Incorporating antagonistic pleiotropy into models for molecular replicators

Abstract

In modern cells, chromosomal genes composed of DNA encode multi-subunit protein/RNA complexes that catalyze the replication of the chromosome and cell. One prevailing theory for the origin of life posits an early stage involving self-replicating macromolecules called replicators, which can be considered genes capable of self-replication. One prevailing theory for the genetics of aging in humans and other organisms is antagonistic pleiotropy, which posits that a gene can be beneficial in one context, and detrimental in another context. We previously reported that the conceptual simplicity of molecular replicators facilitates the generation of two simple models involving antagonistic pleiotropy. Here a third model is proposed, and each of the three models is presented with improved definition of the time variable. Computer simulations were used to calculate the proliferation of a hypothetical two-subunit replicator (AB), when one of the two subunits (B) exhibits antagonistic pleiotropy, leading to an advantage for B to be unstable. In model 1, instability of B yields free A subunits, which in turn stimulate the activity of other AB replicators. In model 2, B is lost and sometimes replaced by a more active mutant form, B′. In model 3, B becomes damaged and loses activity, and its instability allows it to be replaced by a new B. For each model, conditions were identified where instability of B was detrimental, and where instability of B was beneficial. The results are consistent with the hypothesis that antagonistic pleiotropy can promote molecular instability and system complexity, and provide further support for a model linking aging and evolution.

Graphical Abstract

1. Introduction

There is currently no consensus definition for life, however, the ability to replicate and to undergo Darwinian evolution are often cited as key features (Cleland and Chyba, 2002; Higgs, 2017; Takeuchi et al., 2017; Trifonov, 2011). The cell is generally considered to be the basic unit of life, based upon its ability to replicate and undergo Darwinian evolution, as well as related features including the ability to carry out metabolism and the storage of information required to direct replication. In modern cells, chromosomal genes composed of DNA encode multi-subunit protein/RNA complexes that catalyze the replication of the chromosome and cell (Hunt et al., 2011; Sclafani and Holzen, 2007).

One prevailing theory for the origin of life posits an early stage involving self-replicating macromolecules called replicators, which can be considered genes capable of self-replication (Joyce, 2002). The discovery of catalytic RNAs, and the conserved role of RNA molecules in fundamental cellular processes suggests these ancient replicators may have been composed of RNA. As such, evolution may have proceeded through a stage dominated by functional RNAs, often called “the RNA world” (Gilbert, 1986; Joyce, 2002). Mechanisms for self-replication might be non-enzymatic, where the RNA polymer acts as a scaffold that directs the arrangement and covalent attachment of activated monomers (Zhou et al., 2020a, b). Alternatively, the replicator might be an RNA polymerase, capable of incorporating substrate monomers into a new copy of itself, using itself or another molecule as a template. Consistent with these ideas, systems have been created where two RNA ligases catalyze each other’s synthesis using four distinct oligonucleotide substrates (Lincoln and Joyce, 2009), and where one RNA ligase catalyzes its own replication using two oligonucleotide substrates (Olea and Joyce, 2016). In addition, RNA polymerase ribozymes have been identified that can synthesize functional, non-self RNA molecules using an RNA template (Horning and Joyce, 2016). However, no single RNA or other polymer-based polymerase has yet been identified that is capable of complete self-replication. Other possible chemical structures for early replicators have also been hypothesized, including polypeptides and combination nucleic acid/peptide polymers (Kosikova and Philp, 2017; Piette and Heddle, 2020).

The modern cell includes an enveloping lipid bi-layer membrane that serves several functions, including acting as a semi-permeable barrier that maintains the necessary concentration of molecules within the cell to support replication. Replicators functioning at early stages of evolution might have been free in solution, or perhaps more likely, enclosed in proto-cell membranes or other proto-cellular enclosures, such as nooks in the walls of undersea hydrothermal vents (Lane and Martin, 2012; Schrum et al., 2010). In this way, natural selection might act on groups of collaborating replicators.

Aging of living organisms is more correctly termed senescence (Finch, 1990), however the colloquial term aging is used here for convenience and for consistency with other studies. Similar to the situation with defining life, there is no universally accepted definition of aging. However, most investigators would likely agree with a definition of aging involving an increase in age-specific mortality rate, and a decrease in age-specific reproductive capacity (Flatt, 2012; Tower, 2009). The prevailing model for the genetics of aging in humans and other organisms is based on the declining force of natural selection with age, mutation accumulation, and antagonistic pleiotropy (Fabian and Flatt, 2011; Medawar, 1952; Rose and Charlesworth, 1980; Williams, 1957). In young animals, a deleterious mutation is strongly selected against, because it reduces reproductive fitness and is usually not passed on to the next generation (Fig. 1a). In contrast, a “late acting” mutation, where the deleterious effect is not expressed until late ages, is not strongly selected against, because by the time the negative effect is expressed, the mutation has typically already been passed on to the next generation. This phenomenon is called the declining force of natural selection with age, and is thought to lead to the accumulation of deleterious mutations in the genome that contribute to the aging phenotype. The same mutation that has a late-acting deleterious effect may also have a beneficial effect early in life, such as increasing reproductive fitness, and therefore the mutation may be maintained by positive selection, despite the fact that it contributes to the aging phenotype (Fig. 1b). This phenomenon is referred to as antagonistic pleiotropy, in that the same mutation has a benefit in one context, but is deleterious in another context. There is significant support for the role of antagonistic pleiotropy in aging in humans and other organisms (Bartke, 2011; Byars and Voskarides, 2020; Hughes and Reynolds, 2005). Antagonistic pleiotropic mutations may have effects that are common to both sexes, or might be sex-biased or sex-specific. Sexually antagonistic mutations are ones that are beneficial to one sex and deleterious to the other sex, or deleterious in different ways to both sexes (Fig. 1c) (Brooks and Garratt, 2017; Hosken et al., 2009; Immler and Otto, 2018; Immonen et al., 2018; Maklakov and Lummaa, 2013; Mank, 2017; Tower, 2006). Sexually antagonistic mutations may be maintained by sex-specific selective pressures, such as sex-differences in optimal reproductive strategies.

Fig. 1.

