Temperature-Dependent Enhanced Speciation in Ecosystems with Conserved Symmetries

doi:10.21203/rs.3.rs-1964245/v1

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Temperature-Dependent Enhanced Speciation in Ecosystems with Conserved Symmetries

https://doi.org/10.21203/rs.3.rs-1964245/v1

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The impact of climate change on biodiversity needs to be understood from a multidisciplinary approach. Using an analytical framework, we investigate the species response to rising temperatures. Common traits and characteristics among species that allow classification at different taxonomic levels imply an underlying symmetry that gives rise to invariances behind the biodiversity observed in nature. Changing temperatures that go beyond a critical limit break this underlying symmetry which could lead to enhanced speciation.

Conserved symmetries

Critical temperatures

Enhanced speciation

Biodiversity

Ecosystems

The impact of climate change and elevated temperatures lead to intriguing questions on the evolution of species (Nadeau and Urban, 2019; Pauls et al, 2013; Wiens et al, 2009). Although declining biodiversity could be perceived as a consequence of climate change, it is difficult to dismiss the prospect that species richness could result from rising temperatures. After all, there have been studies on the effect of temperature on rates of spontaneous mutation that could drive speciation (Husby et al, 2011; Waldvogel and Pfenninger, 2021; Z Lu et al, 2021; Gillman and Wright, 2014; Chu et al, 2018). What factors drive mutation and the divergence of species have been a recurring experimental and theoretical question (Jablonski, 2022; Livnat, 2017; Smith et al, 2014). In this paper, the effect of changing temperatures as it perturbs the balance in nature is investigated from an analytical framework.

The observed biodiversity in nature indicates that the evolution and proliferation of myriads of species is far from purely random. The fact that one can categorize and classify species within a genus that belongs to a family within an order for a given class in a phylum shows that certain patterns and traits are preserved and maintained at different taxonomic levels. In science, conserved quantities in time underscore symmetries possessed by a system. Motivated by the existence of common traits that manifest in a given taxonomic level, we shall investigate symmetries and conservation properties that may characterize a biological system as it evolves in time. We then present a temperature-dependent mechanism that breaks this symmetry but compatible with both sympatric (Mallet et al, 2009) and allopatric speciations (Hoskin et al, 2005; Moreno-Contreras, 2020). Complementing sympatric speciation, the diversification of species described here does not require large scale geographic distance for speciation to occur. On the other hand, the mechanism we present is also congruent with allopatric speciation, especially if two geographic areas differ in temperature. Whether a population is isolated or not, we analytically show that there could be critical temperatures that can break an inherent symmetry in a population that triggers mutation and speciation. The concept of symmetry breaking is not new and often manifests in nature. Symmetry breaking is found not only in ferromagnets and superconductivity, but also in biological systems such as in predator-prey populations (Silva-Dias and López-Castillo, 2018) and the left-right asymmetry observed in drosophila (Coutelis, et al, 2013), snails, and frogs (Davison et al, 2016). We, therefore, use the concept of symmetry breaking to discuss speciation events. Models where symmetry breaking leads to speciation exists, such as those inspired by bifurcation theories in the mathematical studies of dynamical systems (Stewart et al, 2003). In this paper, we derive insights from spontaneously broken symmetries that have been experimentally observed in nature (Goldstone et al, 1962). We begin in Section 2 by discussing the importance of temperature in evolution, and by defining in Section 3 representations of a species and its corresponding changes in time and location. Succeeding sections investigate conserved properties and corresponding symmetries, as well as how these symmetries may be broken allowing for enhanced speciation to occur and thus propelling biodiversity.

The Miocene epoch, 23.03 to 5.3 million years ago, is characterized by a prolonged period of warm global climate compared to that of the following Pliocene period (Holbourn et al, 2013). The ecosystems of kelp forests and grasslands are known to have first appeared in the Miocene epoch (Kürschner et al, 2008; Cerling et al, 1993). Likewise, the Miocene epoch witnessed the rise and diversification of apes (Moyà-Solà et al, 2004; Kunimatsu et al, 2007; Moyà-Solà et al, 2009; Suwa et al, 2007), and a transition from worm-eating to fish-eating cone snails from the genus Conus (Aman et al, 2015).

