The virial theorem and the Price equation

The virial theorem was first described by Rudolf Clausius in connection with his studies on heat transfer [5]. In its simplest form, it relates the time averaged kinetic energy $\langle T\rangle_{\tau}=\langle\frac{1}{2}\sum_{i=1}^{n}m_{i}v_{i}(t)^{2}\rangle_{\tau}=\frac{1}{\tau}\int_{0}^{\tau}\frac{1}{2}\sum_{i=1}^{n}m_{i}v_{i}(t)^{2}dt$ of $n$ objects with masses $m_{1},\ldots,m_{n}$ and velocities $v_{1}(t),\ldots,v_{n}(t)$, to their time averaged potential energy $\langle U\rangle_{\tau}=\langle\sum_{i=1}^{n}F_{i}(t)z_{i}(t)\rangle_{\tau}$, where $F_{1}(t),\ldots,F_{n}(t)$ are the forces acting on the $n$ objects and $z_{1}(t),\ldots,z_{n}(t)$ are their respective positions:

The mathematical underpinning of the virial theorem is the product rule from calculus. The derivative of the Clausius virial $S(t)=\sum_{i=1}^{n}p_{i}(t)z_{i}(t)$ where $p_{i}(t)=m_{i}v_{i}(t)$ is

	$\displaystyle\frac{dS(t)}{dt}$	$\displaystyle=$	$\displaystyle\sum_{i=1}^{n}m_{i}\frac{dv_{i}(t)}{dt}z_{i}(t)+\sum_{i=1}^{n}p_{i}(t)\frac{dz_{i}(t)}{dt}$		(2)
		$\displaystyle=$	$\displaystyle\sum_{i=1}^{n}F_{i}(t)z_{i}(t)+\sum_{i=1}^{n}m_{i}v_{i}(t)^{2}$
		$\displaystyle=$	$\displaystyle U+2T,$
	$\displaystyle\implies\left\langle\frac{dS(t)}{dt}\right\rangle_{\tau}$	$\displaystyle=$	$\displaystyle\langle U\rangle_{\tau}+2\langle T\rangle_{\tau}.$		(3)

For stably bound systems the velocities and positions of objects have upper and lower bounds, so the average of the derivative of $S(t)$ over a period of time $\tau$ will be zero in the limit of large $\tau$, i.e. for large $\tau$, $\langle\frac{dS(t)}{dt}\rangle_{\tau}\approx 0$. When $\left\langle\frac{dS(t)}{dt}\right\rangle_{\tau}=0$, we obtain from (3) the virial theorem (1): $\langle T\rangle_{\tau}=-\frac{1}{2}\langle U\rangle_{\tau}$.

The virial theorem was well-known to physicists in the late 19th and early 20th centuries [31, 8], however its power as a discovery tool for astrophysics was first highlighted by Fritz Zwicky [46]. Zwicky used the virial theorem to estimate the mass of the Coma cluster, thereby identifying a mass deficit in comparison to luminosity estimates, leading him to posit the existence of what he called dunkle materie (dark matter) [46]. Although Zwicky’s mass estimates were inaccurate [37, 23], the principle of using the virial theorem to identify a measurement gap was sound, and the virial theorem has become widely used in physics and astrophysics. It can be used to derive classic laws such as the ideal gas law [12, 15], and extensions are applicable in many settings, including quantum mechanics [14], astrophysical hydrodynamics [34], and fluid mechanics [25, 1].

The Price equation [29] pertains to selection in evolutionary processes. It was motivated by a desire to understand the evolution of altruism [16], and has been described as a “fundamental theorem of evolution” [30] due to its generalization and unification of many results in evolutionary biology. For example, Fisher’s fundamental theorem of natural selection [11] is a special case of the Price equation [30, 13].