Overview of antagonistic pleiotropy theory. (a) The declining force of natural selection with age and mutation accumulation. In young animals, a deleterious mutation is strongly selected against, because it reduces reproductive fitness and is not passed on to the next generation. In contrast, a “late acting” mutation, where the deleterious effect is not expressed until late ages, is not strongly selected against, because by the time the negative effect is expressed, the mutation has typically already been passed on to the next generation. (b) Antagonistic pleiotropy. The same mutation that has a late-acting deleterious effect may also have a beneficial effect early in life, such as increasing reproductive fitness, and therefore the mutation may be maintained by positive selection, despite the fact that it contributes to the aging phenotype. Antagonistic pleiotropic mutations may have effects that are common to both sexes, or might be sex-biased or sex-specific. (c) Sexual antagonistic pleiotropy (SAP). Sexually antagonistic mutations might be beneficial to one sex and deleterious to the other sex, or might be deleterious in different ways to both sexes. Sexually antagonistic mutations may be maintained by sex-specific selective pressures, such as sex-differences in optimal reproductive strategies.

We have previously hypothesized that when two genes collaborate to create a replicator, it may be selectively advantageous for one gene to have a shorter half-life than the other, i.e., to be unstable (Tower, 2006). This situation was proposed to be selectively advantageous because it leads to increased complexity of the system (Fig. 2), allows the shorter-lived gene to evolve as a separate entity from the longer-lived gene, and facilitates asymmetric inheritance mechanisms. We previously reported that the conceptual simplicity of molecular replicators facilitates the generation of two simple models involving antagonistic pleiotropy (Li et al., 2018). A molecular replicator was considered, that is composed of two subunits, A and B. In model 1, instability of B yields free A subunits, which in turn bind-to and stimulate the activity of other AB replicators (Fig. 3a). In model 2, B is lost and sometimes replaced by a more active mutant form, B′ (Fig. 3b). Here a Model 3 is proposed, in which B becomes damaged and loses activity, and its instability allows it to be replaced by a new B (Fig. 3c). In each model, B exhibits antagonistic pleiotropy. B is beneficial in one context because it is required for the replication activity of the AB replicator. B is detrimental in another context because it sequesters A subunits that might otherwise stimulate other replicators (Model 1), or create the more active form B′ (Model 2), or re-create undamaged B (Model 3).

Fig. 2.

Differential replicator subunit stability can lead to a more complex system. (a) Model for a two-subunit replicator. The two-subunit replicator AB utilizes N number of substrate molecules (S) to generate a new copy of AB. (b) Instability of AB. If the A and B subunits of the AB replicator are unstable, and are degraded at the same time, this leads to a reduced population of AB replicators. (c) If the B subunit is less stable than the A subunit, this leads to a reduced population of AB replicators and free A subunits; this represents a more complex system because it is composed of two distinct species.

Fig 3.

Models for replicator proliferation involving antagonistic pleiotropy. A two-subunit replicator (AB) is considered, where the B subunit exhibits antagonistic pleiotropy, which in turns leads to a selective advantage for B to be unstable. (a) Model 1: Stimulatory effect of free A subunits. The B subunit degrades, leading to free A subunits. The free A subunits bind to AB replicators and stimulates their activity. (b) Model 2: B is sometimes replaced by a more active form B′. In this model, A has the ability to generate either a new B or B′ subunit. (c) Model 3: B sustains damage and is replaced by a new B. In this model, B sustains damage (B*) which reduces the activity of the AB replicator. Instability of B leads to free A subunit, and A is able to generate a new B. In these models, B exhibits antagonistic pleiotropy because B is beneficial in one context (it is required for the self-replication activity of the AB replicator), but B is detrimental in another context: In Model 1, B sequesters A subunits that might otherwise stimulate the activity of other AB replicators. In Model 2, B sequesters A subunits that might otherwise create a higher-activity form of B. In Model 3, B sustains damage that reduces the activity of the AB replicator, and damaged B sequesters A subunits that might otherwise create a new B.

Here the proliferation of the hypothetical AB replicator was calculated under each of the three models, and with improved definition of the time variable. Simulations were conducted using stochastic Python models, as well as using differential rate equations. For each of the three models, conditions were identified where instability of B was detrimental, and where instability of B was beneficial. For both models 1 and 2, conditions were identified where moderate instability of B was optimal, whereas more extreme stability or instability of B was detrimental, reminiscent of modern cellular macromolecules. The models are discussed relative to the behavior of modern replicators, including genes and cells.

2. Methods

Simulations for each of the three models were created using Python (version 3.7.3) and the Python matplotlib library. The input variables and steps for analysis for each of the models are as follows, with the name of the variable in the software presented in parentheses. Detailed user manuals are provided for the software for each model.

2.1. Model 1

The name of the software is RecordResult_1.py and GraphDrawing_1.py, and the name of the user manual is UserManual_Model1.docx.

2.1.1. Model 1 input variables

1. Time (time): the number of seconds this model will simulate.

2. Initial number of gene AB (initialAB): the number of AB genes at the start of the simulation.

3. Replicating activity of gene AB (activityAB): the number of new AB replicators created by AB per second.

4. Replicating activity of gene AAB (activityAAB): the number of new AB replicators created by AAB per second.

5. Stability of gene B (stabilityB): the percent of current AB genes that will lose B and become A per second. For example, if stabilityB = 20, then 20 percent of current AB genes will become A per second, and the A genes will generate new AAB genes.

2.1.2. Model 1 steps for analysis

1. The analysis begins with the values generated from the previous second: [AB], [AAB], [A]. The first second begins with [AB] = 10.

2. Calculate new AB genes generated from replication of AB and AAB.

3. Calculate new AAB genes generated from A binding to AB.

4. Calculate new A genes generated by loss of B from AB.

5. Calculate loss of AB genes due to binding to A to create AAB genes

6. Calculate the number of gene AB, AAB and A for the next second.

2.2. Model 2

The name of the software is RecordResult_2.py and GraphDrawing_2.py, and the name of the user manual is UserManual_Model2.docx.