Changes in temperature have significant effects on rates of evolution (Allen et al, 2006; Ivany et al, 2000). At the cell level, biochemical reactions and processes have been shown to be highly sensitive to temperature (Chen and Shakhnovich, 2010; Matthews et al, 1987; Dill et al, 2011). Rates of diffusion of proteins throughout the cell has also been shown to be temperature dependent. Myoglobin, which is important in intracellular oxygen transport, has diffusion coefficient value 1.8 times higher at 37°C than that determined at 22°C (Papadopoulos et al, 2000). Transport of proteins within cells, on the other hand, depend on diffusion constants which are linearly dependent on temperature $T$ as shown by Stokes-Einstein equation, $D=kT/6\pi \eta {R}_{a}$, and modifications thereof (Dill et al, 2011; Tyn and Gusek, 1990). There is, for example, an optimal growth temperature where cells can grow fastest (Dill et al, 2011). Effects of temperature are further evident as shown by mesophiles that grow at temperatures between 25 and 40°C, while thermophiles grow at temperatures higher than 40°C (Concha-Marambio et al, 2017; Sawle and Ghosh, 2011; Dominy et al, 2004; Ge et al, 2008). Even at the DNA level, a temperature-dependent distinction for an AT base pair and GC base pair can be made, the latter having a higher thermal stability. Thus, the genomic GC content has also been investigated for its evolutionary significance (Šmarda et al, 2014). A correlation is found, for instance in bacteria, between a higher GC content and a higher temperature optimum (Musto, et al, 2006; Nihio et al, 2003). At the protein level, biological functions can also be disrupted when proteins lose their native conformation due to higher temperatures. It would be instructive, therefore, to have an analytical framework for biodiversity and speciation where the temperature $T$ is a crucial parameter.

Let us start by representing a given species by, $\varphi =\varphi \left(\mathbf{r},t\right)$, which is a function of its location, $\mathbf{r}=\left(x,y,z\right)$, and time $t$. To track possible changes in time we define the rate of change of a species as, $\partial \varphi /\partial t$. Possible variations of species as a function of location, on the other hand, can be represented by, e.g., $\partial \varphi /\partial x$, where $x$ is a coordinate in space $\mathbf{r}$. We can designate the space and time components as, ${x}^{\mu }=\left(t,x,y,z\right)$, where $\mu$ can take the values, $\mu =\text{0,1},\text{2,3}$, and use a compact notation, ${\partial }_{\mu }\varphi =\partial /\partial {x}^{\mu }$, where, ${\partial }_{\mu }=\left(\partial /\partial t,\partial /\partial x,\partial /\partial y,\partial /\partial z\right)$. Hence, ${\partial }_{0}=\partial /\partial t$, describes changes in time and, ${\partial }_{3}=\partial /\partial z$, allows description for changes in altitude. The changes, ${\partial }_{\mu }\varphi$, may be due to mutation, genetic drift, or evolution in general as a function of location and time.

Having defined $\varphi$, we next represent interactions between the same species as, ${\varphi }^{2}$. To account for the effect of changes in temperature $T$ on this interaction, we define a parameter, ${\beta }^{2}=a\left({T}_{C}-T\right)$, with $a>0$, and couple this with ${\varphi }^{2}$ as,

$${\beta }^{2}{\varphi }^{2}=a\left({T}_{C}-T\right){\varphi }^{2}$$

In Eq. (1), a critical temperature ${T}_{C}$ is introduced in view of the above normal biodiversity observed in warmer regions such as marine life in the Coral Triangle near the equator. Hence, depending on the value of temperature $T$, the coefficient ${\beta }^{2}$ can be positive $\left({T}_{C}>T\right)$, or negative $\left({T}_{C}<T\right)$ for the same species interaction ${\varphi }^{2}$.

On the other hand, for the rate of change ${\partial }_{\mu }\varphi$, we note that mutations can have different effects when taken in combination rather than individually. To account for mutations which interact with one another, such as by epistasis (Phillips, 2008) in an interconnected network of genes, or the nonrandom association of different alleles in linkage disequilibrium (Slatkin, 2008), we express the interaction of two $\left({\partial }_{\mu }\varphi \right)$ as,

$$\sum _{\mu =0}^{3}\left({\partial }_{\mu }\varphi \right)\left({\partial }_{\mu }\varphi \right)=\left({\partial }_{0}\varphi \right)\left({\partial }_{0}\varphi \right)+\left({\partial }_{1}\varphi \right)\left({\partial }_{1}\varphi \right)+\left({\partial }_{2}\varphi \right)\left({\partial }_{2}\varphi \right)+\left({\partial }_{3}\varphi \right)\left({\partial }_{3}\varphi \right)$$

For notational convenience, we simply write, $\sum _{\mu =0}^{3}\left({\partial }_{\mu }\varphi \right)\left({\partial }_{\mu }\varphi \right)=\left({\partial }_{\mu }\varphi \right)\left({\partial }_{\mu }\varphi \right)$, where a repeated index such as $\mu$ implies summation from 0 to 3. A more complicated situation includes a combined interaction of rates in time and space such as, $\left({\partial }_{t}\varphi \right)\left({\partial }_{y}\varphi \right)$, which for simplicity we presently ignore.