The Price equation relates the change in a trait in a population over time, to fitness values in subpopulations. Formally, the (discrete) Price equation as published in [29] (we follow notation from [13]) considers a numerical trait in $n$ subpopulations at time $t$ denoted ${\bf z}(t)=(z_{1}(t),\ldots,z_{n}(t))$. The subpopulations have sizes $p_{1}(t),\ldots,p_{n}(t)$, and have (Wrightian) fitness ${\bf w}(t)=(w_{1}(t),\ldots,w_{n}(t))$ defined by $w_{i}(t)=\frac{p_{i}(t+\Delta t)}{p_{i}(t)}$ where $\Delta t$ denotes the time interval of one generation [40]. Let $q_{i}(t)=\frac{p_{i}(t)}{\sum_{j=1}^{n}p_{j}(t)}$ be the relative size of the $i$th population, and define the population average of fitness to be $\overline{{\bf w}}(t)=\sum_{i=1}^{n}q_{i}(t)w_{i}(t)$. Note that ${\bf q}(t)$ forms a probability distribution for ${\bf w}(t)$ viewed as a random variable, and $\mathbb{E}({\bf w}(t))=\overline{{\bf w}}(t)$. Let $\Delta z_{i}(t)=z_{i}(t+\Delta t)-z_{i}(t)$, $\Delta{\bf z}(t)={\bf z}(t+\Delta t)-{\bf z}(t)$, and $\overline{{\bf z}}(t)=\sum_{i=1}^{n}q_{i}(t)z_{i}(t)$ with $\Delta\overline{{\bf z}}(t)=\overline{{\bf z}}(t+\Delta t)-\overline{{\bf z}}(t)$.

Intuitively, if subpopulation fitness has positive covariance with trait values, then the trait is beneficial, and the trait value, averaged across populations, will increase after a generation. However, if the covariance between subpopulation fitness and trait values is negative, higher trait values are detrimental and the trait value averaged across populations will decrease after a generation.

The Price equation as published in [29] is discrete in time, and proof of the identity uses basic properties of expectation and covariance along with the fact that $q_{i}(t+\Delta t)=\frac{q_{i}(t)w_{i}(t)}{\overline{{\bf w}}(t)}$, which we leave as an exercise for the reader. Note that

	$\displaystyle\overline{{\bf w}}(t)\Delta\overline{{\bf z}}(t)$	$\displaystyle=$	$\displaystyle\overline{{\bf w}}(t)\overline{{\bf z}}(t+\Delta t)-\overline{{\bf w}}(t)\overline{{\bf z}}(t)$
		$\displaystyle=$	$\displaystyle\overline{{\bf w}}(t)\sum_{i=1}^{n}q_{i}(t+\Delta t)z_{i}(t+\Delta t)-\overline{{\bf w}}(t)\overline{{\bf z}}(t)$
		$\displaystyle=$	$\displaystyle\sum_{i=1}^{n}q_{i}(t)w_{i}(t)z_{i}(t+\Delta t)-\overline{{\bf w}}(t)\overline{{\bf z}}(t)$
		$\displaystyle=$	$\displaystyle\sum_{i=1}^{n}q_{i}(t)w_{i}(t)z_{i}(t)-\overline{{\bf w}}(t)\overline{{\bf z}}(t)+\sum_{i=1}^{n}q_{i}(t)w_{i}(t)z_{i}(t+\Delta t)-\sum_{i=1}^{n}q_{i}(t)w_{i}(t)z_{i}(t)$
		$\displaystyle=$	$\displaystyle\mathrm{cov}({\bf w}(t),{\bf z}(t))+\mathbb{E}({\bf w}(t)\odot\Delta{\bf z}(t)),$
	$\displaystyle\implies\Delta\overline{{\bf z}}(t)$	$\displaystyle=$	$\displaystyle\frac{1}{\overline{{\bf w}}(t)}\mathrm{cov}({\bf w}(t),{\bf z}(t))+\frac{1}{\overline{{\bf w}}(t)}\mathbb{E}({\bf w}(t)\odot\Delta{\bf z}(t)).$

The discrete time Price equation has a continuous time analog [28, 9]. It is formulated using the Malthusian fitness ${\bf r}(t)=r_{1},\ldots,r_{n}$ given by $r_{i}(t)=\frac{1}{p_{i}(t)}\frac{dp_{i}(t)}{dt}$ instead of the Wrightian fitness ${\bf w}(t)$.