2.2.1. Model 2 input variables

1. Time (time): the number of seconds this model will simulate.

2. Initial number of gene AB (initialAB): the number of AB genes at the start of the simulation.

3. Replicating activity of gene AB (activityAB): the number of new AB replicators created by AB per second.

4. Stability of gene B and B′ (stability): the percent of current AB or AB′ genes that will lose B or B′ and become A per second. For example, if stabilityB = 20, then 20 percent of current AB genes as well as 20 percent of current AB′ genes will become A each second.

5. Ratio of AB to AB′ (ratio): the ratio of newly generated genes AB and AB′ per second. For example, if the ratio is 1.0, then for all A genes that are not under hibernation, 50 percent of them will become AB and the other 50 percent will become AB′ each second.

6. Replicating activity of gene AB′ (activityABprime): the number of new AB′ replicators created by AB′ per second.

7. Hibernation time of gene A (hibernation): the percent of current A that remains free A per second. Therefore 100-hibernation is the percent of current A that becomes AB and AB′.

2.2.2. Model 2 steps for analysis

1. The analysis begins with the values generated from the previous second: [AB], [AB′], [A]. The first second begins with [AB] = 10.

2. Calculate new AB genes generated by AB and by A.

3. Calculate new AB′ genes generated from AB′ and A.

4. Calculate new A genes generated from AB and AB′.

5. Calculate the number of gene AB, AB′ and A for the next second.

2.3. Model 3

The name of the software is RecorResult_3.py and GraphDrawing_3.py, and the name of the user manual is UserManual_Model3.docx.

2.3.1. Model 3 input variables

1. Time (time): the number of seconds this model will simulate.

2. Initial number of gene AB (initialAB): the number of AB genes at the start of the simulation.

3. Replicating activity of gene AB (activityAB): the number of new AB replicators created by AB per second.

4. Damage rate of gene B (damageB): the percent of current AB genes that will be damaged and become AB*. For example, if damageB = 20, then 20 percent of existing AB genes will become AB* each second.

5. Stability of gene B (stabilityB): the percent of current AB genes that will lose B and become A per second.

6. Stability of gene B* (stabilityBstar): the percent of current AB* genes that will lose B* and become A per second.

7. Hibernation percent of gene A (hibernation): the percent of current A that remains free A per second. Therefore 100-hibernation is the percent of current A that becomes AB.

8. Replicating activity of gene AB* (activityABstar): the number of new AB replicators created by AB* per second.

2.3.2. Model 3 steps for analysis

1. The analysis begins with the values generated from the previous second: [AB], [AB*], [A]. The first second begins with [AB] = 10.

2. Calculate new AB genes generated by AB, AB* and A.

3. Calculate AB* genes generated from AB.

4. Calculate A genes generated from AB and AB*.

5. Calculate the number of AB, AB* and A for the next second.

2.4. Models 1–3 incorporating the Gompertz equation

For each of the three models, the simulations were modified such that subunit instability proceeds according to the Gompertz equation (μ_x = ae^bx)(Kirkwood, 2015). μ_x = mortality rate at age x, a = initial mortality rate, b = mortality rate acceleration with age, and e = Euler’s number. In each model the “initial stability” corresponds to the Gompertz parameter a (initial mortality rate), and “expoRate” corresponds to Gompertz parameter b (mortality rate acceleration with age). Gompertz-versions of software and user manuals are available upon request.

2.5. Models 1–3 using differential equations

The differential equation approximations are slightly different than the Python simulations. First, the differential equations are deterministic rather than stochastic. Second, the parameters are rates rather than probabilities. For example, consider the baseline model $\dot{AB}=g_{1}AB$ with replicator parameter g₁ (where g₁ > 0). The dot indicates the derivative with respect to time. The solution is AB(t) = c₀ exp(g₁t) where t represents time and c₀ is the initial condition. In one unit of time the amount of AB increases by a multiplicative factor of exp(g₁). Similarly, a simple model for decay is $\dot{AB}=-dAB$ with decay parameter d (where d > 0) has solution AB(t) = c₀ exp(−dt). In one unit of time the amount of AB decreases by a multiplicative factor of exp(−d). The models were modified so that A instantly combines with the other genes. Each of the models are two-dimensional systems of linear homogeneous differential equations. Equations of this form have solutions that are the sum of two exponential functions. For each of the models, the greater of the two exponential rates was calculated. In the figures 7, 8 and 9, this rate was subtracted from the rate of the baseline model (g₁).

Fig. 7.

Model 1 using differential equations. The continuum of colors represents the difference of the rate of the exponential solution from the baseline model. In the baseline, in one unit of time, the amount of AB increases by a multiplicative factor of exp(g₁). The darkest red indicates the rate is 0.4 greater than the baseline, white indicates the rate is equal to the baseline, and the deepest blue indicates the rate is 0.2 less than the baseline. The replicator rate of AB (g₁) is fixed at 1.0, the replicator rate of AAB (g₂) varies from 0.0 to 4.0 (corresponding to a 0 to 55 fold increase per unit time), and the decay rate of B (d) varies from 0.0 to 2.0 (corresponding to a 0% to 84% reduction per unit time).

Fig. 8.

Model 2 using differential equations. The continuum of colors represents the difference of the rate of the exponential solution from the baseline model. In the baseline, in one unit of time, the amount of AB increases by a multiplicative factor of exp(g₁). The fraction of A genes that become AB (f₁) varies in the different panels: (a) 10%, (b) 50%, and (c) 90%. The darkest red indicates the rate is 1.0 greater than the baseline, white indicates the rate is equal to the baseline, and the deepest blue indicates the rate is 0.8 less than the baseline. Note for panel (a) the lowest rate is 0.8 less than the baseline, for panel (b) the lowest rate is 0.4 less than the baseline, and for panel (c) the lowest rate is 0.1 less than the baseline. The replicator rate of AB (g₁) is fixed at 1.0, the replicator rate of AB′ (g₂) varies from 0.0 to 2.0 (corresponding to a 0 to 7.4 fold increase per unit time), and the decay rate of B (d) varies from 0.0 to 2.0 (corresponding to a 0% to 84% reduction per unit time).

Fig. 9.