One way of tracking biological evolution is to view it as a competition between a state where a species remains the same, or does not mutate, and a state which allows changes to occur. We track and represent this competition by defining a function $\mathbb{B}$ which measures the difference between rates of change of a species, Eq. (2), and its ability to stay the same as $\varphi$ and experiencing only same-species interaction ${\beta }^{2}{\varphi }^{2}$, Eq. (1). We define the competition between states as the difference,

$$\mathbb{B}=\frac{1}{2}\left({\partial }_{\mu }\varphi \right)\left({\partial }_{\mu }\varphi \right)-\left(\frac{1}{2}{\beta }^{2}{\varphi }^{2}+\frac{1}{4}\lambda {\varphi }^{4}\right)$$

We refer to the difference $\mathbb{B}$ as the balance function. The last term ${\varphi }^{4}$ in Eq. (3) allows interactions with other species when there is more than one in the same taxonomic group (see, e.g., Eq. (6)) but without experiencing mutation. We take $\lambda >0$ to preserve the difference in $\mathbb{B}$ in case ${\varphi }^{4}$ takes large values. Take for instance $n$ different species, ${\varphi }_{1}, {\varphi }_{2}, \dots ,{\varphi }_{n}$, in the same genus which we represent as elements in a column matrix,

$$\varphi =\left(\begin{array}{c}{\varphi }_{1}\\ \begin{array}{c}{\varphi }_{2}\\ ⋮\\ {\varphi }_{n}\end{array}\end{array}\right)$$

For $n$ species represented by Eq. (4), the balance function Eq. (3) for $\varphi$ becomes,

$$\mathbb{B}=\frac{1}{2}{\left({\partial }_{\mu }\varphi \right)}^{†}\left({\partial }_{\mu }\varphi \right)-\left[\frac{1}{2}{\beta }^{2}{\varphi }^{†}\varphi +\frac{1}{4}\lambda {\left({\varphi }^{†}\varphi \right)}^{2}\right]$$

Here, ${\varphi }^{†}$ is the transpose of Eq. (4) and represents a row matrix, such that, ${\varphi }^{†}\varphi ={\left({\varphi }_{1}\right)}^{2}+{\left({\varphi }_{2}\right)}^{2}+\dots +{\left({\varphi }_{n}\right)}^{2}$, and, ${\left({\partial }_{\mu }\varphi \right)}^{†}\left({\partial }_{\mu }\varphi \right)=\left({\partial }_{\mu }{\varphi }_{1}\right)\left({\partial }_{\mu }{\varphi }_{1}\right)+\left({\partial }_{\mu }{\varphi }_{2}\right)\left({\partial }_{\mu }{\varphi }_{2}\right)+\dots +\left({\partial }_{\mu }{\varphi }_{n}\right)\left({\partial }_{\mu }{\varphi }_{n}\right).$ The last term, ${\left({\varphi }^{†}\varphi \right)}^{2}$ in Eq. (5), which corresponds to ${\varphi }^{4}$ in Eq. (3), would then allow for interactions between different species, i.e.,

$${\left({\varphi }^{†}\varphi \right)}^{2}={\left[{\left({\varphi }_{1}\right)}^{2}+{\left({\varphi }_{2}\right)}^{2}+\dots +{\left({\varphi }_{n}\right)}^{2}\right]}^{2}$$

$${=\left({\varphi }_{4}\right)}^{2}+{\left({\varphi }_{1}{\varphi }_{2}\right)}^{2}+{\left({\varphi }_{1}{\varphi }_{3}\right)}^{2}+{\left({\varphi }_{1}{\varphi }_{4}\right)}^{2}\dots +{\left({\varphi }_{n-1}{\varphi }_{n}\right)}^{2}$$

We shall now look at some of the properties of Eq. (5) in the succeeding sections.

If we wish to consider the net effect of the balance function $\mathbb{B}$ over space and time, we can define a sum or integral of Eq. (5) as,

$$I=\int \mathbb{B} d{x}^{0}d{x}^{1}d{x}^{2}d{x}^{3}$$

$$=\int \mathbb{B} {d}^{4}x$$

Noting that, regardless of differences in location and time, there remain similar characteristics shared overall by specimens belonging to the same taxonomic class, we now designate $I$ to represent this invariance in specimens in view of its universal nature being a sum or integral over space and time.