The continuous time Price equation (5) is the continuum limit of the discrete Price equation (4). To see this, we begin by multiplying the Price equation by $\frac{\overline{{\bf w}}(t)}{\Delta t}$:

$\displaystyle\frac{\overline{{\bf w}}(t)\Delta\overline{{\bf z}}(t)}{\Delta t}$

$\displaystyle=$

$\displaystyle\frac{1}{\Delta t}\mathrm{cov}({\bf w}(t),{\bf z}(t))+\frac{1}{\Delta t}\mathbb{E}({\bf w}(t)\odot\Delta{\bf z}(t)).$

We will now see why, contrary to convention, we have indexed the variables in equation (4) with time. Starting with the left hand side, we observe that

	$\displaystyle\lim_{\Delta t\rightarrow 0}\overline{{\bf w}}(t)$	$\displaystyle=$	$\displaystyle lim_{\Delta t\rightarrow 0}\sum_{i=1}^{n}q_{i}(t)w_{i}(t)$
		$\displaystyle=$	$\displaystyle\lim_{\Delta t\rightarrow 0}\sum_{i=1}^{n}\frac{p_{i}(t)p_{i}(t+\Delta t)}{\left(\sum_{j=1}^{n}p_{j}(t)\right)p_{i}(t)}$
		$\displaystyle=$	$\displaystyle\frac{1}{\sum_{j=1}^{n}p_{j}(t)}\lim_{\Delta t\rightarrow 0}\sum_{i=1}^{n}p_{i}(t+\Delta t)\,=\,1.$

Therefore,

	$\displaystyle\lim_{\Delta t\rightarrow 0}\frac{\overline{{\bf w}}(t)\Delta\overline{{\bf z}}(t)}{\Delta t}$	$\displaystyle=$	$\displaystyle\lim_{\Delta t\rightarrow 0}\frac{\sum_{i=1}^{n}q_{i}(t+\Delta t)z_{i}(t+\Delta t)-\sum_{i=1}^{n}q_{i}(t)z_{i}(t)}{\Delta t}$
		$\displaystyle=$	$\displaystyle\frac{d}{dt}\mathbb{E}({\bf z}(t)).$

The covariance term, in the limit as $\Delta t\rightarrow 0$, is given by

$\displaystyle\lim_{\Delta t\rightarrow 0}\frac{1}{\Delta t}\mathrm{cov}({\bf w}(t),{\bf z}(t))$

$\displaystyle=$

$\displaystyle\lim_{\Delta t\rightarrow 0}\frac{\sum_{i=1}^{n}q_{i}(t)w_{i}(t)z_{i}(t)-\overline{{\bf w}}(t)\overline{{\bf z}}(t)}{\Delta t}.$

Let $g_{i}(t)=\frac{1}{\Delta t}ln(w_{i}(t))$. Note that $w_{i}(t)=e^{g_{i}(t)\Delta t}$ and that $\lim_{\Delta t\rightarrow 0}g_{i}(t)=r_{i}(t)$. Substituting $e^{g_{i}(t)\Delta t}$ for $w_{i}(t)$ yields

	$\displaystyle\lim_{\Delta t\rightarrow 0}\frac{1}{\Delta t}\mathrm{cov}({\bf w}(t),{\bf z}(t))$	$\displaystyle=$	$\displaystyle\lim_{\Delta t\rightarrow 0}\frac{\sum_{i=1}^{n}q_{i}(t)e^{g_{i}(t)\Delta t}z_{i}(t)-\sum_{i=1}^{n}q_{i}(t)e^{g_{i}(t)\Delta t}\overline{{\bf z}}(t)}{\Delta t}$
		$\displaystyle=$	$\displaystyle\lim_{\Delta t\rightarrow 0}\frac{\sum_{i=1}^{n}q_{i}(t)e^{g_{i}(t)\Delta t}(z_{i}(t)-\overline{{\bf z}}(t))}{\Delta t}$
		$\displaystyle=$	$\displaystyle\lim_{\Delta t\rightarrow 0}\sum_{i=1}^{n}q_{i}(t)g_{i}(t)e^{g_{i}(t)\Delta t}(z_{i}(t)-\overline{{\bf z}}(t))\quad\mbox{by the L'H\^{o}pital-Bernoulli rule \cite[cite]{[\@@bibref{}{lhospitale1696}{}{}]}}$
		$\displaystyle=$	$\displaystyle\sum_{i=1}^{n}q_{i}(t)r_{i}(t)(z_{i}(t)-\overline{{\bf z}}(t))$
		$\displaystyle=$	$\displaystyle\sum_{i=1}^{n}q_{i}(t)r_{i}(t)z_{i}(t)-\sum_{i=1}^{n}q_{i}(t)r_{i}(t)\overline{{\bf z}}(t)$
		$\displaystyle=$	$\displaystyle\mathrm{cov}({\bf r}(t),{\bf z}(t)).$