Model 3 using differential equations. The continuum of colors represents the difference of the rate of the exponential solution from the baseline model. In the baseline, in one unit of time, the amount of AB increases by a multiplicative factor of exp(g₁). The darkest red indicates the rate is 0.7 greater than the baseline, white indicates the rate is equal to the baseline, and the deepest blue indicates the rate is 1.0 less than the baseline. The replicator rate of AB (g₁) is fixed at 1.0, the rate of AB to AB* (m) is fixed at 3 corresponding to a 20 fold increase per unit time, the replicator rate of AB* (g₂) varies from 0.0 to 2.0 (corresponding to a 0 to 7.4 fold increase per unit time), and the decay rate of B* (d) varies from 0.0 to 2.0 (corresponding to a 0% to 84% reduction per unit time).

2.5.1. Model 1 using differential equations

The parameters are g₁ the replicator rate of AB, g₂ the replicator rate of AAB, and d the decay rate of B. The system of equations is,

$$\begin{gathered} \dot{AB}=\left(g_1-2d\right)AB+g_{2}AAB \\ \dot{AAB}=dAB \end{gathered}$$

The larger of the two exponential rates is,

$$\lceil g_1-2d+\sqrt{{\left(g_1-2d\right)}^2+4g_{2}d}\rceil /2$$

2.5.2. Model 2 using differential equations

The parameters are g₁ the replicator rate of AB, g₂ the replicator rate of AB′, d the decay rate of B and B′, f₁ the fraction of A genes that become AB, and f₂ the fraction of A genes that become AB′. Note: f₁ + f₂ = 1. The system of equations is,

$$\begin{gathered} \dot{AB}=\left(g_1-d{f}_2\right)AB+d{f}_{1}A{B}^{\prime } \\ {\dot{AB}}^{\prime }=d{f}_{2}AB+\left(g_2-d{f}_1\right)A{B}^{\prime } \end{gathered}$$

The larger of the two exponential rates is,

$$\left\{g_1+g_2-d+\sqrt{{\left[g_1-g_2+d\left(f_1-f_2\right)\right]}^2+4d^2{f}_1{f}_2}\right\}/2$$

2.5.3. Model 3 using differential equations

The parameters are g₁ the replicator rate of AB, g₂ the replicator rate of AB*, d₂ the decay rate of B*, and m the rate AB becomes AB*. Note: since in this model AB decays to A, but then instantly becomes AB again, there is no d₁ decay rate parameter. The system of equations is,

$$\begin{gathered} \dot{AB}=\left(g_1-m\right)AB+\left(g_2+d_2\right)A{B}^{*} \\ {\dot{AB}}^{*}=mAB-d_{2}A{B}^{*} \end{gathered}$$

The larger of the two exponential rates is,

$$\lceil g_1-m-d_2+\sqrt{{\left(g_1-m+d_2\right)}^2+4m\left(g_2+d_2\right)}\rceil /2$$

3. Results

Various combinations of variables were tested for each model using the Python simulations, and for each model an example is presented where instability of B was detrimental, and an example is presented where instability of B was beneficial. The units on the X axis are given in seconds, and a correction factor is required to bring this scale in line with the anticipated properties of a hypothetical replicator. Assuming the replicator is an RNA polymerase 1,000 nucleotides in length, and the rate of polymerization is similar to modern RNA polymerases (approximately 60 nts/sec; (Fuchs et al., 2014)), it would take at least 17 seconds to replicate one new copy, and therefore the scale would be approximately seconds x17. However, it seems unlikely that the ancestral replicator would have a speed equal to modern RNA polymerase and its many co-factors. Assuming the rate of polymerization is similar to the 24-3 RNA polymerase ribozyme (0.02 nts/sec; (Horning and Joyce, 2016), it would take about 50,000 seconds to replicate, and the scale would be seconds x50000; however, it also seems likely a successful ancestral replicator would be more efficient than the 24-3 ribozyme.

3.1. Model 1

In Model 1, the instability of B generates free A subunits, and these free A subunits then bind to AB replicators to generate AAB replicators, potentially conferring greater activity. In this model, B exhibits antagonistic pleiotropy, because B is beneficial in one context, in that it is essential for the replication activity of AB. At the same time, B is detrimental in another context, in that B sequesters A subunits that might otherwise bind to AB replicators and stimulate their activity. Simulations were conducted where the stability of B was varied from most stable (stabilityB=10; 10% of B is lost each second; orange line), to most unstable (stabilityB=90; 90% of B is lost each second; teal blue line; Fig. 4). When the activity of AB=1 (AB makes one new AB replicator per second) and the activity of AAB=5 (AAB makes five new AB replicators per second), this represents a 5-fold stimulation. Under these conditions, instability of B is detrimental (Fig. 4a); as the stability of B was varied from most stable to least stable there was a progressive decrease in replicator proliferation. Similarly, instability of B was observed to be detrimental under conditions where the stimulation caused by A was varied from 2-fold to 7-fold (data not shown). However, then the stimulation caused by A was increased to 8-fold, a more complex pattern emerged (Fig. 4b). Magnification of the data at an early time point in the simulation (6 seconds), shows that instability of B is progressively detrimental (Fig. 4c). However, by the end of the simulation at 10 seconds, a different pattern has emerged. The greatest proliferation of replicators was observed for intermediate stabilities of B (stabilityB=20, stabilityB=30, stabilityB=40), whereas both greater stability of B (stabilityB=10) and greater instability of B (stabilityB=50 to 90) yielded less proliferation (Fig. 4d). Increasing the stimulatory effect of A to 10-fold (Fig. 4e), or 20-fold (Fig. 4f), also produced a pattern where intermediate stabilities of B were optimal, and more extreme stability or instability of B was detrimental. These results indicate that moderate instability of B becomes beneficial when the stimulatory effect of free A subunits is of sufficiently large magnitude.

Fig. 4.

Computer simulations of Model 1, wherein free A subunits bind to other AB replicators to create AAB replicators with potentially greater activity. Stability of B is varied from most stable (stabilityB=10, where 10% of B species are lost each second), to most unstable (stabilityB=90, where 90% of B species are lost each second). Baseline is a plot of activityAB=1 (AB creates one new copy of itself each second; blue dashed line). (a) Five-fold stimulation. Instability of B is detrimental. activityAB=1, activityAAB=5. (b) Eight-fold stimulation. Intermediate instability of B is optimal. activityAB=1, activityAAB=8. (c) Magnification around the 6^th second in (b). (d) Magnification around the 9^th second in (b). (e) Ten-fold stimulation. Intermediate stability of B is optimal. ActivityAB=1, activityAAB=10. (f) Twenty-fold stimulation. Intermediate stability of B is optimal. ActivityAB=1, activityAAB=20.