What happens then when transformations brought about by mutations, genetic drift, or evolution in general occur where both $\varphi$ and ${\partial }_{\mu }\varphi$ vary? Representing possible variations affecting both $\varphi$ and ${\partial }_{\mu }\varphi$ in Eqs. (5) and (7) by a $\delta$ we write,

$$\delta I=\delta \int \mathbb{B} {d}^{4}x$$

$$=\int \left(\delta \mathbb{B}\right) {d}^{4}x+\int \mathbb{B}\delta \left({d}^{4}x\right)$$

$$=\int \left(\delta \mathbb{B}\right){ d}^{4}x$$

Here, $\delta \left({d}^{4}x\right)=0$, since there are no transformations for the time and space coordinates. Noting that, regardless of location and time some traits remain the same in a taxonomic class, we now take the case, $\delta I=0$, which means the integral $I$ remains invariant. Eq. (8) then becomes, $0=\int \left(\delta \mathcal{B}\right){ d}^{4}x$, or that,

$$\delta \mathbb{B}=0$$

This tells us that to investigate invariance properties of $I$ defined as the balance function $\mathcal{B}$ integrated over space and time, we can simply look at cases where Eq. (9) is satisfied (Ramond, 2001). We can then consider possible transformations of a species denoted by, $\delta \varphi$ and $\delta \left({\partial }_{\mu }\varphi \right)$, but preserve the condition, Eq. (9). The symmetry possessed by the system in terms of the similarities or invariant traits defining a taxonomic group regardless of their location in time and space is embodied in Eq. (9). We look at specific examples in the next section.

Since we are looking at features common to different members in the same taxonomic level, we now consider a transformation which interchanges the different ${\varphi }_{i}$’s $\left(i=\text{1,2},\dots ,n \right)$ in Eq. (4). Suppose, for example, we take $\mathbb{B}$ as a balance function for a given genus. We can then rearrange the elements in Eq. (4) without affecting the genus where all the species ${\varphi }_{i}$ belong to. The genus for all the species remains the same, or is invariant, to this rearrangement or transformation such that, $0=\delta I=\int \left(\delta \mathbb{B}\right){ d}^{4}x$. In other words, the balance function $\mathbb{B}$ is preserved, i.e., Eq. (9). Invariance under these transformations can be represented as, $\delta \mathbb{B}={\mathbb{B}}^{{\prime }}-\mathbb{B}=0$, where $\mathbb{B}$ and $\mathbb{B}{\prime }$ denote the balance function before and after the transformation, respectively, which preserves $\mathbb{B}$.

What is an example of a transformation or variation $\delta$ which leaves the balance function $\mathbb{B}$ invariant? An illustration of a group of transformations which satisfies this requirement is the rotation group $O\left(n\right)$ which leaves $\mathbb{B}$, Eq. (5), invariant. Let us demonstrate this for the case of two species ${\varphi }_{1}$ and ${\varphi }_{2}$. In particular, we take the $O\left(2\right)$ group of transformations which rotates species ${\varphi }_{1}$ and ${\varphi }_{2}$ but leaves Eq. (5) unchanged or invariant. Consider again $\mathbb{B}$ as a balance function for a given genus. Exchanging the roles of two species ${\varphi }_{1}$ and ${\varphi }_{2}$ does not change the genus they belong to leaving the balance function $\mathbb{B}$ invariant. We mathematically represent this by designating species ${\varphi }_{1}{\prime }$ and ${\varphi }_{2}{\prime }$ as the result of the transformation given by,

$$\left(\begin{array}{c}{\varphi }_{1}{\prime }\\ {\varphi }_{2}{\prime }\end{array}\right)=\left(\begin{array}{cc}cos\left(\alpha \right)& sin\left(\alpha \right)\\ -sin\left(\alpha \right)& cos\left(\alpha \right)\end{array}\right)\left(\begin{array}{c}{\varphi }_{1}\\ {\varphi }_{2}\end{array}\right)$$

Here, small changes are represented by small values of $\alpha$. Clearly, however for $\alpha =\pi /2$, the roles of ${\varphi }_{1}$ and ${\varphi }_{2}$ are interchanged after transformation but $\mathbb{B}$, Eq. (5), remains unaffected by this rotation. We, therefore, say that the balance function $\mathbb{B}$ possesses an $O\left(2\right)$ symmetry.