Finally, we have that

	$\displaystyle\lim_{\Delta t\rightarrow 0}\frac{1}{\Delta t}\mathbb{E}({\bf w}(t)\odot\Delta{\bf z}(t))$	$\displaystyle=$	$\displaystyle\lim_{\Delta t\rightarrow 0}\frac{\sum_{i=1}^{n}q_{i}(t)e^{g_{i}(t)\Delta t}\Delta z_{i}(t)}{\Delta t}$
		$\displaystyle=$	$\displaystyle\sum_{i=1}^{n}q_{i}(t)\frac{dz_{i}(t)}{dt}$
		$\displaystyle=$	$\displaystyle\mathbb{E}\left(\frac{d{\bf z}(t)}{dt}\right).$

In summmary,

	$\displaystyle\frac{\overline{{\bf w}}(t)\Delta{\bf z}(t)}{\Delta t}$	$\displaystyle\quad=\quad\frac{1}{\Delta t}\mathrm{cov}({\bf w}(t),{\bf z}(t))+\frac{1}{\Delta t}\mathbb{E}({\bf w}(t)\odot\Delta{\bf z}(t))$	$\displaystyle(\mbox{discrete Price equation (\ref{eq:discreteprice}}))$
	$\displaystyle\Big{\downarrow}\scriptstyle{\lim_{\Delta t\rightarrow 0}}$	$\displaystyle\qquad\qquad\qquad\qquad\Big{\downarrow}\scriptstyle{\lim_{\Delta t\rightarrow 0}}$
	$\displaystyle\frac{d}{dt}\mathbb{E}({\bf z}(t))$	$\displaystyle\quad=\quad\mathrm{cov}({\bf r}(t),{\bf z}(t))+\mathbb{E}\left(\frac{d{\bf z}(t)}{dt}\right)$	$\displaystyle(\mbox{continuous Price equation (\ref{eq:contprice}}))$

In the physics setting, recall that the momentum $p_{i}(t)=m_{i}v_{i}(t)=m_{i}\frac{dz_{i}(t)}{dt}$. Let $r_{i}=\frac{1}{{p_{i}(t)}}\frac{dp_{i}(t)}{dt}$, i.e. acceleration divided by velocity. If all the momenta are greater than zero, i.e., $p_{i}(t)>0$ for all $i$, we can define relative momentum as $q_{i}(t)=\frac{p_{i}(t)}{\sum_{j=1}^{n}p_{j}(t)}$. Consider the virial density $\tilde{S}(t)=\sum_{i=1}^{n}q_{i}(t)z_{i}(t)$ [10], whose derivative is $\frac{d\tilde{S}(t)}{dt}=\frac{d}{dt}\mathbb{E}({\bf z}(t))$. The product rule applied to the virial density is

	$\displaystyle\frac{d}{dt}\mathbb{E}({\bf z}(t))$	$\displaystyle=$	$\displaystyle\sum_{i=1}^{n}\frac{d}{dt}\left(\frac{p_{i}(t)}{\sum_{j=1}^{n}p_{j}(t)}\right)z_{i}(t)+\sum_{i=1}^{n}q_{i}(t)\frac{dz_{i}(t)}{dt}$
		$\displaystyle=$	$\displaystyle\sum_{i=1}^{n}\frac{\frac{dp_{i}(t)}{dt}\sum_{j=1}^{n}p_{j}(t)-p_{i}(t)\sum_{j=1}^{n}\frac{dp_{j}(t)}{dt}}{(\sum_{j=1}^{n}p_{j}(t))^{2}}z_{i}(t)+\mathbb{E}\left(\frac{d{\bf z}(t)}{dt}\right)$
		$\displaystyle=$	$\displaystyle\sum_{i=1}^{n}\frac{p_{i}(t)}{\sum_{j=1}^{n}p_{j}(t)}\left(r_{i}(t)-\frac{\sum_{j=1}^{n}r_{j}(t)p_{j}(t)}{\sum_{j=1}^{n}p_{j}(t)}\right)z_{i}(t)+\mathbb{E}\left(\frac{d{\bf z}(t)}{dt}\right)$
		$\displaystyle=$	$\displaystyle\sum_{i=1}^{n}q_{i}(t)r_{i}(t)z_{i}(t)-\sum_{j=1}^{n}q_{j}(t)r_{j}(t)\sum_{i=1}^{n}q_{i}(t)z_{i}(t)+\mathbb{E}\left(\frac{d{\bf z}(t)}{dt}\right)$
		$\displaystyle=$	$\displaystyle\mathbb{E}({\bf r}(t)\odot{\bf z}(t))-\mathbb{E}({\bf r}(t))\mathbb{E}({\bf z}(t))+\mathbb{E}\left(\frac{d{\bf z}(t)}{dt}\right),$
	$\displaystyle\implies\quad\frac{d}{dt}\mathbb{E}({\bf z}(t))$	$\displaystyle=$	$\displaystyle\mathrm{cov}({\bf r}(t),{\bf z}(t))+\mathbb{E}\left(\frac{d{\bf z}(t)}{dt}\right).$		(6)