3.2. Model 2

In Model 2, B is replaced by a new B, or by a mutant form B′ that confers greater activity. In this model, B exhibits antagonistic pleiotropy, because B is beneficial in one context, in that it is essential for the replication activity of AB. At the same time, B is detrimental in another context, in that B sequesters A subunits that might otherwise be free to generate B′ subunits with increased activity. In each simulation, the ratio is set to 1.0, meaning there is an equal chance of B being replaced by B or B′. The stability of B was again varied from most stable (B=10, where 10% of B species, both B and B′, are lost each second), to most unstable (B=90, where 90% of B species, both B and B′, are lost each second). When activity of AB=1 and activity AB′=2, this means AB′ is two-fold more active than AB. Under these conditions, instability of B is detrimental (Fig. 5a); as the stability of B was varied from most stable to least stable there was a progressive decrease in replicator proliferation. Similarly, when AB′ is three-fold more active than AB, instability of B is again detrimental (Fig. 5b). However, when AB′ is 4-fold more active than AB, a more complex pattern emerges (Fig. 5c). Under these conditions, the intermediate stabilities for B perform best (stability = 20, stability = 30), while more extreme stability or instability for B is detrimental. Similarly, when AB′ is 5-fold more active than AB, again the intermediate stabilities for B perform best (stability = 20, stability = 30), while more extreme stability or instability for B is detrimental (Fig. 5d). The same pattern, where intermediate stability of B was optimal, was also observed when the activity of AB′ was progressively increased up to a value where AB′ is 20-fold more active than AB (data not shown). These results indicate that moderate instability of B becomes beneficial when the activity of the mutant form B′ is of sufficiently large magnitude

Fig. 5.

Computer simulations of Model 2, wherein B is replaced by a new B, or by a mutant form B′ that confers greater activity. Stability of B is varied from most stable (stabilityB=10, where 10% of B species are lost each second), to most unstable (stabilityB=90, where 90% of B species are lost each second). Baseline is a plot of activityAB=1 (AB creates one new copy of itself each second; blue dashed line). (a) B′ confers two-fold increase in activity. Instability of B is detrimental. ActivityAB=1, activityAB′=2, ratio=1.0, hibernation=0. (b) B′ confers three-fold increase in activity. Instability of B is detrimental. ActivityAB=1, activityAB′=3, ratio=1.0, hibernation=0. (c) B′ confers four-fold increase in activity. Moderate instability of B is optimal. ActivityAB=1, activityAB′=4, ratio=1.0, hibernation=0. (d) B′ confers five-fold increase in activity. Moderate instability of B is optimal. ActivityAB=1, activityAB′=5, ratio=1.0, hibernation=0.

3.3. Model 3

In Model 3, B sustains damage that potentially reduces the activity of the AB replicator, and the damaged form of B is called B*. When B* is lost due to instability, it is replaced by a new B. In this way, instability of B is potentially beneficial because it will facilitate the replacement of the damaged form B*. In this model, B exhibits antagonistic pleiotropy, because B is beneficial in one context, in that it is essential for the replication activity of AB. B is detrimental in another context, in that it can sustain damage and become B*, thereby reducing replicator activity, and it sequesters A, which might otherwise be free to create a new B. In these simulations, the stability of B* was varied from most stable (stabilityB*=10, where 10% of B* is lost each second), to most unstable (stabilityB*=90, where 90% of B* is lost each second). In each simulation, the stability of B is set to stabilityB=10. When activityAB=3 and activityAB*=2, this means B* reduces the activity of the replicator by 1/3. Under these conditions, instability of B was found to be detrimental (Fig. 6a); magnification of the data around 10 seconds reveals a progressive decrease in replicator activity as B* is varied from most stable (stabilityB*=10, orange line) to most unstable (stabilityB*=90, teal blue line; Fig. 6b). In contrast, when activityAB=3 and activityAB*=0, this means B* completely eliminates the activity of the replicator. Under these conditions, instability of B was found to be beneficial (Fig. 6c); magnification of the data around 10 seconds reveals a progressive increase in replicator activity as B* is varied from most unstable (stabilityB*=90, teal blue line) to most unstable (stabilityB*=10, orange line; Fig. 6d). These results indicate that instability of B* becomes beneficial when the detrimental effect of the damage becomes sufficiently great in magnitude.

Fig. 6.

Computer simulations of Model 3, wherein B sustains damage and is replaced by a new B. (a-d) In each plot, stability of B* is varied from most stable (stabilityB*=10, where 10% of B* species are lost each second), to most unstable (stabilityB*=90, where 90% of B* species are lost each second). Baseline is a plot of activityAB=3 (AB creates 3 new copies of itself each second; blue dashed line). (a) B* reduces activity by 1/3. Instability of B* is detrimental. DamageB=20, stabilityB=10, hibernation=10, activityAB=3, activityAB*=2. (b) Magnification of last second of plot in (a). (c) B* reduces activity to zero. Instability of B* is beneficial. DamageB=20, stabilityB=10, hibernation=0, activityAB=3, activityAB*=0. (d) Magnification of last second of plot in (c).