We note that an alternative way of studying the species ${\varphi }_{1}$ and ${\varphi }_{2}$ can make use of the definitions,

$${\Phi }=\frac{1}{\sqrt{2}}\left({\varphi }_{1}+i{\varphi }_{2}\right)$$

$${{\Phi }}^{*}=\frac{1}{\sqrt{2}}\left({\varphi }_{1}-i{\varphi }_{2}\right)$$

Since ${{\Phi }}^{*}{\Phi }=\frac{1}{2}\left[{\left({\varphi }_{1}\right)}^{2}+{\left({\varphi }_{2}\right)}^{2}\right]$, Eq. (5) for $n=2$, is the same as the balance function for ${{\Phi }}^{*}$ and ${\Phi }$ given by,

$$\mathbb{B}={\left({\partial }_{\mu }{\Phi }\right)}^{*}\left({\partial }_{\mu }{\Phi }\right)-\left[{\beta }^{2}{{\Phi }}^{*}{\Phi }+\lambda {\left({{\Phi }}^{*}{\Phi }\right)}^{2}\right]$$

In the Appendix, it is shown that the transformation Eq. (10) is equivalent to the phase transformations for two species for ${\Phi }$ and ${{\Phi }}^{*}$ given by,

$${{\Phi }}^{{\prime }}={e}^{-i\alpha }{\Phi }$$

$${{{\Phi }}^{*}}^{{\prime }}={e}^{i\alpha }{{\Phi }}^{*}$$

Under the transformations given by Eqs. (14) and (15), the balance function Eq. (13) which is equivalent to Eq. (5), clearly remains invariant, i.e.,

$$\mathbb{B}{\prime }={\left({\partial }_{\mu }{\Phi }{\prime }\right)}^{*}\left({\partial }_{\mu }{\Phi }{\prime }\right)-\left[{\beta }^{2}{{\Phi }}^{*}{\prime }{\Phi }{\prime }+\lambda {\left({{{\Phi }}^{*}}^{{\prime }}{\Phi }{\prime }\right)}^{2}\right]$$

$$=\mathbb{B}$$

with the parameter, $\alpha \ne \alpha \left({x}^{\mu }\right)$. This implies a symmetry possessed by the system in spite of transformations undergone by individual species. For convenience, we may also use Eqs. (13) to (15).

In situations where speciations occur, it is possible that a species evolves into a different genus. From our example above, this implies that $\mathbb{B}$ would no longer be invariant under the $O\left(2\right)$ transformation of species. What, in particular, triggers the $O\left(2\right)$ symmetry breaking? For metamorphosis of a species leading to speciation we expect the term $\left({\partial }_{\mu }{\Phi }\right)$ to dominate. We then consider cases where terms in Eq. (13), i.e., $V={\beta }^{2}{{\Phi }}^{*}{\Phi }+\lambda {\left({{\Phi }}^{*}{\Phi }\right)}^{2}$, which signify the ability of species ${\Phi }$ to interact within the same genus, to be minimal when compared with ${\left({\partial }_{\mu }{\Phi }\right)}^{*}\left({\partial }_{\mu }{\Phi }\right)$ which embodies continuous change brought about by mutations, genetic drift, and other factors. A minimum case of $V$ is obtained by calculating, $\left(\partial V/\partial {\Phi }\right)=0$, which yields,

$$\left({{\Phi }}^{*}{\Phi }\right)=-\frac{{\beta }^{2}}{2\lambda }$$

or equivalently, using Eqs. (11) and (12),

$${\left({\varphi }_{1}\right)}^{2}+{\left({\varphi }_{2}\right)}^{2}=-\frac{{\beta }^{2}}{\lambda }$$

The minimum, therefore, is found at values of ${\varphi }_{1}$ and ${\varphi }_{2}$ which satisfy Eq. (18). Recall, however, that ${\beta }^{2}=a\left({T}_{C}-T\right)$ can be positive or negative depending on the value of temperature $T$. Let us look at these two cases vis-à-vis departures from the minimum of $V={\beta }^{2}{{\Phi }}^{*}{\Phi }+\lambda {\left({{\Phi }}^{*}{\Phi }\right)}^{2}$, which represents species interaction without mutation.

6.1 Low Temperature:${\beta }^{2}>0$

To gain an insight into the values of ${\varphi }_{1}$ and ${\varphi }_{2}$ for minimum $V$, we can take the square root of Eq. (18). For ${T}_{C}>T$, given that ${\beta }^{2}=a\left({T}_{C}-T\right)$, we get ${\beta }^{2}>0$ and the minimum obtained from Eq. (18) is given by $\sqrt{-{\beta }^{2}/\lambda }$ which is an imaginary number. Since both ${\varphi }_{1}$ and ${\varphi }_{2}$ are real, this means that the minimum of $V$ occurs at, ${\varphi }_{1}={\varphi }_{2}=0$, or at the origin of the $\left({\varphi }_{1}-{\varphi }_{2}\right)$-space (see, Fig. 1). For this case, therefore, the $O\left(2\right)$ rotation symmetry of Eq. (13) is preserved by the minimum of $V$, as well as for small departures from the minimum of the species interaction ${{\Phi }}^{*}{\Phi }$. Thus, even if ${\left({\partial }_{\mu }{\Phi }\right)}^{*}\left({\partial }_{\mu }{\Phi }\right)$ (mutation, genetic drift, etc.) dominates, the transformed species still belong to the same genus, i.e., preserves the $O\left(2\right)$ invariance of the balance function $\mathbb{B}$.