The virial theorem and the continuous Price equation are mathematically identical. Therefore, the relationships between traits and fitness in evolutionary biology are not only reminiscent of the relationships between physical quantities like distance, velocity, and acceleration; they are the same. Thus, other versions of the Price equation can engender physical insights. For instance the discrete Price equation (4) can also be understood as a discretization of the virial theorem. Furthermore, equation (6) is a statistical form of the virial. This provides a relationship between group, intrinsic and extrinsic velocities rather than energies and then averages those velocities with respect to relative momentum.

Both equations (5) and (6) can be time averaged, as is natural in the physics setting, to obtain

$$\left\langle\frac{d}{dt}\mathbb{E}({\bf z}(t))\right\rangle_{\tau}=\left\langle\mathrm{cov}({\bf r}(t),{\bf z}(t))\right\rangle_{\tau}+\left\langle\mathbb{E}\left(\frac{d{\bf z}(t)}{dt}\right)\right\rangle_{\tau}.$$

In summary, we have the following relationships:

	Biology	Physics
	Price equation	virial theorem
	$\displaystyle\Big{\downarrow}\scriptstyle{\lim_{\Delta t\rightarrow 0}}$	$\displaystyle\qquad\qquad\qquad\Big{\downarrow}\scriptstyle{p_{i}>0}$
	continuous Price equation	$\displaystyle\qquad=\qquad\mbox{positive momentum virial theorem}$

In genetics, the natural time increment to consider is discrete (generation), whereas in physics continuous time is more natural. Thus, the discrete Price equation pertains to change in a trait after a single generation, whereas the virial theorem is formulated with continuous time, and is additionally time averaged. However, the less intuitive forms of these equations that arise from the correspondences derived above may yield important insights. For example, the perspective of the virial theorem as a special case of the equipartition theorem [27] may be fruitful in evolutionary biology [24]. Translation between biology and physics via the virial theorem and the Price equation may also accelerate discovery of generalizations. While the stochastic Price equation in evolution [33] and the stochastic virial theorem in astronomy [6] were discovered independently, their similarity suggests other generalizations could similarly parallel each other. Moreover, the virial theorem has been applied in a variety of fields (for example economics [2]), meaning that understanding its relationship to the Price equation could be relevant beyond physics and biology.

The dynamical interpretation of evolutionary theory posits a correspondence between theories of evolution and Newtonian mechanics [36, 17]. In this framework, notions such as selection or mutation in biology are associated to forces in physics [36]. The identical form of equations (5) and (6) can constrain such associations and clarify the subsequent analogies. For example, although the standard form of the virial theorem in (3) is an energy equation, equation (6) is a velocity equation, which in biology translates to rates of change of biological quantities. Furthermore, rate of change of a trait or phenotype, i.e., $\frac{d}{dt}\mathbb{E}({\bf z}(t))$ in equation (5) or the finite difference $\Delta{\bf\overline{z}}(t)$ in equation (4), corresponds to the momentum-averaged bulk velocity of a collection of physical objects. The remaining terms in each equation similarly have physical meaning in the context of classical mechanics in equation (6), or evolutionary biology in equation (4). Table 1 shows a list of variables, their meaning in classical mechanics, and their meaning in evolutionary biology.

