeta }+R_{c} U_{0} V_{c}^{-\eta }\right) }{ R_{c}^2 U_{0}\mu _{0}} &{} K_{0}V_{c}^{\beta }\gg R_{c} \end{array}\right. } \end{aligned}$$
(34)
This relationship is similar to the types of results shown in Fig. 4, but gives the ratio between cell and environment concentrations for a single element of interest (rather than as comparisons of ratios of elements), and importantly, does so under the constraints of coexistence. This result leads to particular biosignature possibilities when measuring only a single element, and does so for the more realistic ecosystem conditions of coexistence. If we measure the particle size distribution in an environment, then this is enough to specify the value of \(\alpha =-\zeta \) or \(\alpha =\beta -\zeta \) from Eq. 34, leaving us with \(\gamma \) and \(\eta \) to determine the element ratio scaling between cells and the environment as a function of particle size.
From a biosignatures perspective, the most ambiguous measurement would be particles that perfectly mirror the environmental stoichiometry where \(N^{*}Q^{*}/R^{*}\) equals a constant for all particle sizes. In the first limit, \(K_{0}V_{c}^{\beta }\ll R_{c}\), this would require \(\zeta =\gamma =-\alpha \) and \(\eta =0\). This result would imply that the quota and uptake rates would need to scale with the same exponent and as the negative value of the size exponent, both of which are consistent with the observations of Box 2 and Fig. 3b for extant life. However, this result also requires that there would be no change in growth rate with cell size, which is very unlikely from a variety of biophysical arguments.
In the second limit, \(K_{0}V_{c}^{\beta }\gg R_{c}\), a constant value of \(N^{*}Q^{*}/R^{*}\) requires that \(\zeta -\beta =\gamma =-\alpha \) and \(\eta =0\). Again the absence of changes in growth rate connected with \(\eta =0\) is unlikely. In addition, under this scenario the difference in the uptake and half-saturation scaling, represented by \(\zeta -\beta \), must equal the scaling of the quota and take the opposite value as the size-spectrum scaling, which is a combination that is not consistent with extant life and is a very special case in general. Thus, under both limits \(N^{*}Q^{*}/R^{*}\) is unlikely to have a constant value as a function of cell size, and an observed scaling in this ratio forms a likely biosignature.
This potential biosignature still requires one to measure the cell-size spectrum in detail, which may be challenging in certain settings or with certain devices. However, these relationships can be easily translated into an aggregate ecosystem-level measurement by averaging over all coexisting cells, where the average is given by
$$\begin{aligned} \left\langle \frac{\mathcal {N}^{*} Q^{*}}{R^{*}}\right\rangle= & {} \frac{1}{V_{max}-V_{min}}\int _{V_{min}}^{V_{max}}\frac{\mathcal {N}^{*}\left( V\right) Q^{*}\left( V\right) }{R^{*}}dV \end{aligned}$$
(35)
which, considering the two approximations for \({\mathcal {N}}\), becomes
To fully specify this community level ratio for generalized life we need to constrain the normalizations constants, \(Q_{0}\), \(k_{0}\), \(U_{0}\), and \(\mu _{0}\) given any choice of the exponents. A reasonable way to determine the values of these constants is to match the generalized rates to the observed Terran rates from Box 2 at a particular reference size, \(V_{r}\), which leads to
$$\begin{aligned} U_{0}= & {} U_{t}V_{r}^{\zeta _{t}-\zeta } \end{aligned}$$
(38)
$$\begin{aligned} K_{0}= & {} K_{t}V_{r}^{\beta _{t}-\beta } \end{aligned}$$
(39)
$$\begin{aligned} Q_{0}= & {} Q_{t}V_{r}^{\gamma _{t}-\gamma } \end{aligned}$$
(40)
$$\begin{aligned} \mu _{0}= & {} \mu _{t}V_{r}^{\eta _{t}-\eta }. \end{aligned}$$
(41)
After calibrating the constants to an intermediate cell size of \(V_{r}=10^{-18}\) (\(\hbox {m}^{3}\)), Fig. 5 gives the community level \(\left\langle {\mathcal {N}}^{*} Q^{*}/R^{*}\right\rangle \) as a function of the scaling exponents. When \(K_{0}v^{\beta }\ll R_{c}\) the size exponent \(\alpha \) specifies \(-\zeta \), and when \(K_{0}v^{\beta }\gg R_{c}\) then \(\alpha \) specifies \(\beta -\zeta \). In both approximations we plot \(\left\langle {\mathcal {N}}^{*} Q^{*}/R^{*}\right\rangle \) as a function of \(\eta \) and \(\gamma \) for a range of \(\alpha \) values.
We find that typically \(\left\langle {\mathcal {N}}^{*} Q^{*}/R^{*}\right\rangle \) differs from 1 for a wide range of \(\alpha \), \(\eta \) and \(\gamma \) values. This is true under both limits. For fixed values of \(\alpha \) the \(\left\langle {\mathcal {N}}^{*} Q^{*}/R^{*}\right\rangle =1\) line is a closed curve as a function of \(\eta \) and \(\gamma \) (Fig. 5). This curve defines the regime within which it is possible to find \(\left\langle {\mathcal {N}}^{*} Q^{*}/R^{*}\right\rangle =1\) for any value of either \(\eta \) and \(\gamma \), and this region covers a wide range of exponent values. However, the known values of \(\eta \) and \(\gamma \) for extant life occur fairly far from this curve and would show elemental concentrations that are distinguishable from the environment. It is likely that the full range of \(\alpha \), \(\eta \) and \(\gamma \) combinations explored here are precluded for biophysical reasons, but this requires more detailed work in the future. Finally, it is important to note that most \(\alpha \), \(\eta \) and \(\gamma \) combinations would yield cell-to-environment ratios that significantly differ from 1, and that the gradients are very steep around the \(\left\langle {\mathcal {N}}^{*} Q^{*}/R^{*}\right\rangle \)=1 line. Thus, it is a fairly safe assumption that the elemental abundances of cells should differ from the environment as this would be the expectation for physiological scaling chosen at random.