Introduction

The isotopic composition of the simplest forms of life in the early stages of evolution is largely determined by the isotopic composition of the Earth’s crust, which, in turn, formed during compression of the protoplanetary cloud. Part of the isotopes that were important for the emergence and subsequent evolution of life were radioactive and decayed with a certain period. Another part was continuously formed by nuclear reactions (for example, carbon-14). However, most biologically important isotopes are stable.

In the early stages of evolution, life existed in the simplest forms, such as in replicators, in separate RNA molecules, in hypercycles, and in similar forms. Such forms can barely be detected in minerals and rocks because of their instability with respect to a variety of environmental conditions. At later stages (with the emergence of protocells), the composition of the internal environment of such a simple organism began to differ from that of the external environment. Protocells could have had limited functions compared with those of modern cells. However, the transport of substances through the membrane of protocells is among the most important functions of cells and must have existed during the earliest stages of their origin. In the literature, various options for the possible structure and functioning of such protocells (see, for example Ganti 2003; Melkikh and Chesnokova 2012; Morowitz et al. 1988) have been considered. However, detecting the remains of protocells from this stage in the evolution of protocells is also problematic.

In such a situation, one of the few methods for determining traces of life during the early stages of the evolution is the isotopic method. The method relies on the differences between the content of the isotopes of various elements in a certain region of space in the environment; these differences are considered an indicator of the presence of life. Traces of life have been found in the most ancient rocks (see, for example, Pons et al. 2011) using this method.

However, models of isotope separation (see, for example, Brunner and Bernasconi 2005; Zeebe et al. 1999), are phenomenological, i.e., they lack a microscopic description of the mechanisms of isotope separation. Therefore, it is not possible to form definite conclusions about whether a deviation of the isotope composition in the area of a rock is really an indicator of the early stages of life.

The purpose of this paper is to construct a model of isotope separation for the various elements in the simplest cell. Using this model, we can predict the isotope separation coefficient, which is caused by the transport of ions across the cell membrane.

Deviations of the Isotope Composition as Indicators of the Existence of the Early Stages of Life

The following relation was used as a quantity to characterize the isotope composition of elements (see, e.g., Wolfsberg et al. 2009):

$$ {\updelta}_x=1000*\left({R}_X-{R}_{STD}\right)/{R}_{STD} $$
(1.1)

where R (RSTD is the standard value) is the ratio of the concentration of the heavy isotope to the concentration of the light isotope. The magnitude of δ may vary depending on the environmental conditions around the rock during its formation. In particular, the kinetics of phase transitions (liquid - gas, liquid - solid), generally speaking, differ for different isotopes. However, the measurement of deviations from the natural isotope composition in a small region of space allows us to conclude with high probability that a cell could have existed in this area. This conclusion particularly applies to the isotopes of elements that are important for cell viability.

A deviation of the isotope composition for different elements was considered to be an indicator of the existence of life. For example, the iron-sulfur world hypothesis (Wächtershäuser 2010) proposes that metals are able to retain and concentrate a significant amount of organic matter. Moreover, these metals are known to catalyze some redox reactions, and the author suggests the first biomolecules appeared due to these reactions. Wachtershauser considers iron, cobalt, and nickel to be the most versatile catalysts among the metals. Under anaerobic conditions, the most stable compounds are iron sulfides, which are clusters of metal surrounded by sulfur ligands. Metal catalysis provides the synthesis of compounds of volcanic gas (CO, NH3, HCN, and H2S).

Some authors (Bradley et al. 2009) state that an excess of C13 can be achieved through chemosynthesis.

Another paper (Brasier et al. 2006.) considered a part of the most ancient rocks, the so-called “deep carbon” rocks. Test samples from the Strelley Pool Chert and Dresser Formation showed an excess of C12, which could be formed biotically. A comparison of the microfossils from Apex Chert to samples from other sources indicated that Apex Chert microfossils were from bacteria that were more primitive and more ancient than cyanobacteria. The diameter of the potential sources of microfossils from Apex Chert do not exceed 20 micrometers. Studying of the above samples is complicated by the arbitrary location of the samples and by the lack of the direct connection between the shape, size, and morphology of the cell.

A study of samples from the Isua Supracrustal Belt, Greenland (Pons et al. 2011) from the Archean age showed significant depletion of heavy zinc isotopes in rocks that originated from volcanoes.