Figure 1. The minimum of $V$ is at ${\varphi }_{1}={\varphi }_{2}=0$.

6.2 High Temperature:${\beta }^{2}<0$

We next consider the case where temperature $T$ is greater than a critical temperature ${T}_{C}$, or ${\beta }^{2}=a\left({T}_{C}-T\right)<0$. Here, we shall see how the minimum of $V$ breaks the $O\left(2\right)$ symmetry. With ${\beta }^{2}$ negative, then $\sqrt{-{\beta }^{2}/\lambda }$ is a real number. As seen in Eq. (18), the minimum of $V$ occurs at a ring-like set of values in $\left({\varphi }_{1}-{\varphi }_{2}\right)$-space with radius, $r=\sqrt{-{\beta }^{2}/\lambda }$, (see Fig. 2). Eq. (18), therefore, tells us that the minimum of $V$ is satisfied by many values of ${\varphi }_{1}$ and ${\varphi }_{2}$. We can choose any pair of values for of ${\varphi }_{1}$ and ${\varphi }_{2}$, and as long as it satisfies Eq. (18), this gives a minimum of the interaction ${{\Phi }}^{*}{\Phi }$. For example, ${\varphi }_{1}=r$ and ${\varphi }_{2}=0$, is a point in the ring described by Eq. (18) which is the red dot in Fig. 2.

In reality, since species interactions within a genus may never really be always at the minimum, we allow small deviations from the minimum by defining species ${\varphi }_{1}{\prime }$ and ${\varphi }_{2}{\prime }$ as,

$${\varphi }_{1}^{{\prime }}={\varphi }_{1}-r$$

$${\varphi }_{2}^{{\prime }}={\varphi }_{2}$$

where ${\varphi }_{1}^{{\prime }}$ measures the departure of ${\varphi }_{1}$ from the minimum at $r$. We can then look at $\mathbb{B}$ and express the balance function in terms of these small deviations from the minimum. We do this by first writing Eq. (13), or Eq. (5) for $n=2$ explicitly as,

$$\mathbb{B}=\frac{1}{2}\left[\left({\partial }_{\mu }{\varphi }_{1}\right)\left({\partial }_{\mu }{\varphi }_{1}\right)+\left({\partial }_{\mu }{\varphi }_{2}\right)\left({\partial }_{\mu }{\varphi }_{2}\right)\right]-\frac{1}{2}{\beta }^{2}\left[{\left({\varphi }_{1}\right)}^{2}+{\left({\varphi }_{1}\right)}^{2}\right]$$

$$-\frac{1}{4}\lambda {\left[{\left({\varphi }_{1}\right)}^{2}+{\left({\varphi }_{1}\right)}^{2}\right]}^{2}$$

Expressing this in terms of the small deviations about a minimum using Eqs. (19) and (20) where, ${\varphi }_{1}={\varphi }_{1}^{{\prime }}+r$, and ${\varphi }_{2}={\varphi }_{2}^{{\prime }}$, Eq. (21) becomes,

$$\mathbb{B}=\frac{1}{2}\left[\left({\partial }_{\mu }{\varphi }_{1}^{{\prime }}\right)\left({\partial }_{\mu }{\varphi }_{1}^{{\prime }}\right)+\left({\partial }_{\mu }{\varphi }_{2}^{{\prime }}\right)\left({\partial }_{\mu }{\varphi }_{2}^{{\prime }}\right)\right]+{\beta }^{2}{\left({\varphi }_{1}^{{\prime }}\right)}^{2}$$

$$-\frac{1}{4}\lambda {\left[{\left({\varphi }_{1}^{{\prime }}\right)}^{2}+{\left({\varphi }_{2}^{{\prime }}\right)}^{2}\right]}^{2}-\lambda r{\varphi }_{1}^{{\prime }}\left[{\left({\varphi }_{1}^{{\prime }}\right)}^{2}+{\left({\varphi }_{2}^{{\prime }}\right)}^{2}\right]+\frac{{\beta }^{4}}{4\lambda }$$

Note that for temperature, ${T>T}_{C}$ or ${\beta }^{2}<0$, Eq. (22) for small deviations from the minimum no longer possesses the $O\left(2\right)$ symmetry of Eq. (21). Hence, the minimum of the balance function $\mathbb{B}$ spontaneously breaks the $O\left(2\right)$ symmetry for the case ${\beta }^{2}<0$. For instance, the second from the last term with square brackets breaks the $O\left(2\right)$ invariance. We discuss the consequence of this symmetry breaking in the next section.