3.4. Models 1–3 using the Gompertz equation

Simulations were conducted for each of models 1–3 where the instability of B was modeled using the Gompertz equation. Initial stability of B was varied from most stable (initialStabilityB=10, where 10% of B species are lost each second), to most unstable (initialStabilityB=90, where 90% of B species are lost each second). The exponential increase in B instability with time was set to 5%. For each of the three models, the results were similar to those obtained above without employing Gompertz. For model 1, when the stimulation by free A subunits was equal to or greater than 8-fold, intermediate instability of B became optimal (supplemental Fig. S1a), similar to the results obtained above without Gompertz (Fig. 4b). For model 2, when the activity of AB′ is 5-fold greater than the activity of AB, intermediate stability of B becomes optimal (Fig. S1b), similar to the results obtained above without Gompertz and at a 4-fold increase in activity for AB′ relative to AB (Fig. 5c). Finally, for model 3, when the detrimental effect of damaged B is of sufficiently great magnitude (activityAB=3 and activity AB*=0), instability of B becomes beneficial (Fig. S1c,d), similar to the results obtained above without Gompertz (Fig. 6c,d)

3.5. Models 1–3 using differential equations

Simulations were conducted for each of models 1–3 using differential equations. The differential equation approximations differ from the Python simulations in that they are deterministic rather than stochastic, and the parameters are rates rather than probabilities. In model 1 using differential equations, a benefit for B instability is observed only when the activity of AAB is relatively high. When the activity of AAB varies from 2 to 4, the abundance of AB is observed to increase as the decay rate of B increases (Fig. 7). However, when the activity of AAB varies from 0 to 2, the abundance of AB is observed to decrease as the decay rate of B increases. These results are similar to those obtained above with the Python simulations, where a benefit for B instability was only obtained when the activity of AAB was at least 8-fold greater than the activity of AB (Fig. 4b).

In model 2 using differential equations, no clear benefit for B instability was observed. When the ratio of formation of B:B′ was set at 1.0, i.e., 50% of new B species are B′, instability of B was observed to have a negative effect (Fig. 8b). Similarly, when the ratio of formation of B:B′ was set at 9.0, i.e., only 10% of new B species are B′, instability of B was observed to have an even greater negative effect (Fig. 8c). However, when the ratio of formation of B:B′ was set at 0.10, i.e., 90% of new B species are B′, a different pattern emerged (Fig. 8a). When replication activity of AB′ was low, ranging from 0.0 to 1.0, instability of B was again observed to be detrimental. However, when activity of AB′ was high, ranging from 1.0 to 2.0, instability of B was near neutral. Therefore, a relatively high activity of AB′ combined with a high frequency of generation of B′ could neutralize the costs of B instability, but no conditions were obtained where B instability was favorable. Increasing the activity of AB′ further to 10.0 also did not result in any benefit for B instability (data not shown). In contrast, in the Python simulations presented above, a benefit for B instability was observed when the activity of AB′ was 4-fold or more greater than activity of AB (Fig. 5c,d). The reason for this difference between the results obtained with the Python simulations and the rate equations is not clear at this time.

In model 3 using differential equations, a benefit for B instability is observed only when the activity of the damaged form (AB*) is low. When the replication activity of AB* varies from 0.0 to 1.0, then instability of B* is progressively favorable across the entire range from 0.0 to 2.0. However, when the activity of the damaged form (AB*) varies from 1.0 to 2.0, then instability of B* is progressively detrimental across the entire range from 0.0 to 2.0. These results are similar to those obtained above using the Python simulations, where instability of B was favorable when activity of AB* was equal to 0 (Fig. 6c,d), but instability of B was detrimental when activity of AB* was equal to 2 (Fig. 6a,b) or greater.

4. Discussion

In this study, the proliferation of a hypothetical, two-subunit (AB) molecular replicator was simulated using programs written in Python, as well as using differential equations. The programs are relatively simple, and can be run on personal computer workstations with Windows 10 operating system. The proliferation of AB was simulated under a variety of conditions, for each three non-exclusive models involving antagonistic pleiotropy. Using the Python simulations, for each of the three models, conditions were identified where instability of B relative to A is beneficial in terms of total replicator proliferation. Moreover, for models 1 and 2, conditions were identified where moderate stability of B was optimal, whereas more extreme stability or instability was detrimental, reminiscent of modern cellular macromolecules.

In our previous study, simulations were conducted for models 1 and 2 using Python models where “iterations” (cycles of replication) were used in place of time (Li et al., 2018). In that study, for model 1, instability of B was beneficial then the replication activity of AAB was greater than two times the activity of AB. For model 2, the previous study indicated that instability of B was beneficial when the activity of AB′ was two times greater than the activity of AB, and after a sufficiently large number of iterations. In the present study, the Python models involved improved definition of the time variable, and yielded overall similar results. For model 1, instability of B was beneficial when the replication activity of AAB was greater than three times the activity of AB, and for model 2, instability of B was beneficial when the activity of AB′ was greater than three times the activity of AB.

The simulations using rate equations gave results that were similar to the results from the Python simulations for both models 1 and 3, thereby providing strong support for the conclusion that B instability is favorable in models 1 and 3. The exception was model 2, where the Python simulation yielded a benefit from B instability, but the rate equation simulation did not. Analysis of the rate equation results explains why there is no benefit from B instability (recall variable d is the instability of both B and B′). First, assume g₂ > g₁ and d = 0. Then AB increases exponentially at rate g₁ and AB′ increases exponentially at rate g₂. Second, consider d not equal to zero. For AB, the net change per unit time due to d is (AB′ d f₁) − (AB d f₂). Due to the difference in replicator rates, AB′ will be more abundant than AB as time increases, so this net change will be positive for large times (regardless of the values of f₁, f₂). Moreover, this net increase will be larger for larger d. Similarly, for AB′, the net change per unit time due to d will be negative for large times, and this net loss will be larger for larger d. Therefore, the greatest growth favors AB′, and greater d shifts the abundance of AB′ to AB. These results indicate there is no benefit from B instability in model 2, and the reason for the different result obtained using the Python simulation is not clear at this time.

Limitations of the study include several simplifying assumptions, including the instantaneous and stable association of subunits, unlimited substrate availability and the lack of product inhibition. Future directions might include incorporation of these additional variables, however, this would likely necessitate more complex programs and greater computational capacity. With the present programs, many of the possible parameter combinations remain unexplored. For example, in the Python simulations of model 2, the ratio of B:B′ is kept at 1.0, so there is an equal chance of creating either form; changing to ratio to favor production of B′ would increase the extent to which instability of B is favorable. Similarly, the hibernation time of A is kept at 0, meaning A re-creates B or B′ instantly; increasing the hibernation time of A would increase the cost of losing B, and thereby make instability of B less favorable. In the future, it may be of interest to systematically test all or most of the available parameter combinations, however, this would again likely necessitate more complex programs and greater computational capacity. Finally, it is important to note that because molecular replicators are still hypothetical, we were able to take some liberties in ascribing properties to them. For this reason, we cannot rule out the possibility that some of the similarities between the replicator models and modern biology are trivial. That said, we suggest the potential to apply the underlying model (Fig. 2) to multiple levels of biological organization, along with its simplicity, argues in favor of its potential significance.