One article (Mojzsis et al. 1996) also considered samples that had an excess of C12. Samples (apatite crystals) were taken from the Isua Supracrustal Belt, Greenland and from the nearby Akilia Island. The resulting isotope compositions corresponded to a specific biochemical cycle. The inclusions had a high content of light carbon and were approximately 30 micrometers in size.

It must be emphasized that, of course, the separation of isotopes is a side effect of cell activity, but not a condition of its existence.

Model of Isotope Separation During the Transport of Ions Through a Cell Membrane

We first estimate the minimum characteristic size of the cells after its death and transformation into solid rock. It is known that the characteristic time of diffusion can be estimated on the basis of Einstein”s formula:

$$ \tau =\frac{R^2}{6D} $$

Here D represents the diffusion coefficient, R—is the characteristic size of cell (area containing a number of cells). We assume that the isotope exchange occurs only by diffusion (without cracks, hydrodynamic transport, etc.). If we take the characteristic size of 100 microns, and the characteristic time for 4 billion years, we find that these values are achievable for the diffusion coefficient in the solid, equal to 10−26 m2/s, for the characteristic size of 10 cm (a colony of cells) and a maximum diffusion coefficient equal to 10−20 m2/s etc. These coefficients are consistent with experimental data on diffusion in solids (see, for example, Borg and Dienes 1988; Heitjans and Karger 2005).

It is possible that the conditions under which such slow diffusion takes place after the transformation of living cells in a solid, are quite rare (as well as rarely preserved fossils of animals, whole skeletons, etc.). Yet such conditions can be realized when the cell quickly (before the membrane has not yet been destroyed) becomes part of the solid medium. For example, this may occur as a result of the collapse of rocks caused by landslides, volcanic activity, earthquakes, etc. The shape of the cells, of course, cannot be retained (in contrast to fossils). The speed of further diffusion of substances is essentially dependent on environmental parameters such as temperature, pressure, etc. However, if such conditions, are implemented, then the separation of isotopes in a given area of rock can indicate the presence of a living cell in this area in the past.

Generally speaking, the change in concentration of a substance in a given volume may be caused by several reasons. If we exclude the hydrodynamic transport (in which there is no separation of isotopes), we can write the balance of the substance inside the cell in accordance with (Melkikh and Seleznev 2012):

$$ \begin{array}{c}\hfill \frac{\partial \left({n}_j^iV\right)}{\partial t}=-{\displaystyle \sum_j{J}_j\left({n}_1^i,\dots {n}_j^i,\dots {n}_m^i,\varphi, {n}_1^o,\dots {n}_m^o\right)}\hfill \\ {}\hfill {\displaystyle \sum_j{Z}_j}{n}_j^{in}=0\hfill \end{array} $$
(2.1)

where n i j and n o j are the concentrations of substances inside and outside the cell, V – is the cell volume, φ – is the resting potential, J j – fluxes of ions through the membrane, Z j – is the charge of ion.

Since we are interested in the balance of isotopes of an element, the amount of change in any compartment (cell) does not depend on the chemical reactions in which it takes part. On the other hand, the change in the number of atoms of a given isotope could be due to radioactive decay (nuclear reactions). This process we will also be neglected. Equation (2.1) is based on the assumption that the main diffusion resistance is concentrated on the membrane, and the diffusion inside the cells is fast enough.

If we consider the steady state of the cells, the time derivative in equation (2.1) must be equal to zero. In such a case the flows through a boundary region of each component should be zero.

Models of isotope separation in living systems were built in several papers (see, for example, Brunner and Bernasconi 2005, Zeebe et al. 1999). However, in these models, the process of separation is considered qualitatively (phenomenologically). The processes that result from this effect remain unclear. Is it possible to identify any general patterns in the process of isotope separation in living systems? Quite naturally, the separation of isotopes of any element is related to the spatial transport of these isotopes. The most important form of transport for life is the transport of substances through the cell membrane. In living cells, the transport of substances can be divided into active transport (using external sources of free energy) and passive transport.

In several papers (Melkikh and Seleznev 2005, 2006, 2007, 2009; Melkikh and Sutormina 2013), models of the transport of substances in various cell types were constructed. A distinctive feature of the proposed models is that they allow independent calculations of the concentrations of all of the transported ions inside the cell and of the resting membrane potential.

From the physical point of view transport of substances in prokaryotes and eukaryotes does not differ significantly. However, since we are talking about modeling of the early stages of evolution, it seems that transport in archaea (Melkikh and Seleznev 2009) is the closest model.