Looking at Eqs. (19) and (20), let us consider, ${\varphi }_{1}^{{\prime }}\ll 1$ and ${\varphi }_{2}^{{\prime }}\ll 1$, which would represent small deviations from the minimum. In this case, we may ignore higher powers of ${\left({\varphi }_{1}^{{\prime }}\right)}^{3}$, ${\left({\varphi }_{2}^{{\prime }}\right)}^{3},$ ${{\varphi }_{1}^{{\prime }}\left({\varphi }_{2}^{{\prime }}\right)}^{2}$, etc., such that Eq. (22) can be written as ${\mathbb{B}}_{s.d.}$ representing the balance function for small deviations from its minimum. This is given by,

$${\mathbb{B}}_{s.d.}\approx \frac{1}{2}\left({\partial }_{\mu }{\varphi }_{1}^{{\prime }}\right)\left({\partial }_{\mu }{\varphi }_{1}^{{\prime }}\right)+{\beta }^{2}{\left({\varphi }_{1}^{{\prime }}\right)}^{2}+\frac{1}{2}\left({\partial }_{\mu }{\varphi }_{2}^{{\prime }}\right)\left({\partial }_{\mu }{\varphi }_{2}^{{\prime }}\right)+\frac{{\beta }^{4}}{4\lambda }$$

With ${\beta }^{2}<0$ for high temperatures ${T>T}_{C}$, ${\mathbb{B}}_{s.d.}$ shows that species ${\varphi }_{1}^{{\prime }}$ maintains its presence in the balance function up to ${\left({\varphi }_{1}^{{\prime }}\right)}^{2}$ (see, first two terms in Eq. (23). Curiously, however, the behavior of species ${\varphi }_{2}^{{\prime }}$ has drastically changed. Firstly, ${\varphi }_{2}^{{\prime }}$ has lost its same-species interaction term proportional to ${\left({\varphi }_{2}^{{\prime }}\right)}^{2}$ although the interaction of rates of change, $\left({\partial }_{\mu }{\varphi }_{2}^{{\prime }}\right)\left({\partial }_{\mu }{\varphi }_{2}^{{\prime }}\right)$ remain. Species ${\varphi }_{2}^{{\prime }}$ appears to undergo continuous change, which could be driven by mutation or genetic drift, with no chance of self-interaction, i.e., ${\left({\varphi }_{2}^{{\prime }}\right)}^{2}$ due to constant change. We refer to the disappearance of ${\left({\varphi }_{2}^{{\prime }}\right)}^{2}$ as enhanced speciation where ${\varphi }_{2}^{{\prime }}$ interaction with an individual of the same species is no longer possible. Here, an apparent run-away transformation where $\left({\partial }_{\mu }{\varphi }_{2}^{{\prime }}\right)\left({\partial }_{\mu }{\varphi }_{2}^{{\prime }}\right)$ dominates, triggers speciation or evolution into a different taxonomic class, i.e., Eqs. (22) and (23) no longer has the $O\left(2\right)$ symmetry of the original genus. Secondly, for high temperatures, the balance function Eq. (23) shows that species ${\varphi }_{2}^{{\prime }}$ has lost its dependence on temperature since ${\left({\varphi }_{2}^{{\prime }}\right)}^{2}$ has disappeared, leaving only the result of its transformation $\left({\partial }_{\mu }{\varphi }_{2}^{{\prime }}\right)$. In contrast, ${\varphi }_{1}^{{\prime }}$ retains a term, ${\beta }^{2}{\left({\varphi }_{1}^{{\prime }}\right)}^{2}=a\left({T}_{C}-T\right){\left({\varphi }_{1}^{{\prime }}\right)}^{2}$, in Eq. (23).

