Blame weak molecular interactions, small, often very small, numbers of specific molecules per cell, Brownian motion, and (perhaps) Heisenberg – students are often not taught about the ubiquitous role of stochastic processes in biological systems, and how they produce unpredictable behaviors – perhaps they should be.
Background: Three recent events combined to spur this reflection on stochasticity in biological systems, how it is taught, and why it matters. The first was an article describing an approach to introducing students to homeostatic processes in the context of the bacterial lac operon (Booth et al., 2022), an adaptive gene regulatory system controlled in part by stochastic events. The second were in-class student responses to the question, why do interacting molecules “come back apart” (dissociate). Finally, there is the increasing attention paid to what are presented as deterministic genetic factors, as illustrated by talk by Kathryn Harden, author of the “The Genetic Lottery: Why DNA matters for social equality” (Harden, 2021). Previous work has suggested that students, and perhaps some instructors, find the ubiquity, functional roles, and implications of stochastic, that is inherently unpredictable processes, difficult to recognize and apply. Given their practical and philosophical implications, it seems essential to introduce students to stochasticity early in their educational journey.
What is stochasticity and why is it important for understanding biological systems? Stochasticity results when intrinsically unpredictable events, e.g. molecular collisions, impact the behavior of a system. There are a number of drivers of stochastic behaviors. Perhaps the most obvious, and certainly the most ubiquitous in biological systems is thermal motion. The many molecules within a solution (or a cell) are moving, they have kinetic energy – the energy of motion and mass. The exact momentum of each molecule cannot, however, be accurately and completely characterized without perturbing the system (echos of Heisenberg). Such an impossible measurement will leave us uncertain as to the state of the system’s components going forward.
Through collisions energy is exchanged between molecules. A number of chemical processes are driven by the energy delivered through such collisions. Think about a typical chemical reaction. In the course of the reaction, atoms are rearranged – bonds are broken (a process that requires energy) and bonds are formed (a process that releases energy). Many (most) of the chemical reactions that occur in biological systems require catalysts to bring their required activation energies into the range available within the cell. 
What makes the impact of thermal motion even more critical for biological systems is that many (most) regulatory interactions and macromolecular complexes, the molecular machines discussed by Alberts (1998), are based on regions of non-covalent, that is relatively weak, surface-surface interactions between or within molecules. Such interactions are central to most regulatory processes, from the activation of signaling pathways to the control of gene expression. The specificity and stability of these non-covalent interactions, which include those involved in determining the 3D structure of macromolecules, are directly impacted by thermal motion, and so by temperature – one reason controlling body temperature is important.
So why are these interactions stochastic and why does it matter? A signature property of a stochastic process is that while it may be predictable when large numbers of atoms, molecules, or interactions are involved, the behaviors of individual atoms, molecules, interactions are not. A classic example, arising from factors intrinsic to the atom, is the decay of radioactive isotopes. While the half-life of a large enough population of a radioactive isotope is well defined, when any particular atom will decay is not. This is the reason we cannot accurately predict whether Schrȍdinger’s cat is alive or dead. The same behavior applies to the binding of a regulatory protein to a specific site on a DNA molecule and its subsequent dissociation: predictable in large populations, not-predictable for individual molecules. The situation is exacerbated by the fact that biological (living) systems are composed of cells and cells are, typically, small, and so contain relatively few molecules of each type (Milo and Phillips, 2015). In particular, there are typically one or two copies of each gene in a cell. The expression of a particular gene depends upon the binding of specific proteins, transcription factors, that act to activate or repress gene expression. In contrast to a number of other cellular proteins, “as a rule of thumb, the concentrations of such transcription factors are in the nM range, corresponding to only 1-1000 copies per cell in bacteria or 103-106 in mammalian cells” (Milo and Phillips, 2015). Moreover, while DNA binding proteins bind to specific DNA sequences with high affinity, they also bind to DNA “non-specifically”, that is with low affinity in a largely sequence independent manner. Given that there are many more non-specific (non-functional) binding sites in the DNA than functional binding sites, the effective concentration of a particular transcription factor can be significantly lower than its total cellular concentration would suggest. For example, in the case of the lac repressor of E. coli (discussed further below), there are estimated to be ~10 molecules of the tetrameric lac repressor per cell, but “non-specific affinity to the DNA causes >90% of LacI copies to be bound to the DNA at locations that are not the cognate promoter (sic) site” (Milo and Phillips, 2015); at most only a few molecules are free in the cytoplasm and available to bind to specific regulatory sites . Most transcription factors bind in a functionally significant manner to hundreds to thousands of gene regulatory sites per cell, often with distinct binding affinities. The effective binding affinity can also be influenced by positive and negative interactions with other transcription and accessory factors, chromatin structure, and DNA modifications. Functional complexes can take time to assemble, and once assembled can initiate multiple rounds of polymerase binding and activation, leading to a phenomena known as transcriptional bursting. An analogous process occurs with RNA-dependent polypeptide synthesis (translation), leading to translational bursting. The result, particularly for genes expressed at lower levels, is that stochastic (unpredictable) bursts of transcription/translation can lead to functionally significant changes in protein levels (Raj et al., 2010; Raj and van Oudenaarden, 2008).