The basis of the proposed model is the consideration of active transport as a two-level system. The probability of finding the system at a given level depends on the driving force of the process of transport - the difference between the chemical potentials of ATP-ADP (or other types of ions). Then the flow of ions across the membrane, generated by transport system, can be represented as:

$$ J=C\left({n}^i \exp \left(\varDelta {\mu}_A+\varphi \right)-{n}^o\right) $$

Let us describe the transport of singly charged ion in the steady state, according to (Melkikh and Seleznev 2005)

$$ C\left({n}^i \exp \left(\varDelta {\mu}_A+\varphi \right)-{n}^o\right)+P\left({n}^i \exp \left(\varphi \right)-{n}^o\right)=0 $$
(2.2)

Here, the first term is the active flux; the second term is the passive flux; C and P are values that characterize the intensity of the active and passive fluxes, respectively; φ is the potential of the cell membrane; and Δμ A is the difference between the chemical potentials of ATP and ADP (the driving force of active transport).

If we consider the transport of different isotopes of the same ion, then, generally speaking, both C and P may be different for the two isotopes. Suppose, however, that the value of C is identical for both isotopes and that the value of P is different. This assumption is based on the following arguments. Ions are actively transported after the arrival of the ion at the sorption center. Moreover, each separate sorption center should contain an ATP molecule. In this case, the reaction can only occur when these two particles both participate, which results in the movement of the ion from one side of membrane to the other. Naturally, the reverse reaction is also possible. All of these reactions have been extensively discussed previously (Melkikh and Seleznev 2006, 2012). Generally speaking, the rate of this reaction is proportional to the arrival rates for both the ion and the ATP at their sorption centers. The experimentally measured activity of various ATP-ases is approximately 103–104 s−1, but the concentration of the transferred ions can vary much more significantly by a factor of 105–106 (for example, for sodium ions and protons). This result indirectly confirms the assumption that the activity of ATP-ase is determined mainly by the rate at which ATP arrives at its sorption center.

The ion concentrations outside and inside the cell can be expressed using the following formulas (2.2):

$$ {n}^i\left(C \exp \left(\varDelta {\mu}_A+\varphi \right)+P \exp \left(\varphi \right)\right)={n}^o\left(C+P\right) $$
(2.3)
$$ {n}^i={n}^o\frac{C+P}{C \exp \left(\varDelta {\mu}_A\right)+P} \exp \left(-\varphi \right) $$
(2.4)

Consider the external and internal concentrations of the different isotopes (denote them by the indices 1 and 2) of one ion:

$$ {n^i}_1={n^o}_1\frac{C+{P}_1}{C \exp \left(\varDelta {\mu}_A\right)+{P}_1} \exp \left(-\varphi \right) $$
(2.5)

Write the coefficient of separation in the form:

$$ \xi =\frac{{n^i}_1}{{n^i}_2}\frac{{n^o}_2}{{n^o}_1} $$
(2.6)

If ξ = 1, then the effect of separation is absent. Using (2.5) we can obtain maximal coefficient of selection (consider С as a variable)

$$ \xi =\frac{\sqrt{\gamma } \exp \left(-\varDelta {\mu}_A/2\right)+1}{\sqrt{\gamma } \exp \left(-\varDelta {\mu}_A/2\right)+\gamma}\frac{\sqrt{\gamma } \exp \left(\varDelta {\mu}_A/2\right)+\gamma }{\sqrt{\gamma } \exp \left(\varDelta {\mu}_A/2\right)+1} $$
(2.7)

where

$$ \gamma =\frac{P_2}{P_1} $$

Thus, we have an expression for the coefficient of selection, which is optimized for the power (frequency) of the transport system. This means that if the active transport is large relative to passive transport (the first term in (2.2) is much larger than the second), then the coefficient of separation will approach unity. Otherwise (C is small, and the active flux is small compared with the passive flux), the partition coefficient will also approach unity. This result can be observed directly from expressions (2.5) and (2.6).

Note, however, that such a regime, under which the separation factor is maximal, is not effective and is not beneficial to the cell because the passive flux is large (relative to the active flux). However, a high efficiency of ion transport can be achieved only if the passive flux is negligibly small. Passive flow in this case would just mean the loss of ion free energy.

Let us analyze the dependence of the maximum separation coefficient (2.7) on the parameters.