Crucial to symmetry breaking, therefore, is a temperature that goes beyond a critical value ${T}_{C}$. In this case, small deviations from the minimum of $\left({\varphi }_{1}-{\varphi }_{2}\right)$-space shows the disappearance of the same-species interaction ${\left({\varphi }_{2}^{{\prime }}\right)}^{2}$ and only a space and time evolution, i.e., $\left({\partial }_{\mu }{\varphi }_{2}^{{\prime }}\right)\left({\partial }_{\mu }{\varphi }_{2}^{{\prime }}\right)$ persists implying continued transformation towards an organism that no longer belongs to the same genus. This runaway speciation, however, is only possible if there are at least two species which allows a continuous symmetry transformation such as Eq. (10) [16]. Moreover, if one starts with $n$ species such as Eq. (4), symmetry breaking would lead to $n-1$ species experiencing enhanced speciation that breeds more biodiversity. This echoes the empirical observation that "diversity builds on diversity" (Bouchet, 2011).

Continued exploration of terrestrial and marine life has witnessed a proliferation of new species with many still undiscovered. It is estimated, for instance, that 70% of life in the seas are not even known to science (Régnier et al, 2009). The identification of newly discovered species and determination of their phylogeny have indeed become a daunting task. Putting specimens in different taxonomic classes, however, could imply an underlying symmetry behind morphological differences. In this paper, we investigate possible symmetries among species and how they may be broken to allow rapid speciation that leads to biodiversity. In particular, we show how changes in temperature beyond a critical value ${T}_{C}$ may trigger accelerated speciation that enhances biodiversity which may partly explain the rich flora and fauna, as well as marine life in the tropics. On the other hand, examples of critical temperatures may be gleaned from the case of mesophiles that grow at temperatures between 25°C and 40°C whereas thermophiles survive at temperatures higher than 40°C (Concha-Marambio et al, 2017; Sawle and Ghosh, 2011; Dominy et al, 2004; Ge et al, 2008). It is also not difficult to imagine that there are species that could proliferate at lower rather than at higher temperatures (Waldvogel and Pfenninger, 2021; Dominy et al, 2004). This scenario can also be accommodated in the present framework by redefining Eq. (1) as, ${\beta }^{2}=a\left({T-T}_{C}\right)$, i.e., interchanging the positions of ${T}_{C}$ and $T$. In this case, symmetry breaking, or runaway speciation, occurs when temperatures are lower than a critical temperature ${T}_{C}$ when ${\beta }^{2}<0$. The mathematical framework presented here, where the impact of temperature $T$ is made explicit, provides a different perspective and insight into the frequent branching that occurs in phylogenetic trees.

Declaration of Competing Interest

The author has no conflicts of interest to declare that are relevant to the content of this article.

Acknowledgement

The author would like to thank Hyunjin Shim for helpful discussions.

The transformation Eq. (10) can be written as,

$${\varphi }_{1}^{{\prime }}={\varphi }_{1}cos\left(\alpha \right)+{\varphi }_{2}sin\left(\alpha \right)$$

$${\varphi }_{2}^{{\prime }}=-{\varphi }_{1}sin\left(\alpha \right)+{\varphi }_{2}cos\left(\alpha \right)$$

Using the definition Eq. (11) together with Eqs. (A1) and (A2), a transformed ${\Phi }$ field becomes,

$${\Phi }{\prime }=\frac{1}{\sqrt{2}}\left({\varphi }_{1}{\prime }+i{\varphi }_{2}{\prime }\right)$$

$${\Phi }{\prime }=\frac{1}{\sqrt{2}}\left[{\varphi }_{1}cos\left(\alpha \right)+{\varphi }_{2}sin\left(\alpha \right)-i{\varphi }_{1}sin\left(\alpha \right)+i{\varphi }_{2}cos\left(\alpha \right)\right]$$

$$=\frac{1}{\sqrt{2}}\left\{{\varphi }_{1}\left[cos\left(\alpha \right)-i sin\left(\alpha \right)\right]+{i\varphi }_{2}\left[cos\left(\alpha \right)-i sin\left(\alpha \right)\right]\right\}$$

$$=\frac{1}{\sqrt{2}}{e}^{-i\alpha }\left[{\varphi }_{1}+{i\varphi }_{2}\right]$$

$$={e}^{-i\alpha }{\Phi }$$

which is Eq. (14) above.

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No competing interests reported.

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Temperature-Dependent Enhanced Speciation in Ecosystems with Conserved Symmetries

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Abstract

Figures

1 Introduction

2 Temperature Impacts Cell Processes

3 Species Representation

4 Invariances Over Space And Time

5 Symmetries Of The Balance Function

6 Symmetry Breaking

6.1 Low Temperature:\({\beta }^{2}>0\)

6.2 High Temperature:\({\beta }^{2}<0\)

7 Enhanced Speciation

8 Conclusion

Declarations

Appendix

References

Additional Declarations

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