Possible molecular structures for the hypothetical AB replicator include RNA, DNA, RNA plus DNA, polypeptide, or other possibilities (Altay et al., 2017; Piette and Heddle, 2020; Szilagyi et al., 2017). Possible mechanisms for AB replication include non-enzymatic templating (Zhou et al., 2020a, b), or enzymatic copying such as an RNA-directed RNA polymerase (Horning and Joyce, 2016). A polymeric structure for A and for B seems most likely, as most modern macromolecules are polymers (and polymers of polymers), and polymeric structures appear most compatible with possible non-enzymatic templating mechanisms and template-dependent polymerization mechanisms for replication. For example, a peptide replicator might bind to monomers and promote their assembly into a new replicator through non-covalent interactions similar to the proliferation of prions (Wickner et al., 2014; Wiltzius et al., 2009) (supplemental Fig. S2). As in Model 1, free A subunits might bind to such a peptide replicator and stimulate its activity by favoring conformation changes that promote product release (Fig, S2a, step (ii)), or perhaps by favoring monomer binding (Fig.S2a, step (i)). Functional, two-subunit peptide replicators have been developed where the two subunits bind to each other through thioester bonds, and self-replicate by binding new peptide subunits from solution through complementary non-covalent interactions (Dadon et al., 2015). Catalytic peptide replicators have also been proposed that would create peptide bonds between bound monomers (Guseva et al., 2017). Another way the free A subunits in model 1 might favor replicator activity is by forming a surrounding proto-cell-membrane, similar to the autopoiesis and chemoton models for proto-cells (Cornish-Bowden and Cardenas, 2020). Notably, Fletcher and coworkers have recently developed a self-replicating micelle composed of unstable surfactant subunits, where one of the breakdown products of the subunits is combined with substrate to create new subunits (Colomer et al., 2018; Morrow et al., 2019).

Another possible two-subunit replicator might be an RNA-dependent RNA polymerase, that is itself composed of RNA (supplemental Fig. S3). In this model, A and B are complementary RNAs that are both enzymatically active, and serve as templates for each other’s synthesis. As in Model 1, free A subunits might bind to such a replicator and help synthesize part of New-A or New-B (Fig. S3b, steps (i) and (iii)), or might favor conformation changes that promote product release (Fig. S3b, steps (iv) and (v)). The ability of A to re-create B is a contrivance of Models 2 and 3. Possibilities include non-enzymatic templating by A using part of its sequence as the template (Fig. S2b), and/or enzymatic polymerization by A using part of its own sequence as the template (Fig. S3c). Conceivably, the re-generation of B by A could proceed through a different mechanism that that used for complete self-replication of AB.

One important question is to what extent can the limited stability of a replicator subunit or of the entire replicator be considered a model for aging? As discussed above, definitions of aging typically include an increased chance of death with time (i.e., increased mortality rate) and a decreased ability to reproduce with time. In the present models, the instability of B leads to the loss of replicative ability for that AB replicator with time, and as such could be considered an example of reproductive aging. At the level of the individual B gene, its instability with time could be interpreted as an example of time-dependent death for the B gene, and potentially an example of aging. In the simulations presented here, the stability of A is set such that A is stable for the duration of the simulations. However, if the stability of A is set to less than the duration of the simulation, A will also be lost, and this could be interpreted as the death of that entire AB replicator; because this is an increased chance of death with time, this might be considered a model for aging of the AB replicator. The exponential increase in mortality rate with age observed in humans and other organisms can be modeled using the Gompertz equation (Kirkwood, 2015). The Gompertz equation was incorporated into the Python simulations for Models 1–3, and allowed the stability of B to be defined by both an initial instability and an exponential increase in instability, similar to organismal aging. The results were similar to those obtained without using Gompertz.

The realization of the importance of cellular macromolecule turnover is generally attributed to Schoenheimer (Schoenheimer, 1942) who observed that “The large molecules, such as the fats and the proteins, are, under the influence of lytic enzymes, constantly being degraded to their constituent fragments. These changes are balanced by synthetic processes…” In the modern cell, virtually all macromolecules are degraded and regenerated at a finite rate. The minimal gene set estimated for a viable cell includes both proteases and ribonuclease (Gil et al., 2004; Hutchison et al., 2016), consistent with the idea that appropriate instability of proteins and RNAs is essential for life. In eukaryotic cells, regulated degradation of cyclin proteins is required for cell replication (Heim et al., 2017), and in E. coli, synthesis and degradation of FtZ protein correlates with cell division (Mannik et al., 2018).

Proteins are often observed to be marginally stable with regard to their folded state, and there are several hypotheses for why this is selectively advantageous, including facilitating folding, providing flexibility for functions like binding and catalysis, and for facilitating degradation of the protein after damage (Kulkarni et al., 2018; Moosa et al., 2020; Williams et al., 2007). Indeed, selection for efficient protein synthesis and folding is thought to be a main driver of coding sequence evolution (Drummond and Wilke, 2009), and protein folding instability is positively correlated with the rate of protein evolution (Agozzino and Dill, 2018). Proteins vary dramatically in their relative degradation rates, and this stability is encoded in their primary sequence through the inherent susceptibility of the component amino acids to damage, and via motifs that bind to protein degradation regulatory factors (Fuertes et al., 2003; Laney and Hochstrasser, 1999; Varshavsky, 2019). Certain transcription factor proteins and signaling proteins have half-lives on the order of minutes, to enable the rapid activation and inactivation of cellular signaling pathways, and therefore a rapid response of the cell to a changing environment. Cellular proteins are susceptible to multiple forms of damage, including hydrolysis, oxidative damage, crosslinking, denaturation and aggregation, and the degradation of such damaged and aggregated proteins is essential for cell viability in response to stress (Sinnige et al., 2020; Tower, 2009, 2011). Moreover, misregulation of protein synthesis, maintenance and degradation pathways is directly implicated in aging and aging-related disease (Kaushik and Cuervo, 2015; Vilchez et al., 2014). Correspondingly, model 3 incorporates variables for the rate of damage to B, and for the rate of degradation of the damaged form B*.