When Δμ A  > > 1, expression (2.7) can be simplified (here, it is assumed that only the thermal velocities contribute to the ratio of permeabilities and that the barriers for different isotopes are equal to each other):

$$ \xi =\frac{1}{\gamma }=\sqrt{\frac{m_1}{m_2}} $$
(2.8)

Such a formula was considered to be the basis for the isotope separation coefficient in many (including non-living) systems, but it should be noted that this is only a limiting case and these conditions will not always be met.

For small values of Δμ A (which corresponds to a lack of free energy in a cell or to cell death), we obtain:

$$ \xi =1-\frac{\sqrt{\gamma }-1}{\sqrt{\gamma }+1}\varDelta {\mu}_A $$

Thus, the isotope separation effect disappears when Δμ A  → 0 and also when γ → 1.

Naturally, isotope separation processes must also occur in modern cells. Table 1 shows the experimental data for the enrichment factor δ = ξ-1 for various ions in living systems, along with the enrichment factors calculated by formula (2.8). Calculations of the partition coefficient depend on which isotope is considered first; thus, we consider value of the modulus δ.

Table 1 Experimental and theoretical values of the enrichment factors for various ions
Full size table

As shown in table 1, the theoretical and experimental values of the enrichment coefficients agree in some cases. However, there are significant differences between the theoretical and experimental data. The main reasons for the differences between theory and experiment are the following:

  1. 1.

    Some ions are transported across the membrane as a part of complexes whose mass can be considerably greater than the mass of the ion. This will decrease the maximum separation coefficient. This can explain the fact that the theoretical results in the table tend to be larger than experimental ones. However, the separation of isotopes during transport complex molecules requires separate consideration, since such a molecule will contain a many isotopes of other elements.

  2. 2.

    For many ions more than one active transport system exists. This is the way of regulation of ion transport when the composition of the environment (see, eg, Melkikh and Sutormina 2013) changes. This means that in some cases the total active flow of any ion will be lower than the same flow generated by one system of transport. Accordingly, the coefficient of isotope separation will reduce.

  3. 3.

    We considered the maximum (optimized) value of selection coefficient. However, the real value may be lower. For example, equation (2.7) is maximized in value of C. Then the formula (2.8) is maximized again. Since the separation of isotopes is considered as a side effect of work of transportation systems, such ideal cases may not be realized in nature.

In monograph (Melkikh and Sutormina 2013), models of ion transport for different types of cells (animal, plant, protozoa, bacteria, and archaea) were constructed. One of the most important conclusions that can be drawn from an analysis of the models is that all positive ions, even those at a low concentration in the environment, must be transported actively. Otherwise, their concentration inside the cell will be high, resulting in a significant drop in the resting membrane potential. This conclusion is particularly relevant for ions with a charge of 2+ or more. For the separation of isotopes, this conclusion means that even trace elements must undergo both passive and active transport (i.e., the expression for full flow (2.2) should be true). Naturally, the stoichiometry varies for different types of ions, and the driving force of the active transport (e.g., it may be cotransported with another ion in a symporter) also varies, but the general form of expression (2.2) is retained.

Maximal Coefficient of Separation of Isotopes in a Cell and Efficiency of Transport System

Let us find the relationship between the coefficient of separation of ions and the energy efficiency of the transport system.

We introduce (in accordance with Melkikh and Seleznev 2006) the efficiency of the system as follows:

$$ \eta =\frac{J_{\varDelta \mu = 0}\cdotp \varDelta \mu }{v_{\varDelta \mu = 0}\cdotp \varDelta {\mu}_A} $$
(3.1)

The index Δμ = 0 indicates that the values of the fluxes in the numerator and in the denominator (ion flux and ATP flux) are for this condition (Δμ = 0). This allows us to select the pure active flux (because only this flux is useful) of the total flux of ions. Divide expression (2.2) by the ion concentration inside the cell:

$$ C\left( \exp \left(\varDelta {\mu}_A+\varphi \right)-\frac{n^o}{n^i}\right)+P\left( \exp \left(\varphi \right)-\frac{n^o}{n^i}\right)=0 $$
(3.2)

Let us introduce the difference in the chemical potentials between the external and internal environment:

$$ \varDelta \mu =\mathit{\hbox{-}}\varphi + In\frac{n^o}{n^i} $$
(3.3)
$$ \frac{n^o}{n^i}= \exp \left(\varDelta \mu +\varphi \right) $$
(3.4)