The momentum-averaged positions $\mathbb{E}({\bf z}(t))$ and velocities $\frac{d}{dt}\mathbb{E}({\bf z}(t))$ are discrete analogs of momentum-averaged position and momentum velocity in electromagnetism, where they emerge from the virial density in an application of the virial theorem to electromagnetic pulses [10]. Whereas $\mathrm{cov}({\bf r}(t),{\bf z}(t))$ is frequently referred to as the selection term in the Price equation [4], the connection to the virial theorem suggests it is better described as a selection rate. Similarly, the transmission term $\mathbb{E}\left(\frac{d{\bf z}(t)}{dt}\right)$ is more accurately a transmission rate. Moreover, the analog to fitness as understood from the perspective of the virial theorem is acceleration normalized by velocity. Most significantly, while the dynamical interpretation typically relies on associating evolutionary force to natural selection, drift, migration, or mutation [17], the equivalence between the virial theorem and the Price equation, suggests that force is more naturally associated to the population growth rate that they can affect. The correspondence of force to a rate of change is not surprising, since force is also a rate of change, specifically the rate of change of momentum. This stands more in line with the statistical interpretation of evolutionary theory [42, 41, 22], which, among several critiques of the dynamical interpretation, finds fault with the analogies of biological processes such as mutation with forces in physics, arguing that the physical forces are causal in a way that processes such as selection or mutation are not [42]. Nevertheless, the analogy of population growth with force can be viewed as consistent with the dynamical interpretation; for example, population growth can directly affect patterns of DNA polymorphism [43]. Moreover, the mathematical correspondence of the Price equation with the virial theorem suggests that the analogies in Table 1 could sharpen the arguments for the dynamical interpretation [17].

Ultimately, analogies between the Price equation and the virial theorem point towards potentially productive directions for exploration in both biology and physics. The statistical framing of the virial theorem in (6) highlights phenomena that may have been overlooked in the physics realm. For example, the first term in the right hand side of (6), namely $\mathrm{cov}({\bf r}(t),{\bf z}(t))$, can be understood to quantify the extent of the Yule-Simpson effect [26, 44, 35], which describes a situation where within group trends can be reversed upon averaging. In biology, the Price equation has potential to be more widely used as a tool. While it has been hailed as a unifying framework for researchers [18], one that “can serve as a heuristic principle to formulate and systematise different theories and models in evolutionary biology” [19], the emphasis on its use has been more oriented towards understanding how it generalizes specific equations, rather than applying it for biological discovery. For example, the Price equation can be used to derive the Breeder’s equation [3, 45], Fisher’s fundamental theorem [30, 28], the house-of-cards approximation for genetic variance at mutation-selection balance [45, 38], and many other formulas and identities in genetics [45, 32]. However, it has been referred to as a tautology and a vacuous statement with no application. In [39] the Price equation is described as a theorem that establishes that “If the left hand side is computed as suggested in [29], and the right hand side too, then they are equal.” This critique of the Price equation, namely that it does not and cannot serve as a tool, stands in contradiction to evidence from physics, where the mathematically equivalent virial theorem has been understood as a powerful tool since its use to discover dark matter in 1933 [46]. The manifold applications of the virial theorem [21] suggest there is much to yet be gained from application of the Price equation as a tool for biology. In fact, the equivalence we have demonstrated between the Price equation and the virial theorem shows that the description of missing heritability as dark matter [20] may be understood to be more than just an informal analogy between mysteries in genetics and astronomy.

SL studied the virial theorem while participating in the “Introduction to Astrophysics” cluster in the COSMOS summer program held at UC Irvine from July 9, 2023 to August 4, 2023. Specifically, she used the virial theorem to repeat Zwicky’s Coma cluster mass estimates using modern measurements of velocity dispersion and galaxy positions. LP learned of the virial theorem from SL, and in discussing its proof with SL, realized that it must be related to the Price equation. SL and LP explored the applications and implications of the connection. LP drafted the initial manuscript; both SL and LP edited the final version.

SL thanks Manoj Kaplinghat and Gopolang Mohlabeng who led the “Introduction to Astrophysics” cluster (cluster 4) at the 2023 UC Irvine COSMOS program. LP relied in part on notes about Fisher’s theorem of natural selection from his April 22, 2008 lecture for UC Berkeley course Math 239: Discrete Mathematics for the Life Sciences that were transcribed and edited by Cynthia Vinzant and Caroline Uhler.