The DNA macromolecule is degraded when cells undergo programmed cell death during development and adult tissue turnover, and ultimately upon the death of the animal (Javan et al., 2015; Tower, 2015b). Partial degradation and regeneration of the DNA occurs in the cell during certain types of DNA repair (Chatterjee and Walker, 2017). However, because of the semi-conservative nature of DNA replication, it is possible that some DNA strands may persist over many generations of a cell lineage, and possibly over many generations of the organism by passage through the immortal germ-line (Yadlapalli and Yamashita, 2013). In addition to considering the physical destruction of the DNA as the metric of stability, it is also relevant to consider sequence changes as the metric of stability. For a given gene, a sufficiently large change in sequence would yield a new gene, and therefore the rate of sequence change could be used a metric of a gene’s stability. In modern cells, genes are replicated by DNA polymerases, and the rate of sequence change (mutation) is controlled by the fidelity of the DNA polymerase combined with DNA mismatch repair systems. Notably, the mutation rate is under selective pressure to be neither too high nor too low, to allow for an optimal mutation rate to support natural selection and evolution, without overwhelming the organism with deleterious mutations (Sherer and Kuhlman, 2020). Correspondingly, model 2 incorporates the chance of creating a more active mutant form of B (called B′), and a moderate stability of B was found to be optimal using the Python simulations. In the future, it may be of interest to also incorporate a chance of creating a less active mutant form of B, which is expected to increase the costs of B instability.

There are several additional ways in which the models for the AB replicator might be applied to the behavior of modern genes, cells and multicellular organisms. During evolution, the target of selection might be an individual replicator such as AB, or might be a cell or proto-cell containing multiple collaborating replicators. For example, the free A subunits generated by the AB replicator in model 1 might interact with and regulate other replicators, and vice versa, thereby allowing for regulatory interactions between collaborating replicators. There are several examples of genes in the eukaryotic cell that are inherently unstable relative to other genes. One well-studied example of genes that are inherently unstable is the eukaryotic telomeric repeat genes, which collaborate with chromosomal genes. In somatic cells that lack telomerase, the telomeric repeat gene sequences are progressively lost each cell division due to the DNA end replication problem (Ohki et al., 2001; Olovnikov, 1973). Telomere loss eventually stops the replication of the entire cell, and serves as an anti-cancer mechanism in mammals and other species (Shay, 2016).

Another example of modern cellular genes that are inherently shorter-lived are those of the mitochondrial genome. Metazoan cells contain both nuclear genome and mitochondrial genome. The nuclear genome typically only replicates when the cell divides, but the mitochondrial genome is continually degraded and regenerated in the cell, and therefore the mitochondrial genome has a shorter half-life than the nuclear genome (Pickles et al., 2018). Both the nuclear genome and the mitochondrial genome are required to generate new mitochondrial genomes, and both the nuclear genome and the mitochondrial genome are required to support cell growth that is turn required for cell division and the generation of new nuclear and mitochondrial genomes. The shorter half-life of the mitochondrial genome relative to the nuclear genome is hypothesized to be an evolutionary strategy that allows the nuclear and mitochondrial genomes to collaborate in the cellular replicator, while still allowing the mitochondrial genome to evolve as a separate entity from the nuclear genome (Tower, 2006). Moreover, the shorter half-life of the mitochondrial genome relative to the nuclear genome is proposed to facilitate asymmetric inheritance of the mitochondrial genome, the evolution of sex, and evolution in general (Tower, 2006). In metazoan species with uni-parental mitochondrial transmission, in terms of transmission across generations, the nuclear genome is stable in males and females, in that it is transmitted to offspring by both. In contrast, the mitochondrial genome is stable in females and is unstable in males, in that it is not effectively transmitted to offspring through the male. This relative instability of the paternal mitochondrial genome is produced by degradation in the male germline cells (Arama et al., 2006; Fabian and Brill, 2012; Fabrizio et al., 1998; Huang et al., 2020; Muro et al., 2006) and in the fertilized zygote (Nishimura et al., 2006; Sato and Sato, 2017). Because the mitochondrial genomes are inherited almost exclusively from the mother, natural selection can only act to optimize mitochondrial gene function and nuclear-mitochondrial gene interactions in the female (Frank and Hurst, 1996; Rand, 2005). As a consequence, the male inherits mitochondrial genomes that are not optimal for his physiology (sometimes called “Mother’s curse” (Gemmell et al., 2004)). It has been hypothesized that natural selection will select for compensatory nuclear alleles in the male, and that these compensatory nuclear alleles may in turn not be optimal for the female in the next generation (Tower, 2006). These ongoing and conflicting sex-specific selective pressures, centered on mitochondrial function, are hypothesized to promote genetic diversity, multicellularity, the evolution of the sexes, and evolution in general (Tower, 2006), as well as contributing to the failure in mitochondrial maintenance observed during aging across species (Tower, 2015a; Ventura-Clapier et al., 2017). In these ways, the shorter half-life of the mitochondrial genome relative to the nuclear genome can be seen to generate increased complexity of the system (similar to Fig. 2). Recently, other groups have also explored the possible rules and mechanisms by which nucleus and mitochondria collaborate as replicators (Havird et al., 2019; Klucnika and Ma, 2019; Radzvilavicius et al., 2016; Rand, 2011), including the idea that uniparental mitochondrial transmission might promote the evolution of sex (Havird et al., 2015).

In summary, the data support an antagonistic pleiotropy model in which both evolution and aging are emergent properties when two genes collaborate as a replicator (Fig. 2). Applying antagonistic pleiotropy to three specific models for replicator proliferation enabled simulations where instability of a subunit was beneficial. In the future, it may be of interest to further develop these models by incorporating additional variables, and to potentially create multi-subunit molecular replicators in the laboratory.

Highlights.

• Antagonistic pleiotropy can be incorporated into models for molecular replicators

• Antagonistic pleiotropy can create a benefit for one subunit of a replicator to be unstable

• Replicator subunit instability increases the complexity of the system