Substitute the expression (3.4) into equation (3.2) and obtain:

$$ \exp \left(\varDelta \mu \right)=\frac{C \exp \left(\varDelta {\mu}_A\right)+P}{C+P} $$

At zero chemical potential difference, the second part of the total flux, which is responsible for the passive transport of ions, becomes equal to zero. As a result, we obtain:

$$ {J}_{\varDelta \mu =0}=C{n}^o\left( \exp \left(\varDelta {\mu}_A\right)-1\right)={\nu}_{\varDelta \mu =0} $$
(3.5)

The expression for the efficiency becomes simpler:

$$ \eta =\frac{\varDelta \mu }{\varDelta {\mu}_A}=\frac{ \ln \left(\frac{C \exp \left(\varDelta {\mu}_A\right)+P}{C+P}\right)}{\varDelta {\mu}_A} $$
(3.6)

Let us express the constant C from (3.6):

$$ C=P\frac{ \exp \left(\eta \varDelta {\mu}_A\right)-1}{ \exp \left(\varDelta {\mu}_A\right)- \exp \left(\eta \varDelta {\mu}_A\right)}=\alpha P $$
(3.7)

where

$$ \alpha =\frac{ \exp \left(\eta \varDelta {\mu}_A\right)-1}{ \exp \left(\varDelta {\mu}_A\right)- \exp \left(\eta \varDelta {\mu}_A\right)} $$

Because С is equal for both isotopes, we can write:

$$ C=\alpha {P}_1 $$

This expression indicates that, in general, η is also different for the two isotopes, but this minor difference will be neglected.

Substitute C into the formula (2.4) and then use the result in (2.6) we obtain:

$$ \xi =\frac{\alpha^2 \exp \left(\varDelta {\mu}_A\right)+\alpha \exp \left(\varDelta {\mu}_A\right)+\alpha \frac{P_1}{P_2}+\frac{P_1}{P_2}}{\alpha^2 \exp \left(\varDelta {\mu}_A\right)+\alpha +\alpha \exp \left(\varDelta {\mu}_A\right)\frac{P_1}{P_2}+\frac{P_1}{P_2}} $$
(3.8)

Consider the limiting cases of (3.8):

  1. 1.

    η approaches unit (there are only active fluxes), then α approaches infinity, and the enrichment factor approaches zero (the separation factor approaches unity). This means that there is no separation of isotopes.

  2. 2.

    η approaches zero (there are only passive streams), then α approaches zero, and the enrichment factor approaches zero (separation factor approaches unity). In this case, there is also no separation of isotopes.

After maximizing the separation factor for α, we obtain the value of η in the following form:

$$ \eta =\frac{1}{\varDelta {\mu}_A} \ln \left(\frac{1+\sqrt{\gamma } \exp \left(\frac{\varDelta {\mu}_A}{2}\right)}{1+\sqrt{\gamma } \exp \left(-\frac{\varDelta {\mu}_A}{2}\right)}\right) $$
(3.9)

For a sufficiently large difference in the chemical potentials of ATP and ADP, Δμ A , and for values γ, close to unity, this expression approaches ½.

Thus, it is confirmed that the transfer processes, which are effective in terms of energy conversion, are inefficient in the sense of separation of isotopes, and vice versa.

Of course, such a mode in which the separation coefficient of isotope is maximal is hardly realized in living cells. However, the resulting formula can be viewed in a broader context, referring to the creation of artificial cells, for which the separation of isotopes of any ion would be a major task.

Note, however, that the term “efficiency” in relation to the processes for transporting substances may have different meanings. A previous study (Melkikh and Sutormina 2013) examined different definitions of the effectiveness. In particular, the efficiency may be associated with the resistance of the internal medium of the cells to changes in the environment. It has been shown that such stability (robustness) may only be provided by active and passive fluxes. With respect to the separation of isotopes, this means that the value η = 1 is not advantageous to the cell, and thus the first limiting case does not occur (i.e., isotope separation would occur).

Conclusion

In this paper, we constructed a model of the active transport of ions; this model provides an understanding of the possible mechanism of isotope separation during the transport of substances through the cell membrane. This allows us to consider the isotope separation process in a broader context, while bearing in mind the various modes of a transportation system (including modes that are not realized in nature). These modes can be implemented, for example, in the creation of artificial cells. The data on the isotope separation coefficients of various ions in a restricted region of space not only potentially indicate the presence of life in this area, but might also provide information on the ion transport systems of protocells in the early stages of the evolution of life.