There are many examples of stochastic behaviors in biological systems. Originally noted by Novick and Weiner (1957) in their studies of the lac operon, it was clear that gene expression occurred in an all or none manner. This effect was revealed in a particularly compelling manner by Elowitz et al (2002) who used lac operon promoter elements to drive expression of transgenes encoding cyan and yellow fluorescent proteins (on a single plasmid) in E. coli. The observed behaviors were dramatic; genetically identical cells were found to express, stochastically, one, the other, both, or neither transgenes. The stochastic expression of genes and downstream effects appear to be the source of much of the variance found in organisms with the same genotype in the same environmental conditions (Honegger and de Bivort, 2018). Beyond gene expression, the non-deterministic effects of stochastic processes can be seen at all levels of biological organization, from the biased random walk behaviors that underlie various forms of chemotaxis (Spudich and Koshland, 1976) and the search behaviors in C. elegans (Roberts et al., 2016) and other animals (Smouse et al., 2010), the noisiness in the opening of individual voltage-gated ion channels in neurons (Braun, 2021; Neher and Sakmann, 1976), and various processes within the immune system (Hodgkin et al., 2014), to variations in the behavior of individual organisms (see the leafhopper example cited by Honegger & de Bivort, 2018). Stochastic events are involved in a range of “social” processes in bacteria (Bassler and Losick, 2006). Their impact serves as a form of “bet-hedging” in populations that generate phenotypic variation in a homogeneous environment (see Symmons and Raj, 2016). Stochastic events can regulate the efficiency of replication-associated error-prone mutation repair (Uphoff et al., 2016) leading to increased variation in a population, particularly in response to environmental stresses. Stochastic “choices” made by cells can be seen as questions asked of the environment, the system’s response provides information that informs subsequent regulatory decisions (see Lyon, 2015) and the evolution of a population (Jablonka and Lamb, 2005). Together stochastic processes introduce a non-deterministic (i.e. unpredictable) element into higher order behaviors (Murakami et al., 2017; Roberts et al., 2016).
Controlling stochasticity: While stochasticity can be useful, it also needs to be controlled. Not surprisingly then there are a number of strategies for “noise-suppression”, ranging from altering regulatory factor concentrations, the formation of covalent disulfide bonds between or within polypeptides, and regulating the activity of repair systems associated with DNA replication, polypeptide folding, and protein assembly via molecular chaperones and targeted degradation. For example, the identification of “cellular competition” effects has revealed that “eccentric cells” (sometimes referred to by the less accurate terms of “winners and losers”) can be induced to undergo apoptosis (die) or migration in response to their “normal” neighbors (Akieda et al., 2019; Di Gregorio et al., 2016; Ellis et al., 2019; Hashimoto and Sasaki, 2020; Lima et al., 2021).
Student understanding of stochastic processes: There is ample evidence that students (and perhaps some instructors as well) are confused by or uncertain about the role of thermal motion, that is the transfer of kinetic energy via collisions, and the resulting stochastic behaviors in biological systems. As an example, Champagne-Queloz et al (2016; 2017) found that few students, even after instruction through molecular biology courses, recognize that collisions with other molecules were responsible for the disassembly of molecular complexes. In fact, many adopt a more “deterministic” model for molecular disassembly after instruction (FIG. 1). In earlier studies, we found evidence for a similar confusion among instructors (FIG. 2) (Klymkowsky et al., 2010).
Introducing stochasticity to students: Given that understanding stochastic (random) processes can be difficult for many (e.g. Garvin-Doxas and Klymkowsky, 2008; Taleb, 2005), the question facing course designers and instructors is when and how best to help students develop an appreciation for the ubiquity, specific roles, and implications of stochasticity-dependent processes, at all levels, in biological systems. Here we suggest that introducing students to the dynamics of non-covalent molecular interactions, prevalent in biological systems in the context of stochastic interactions (i.e.kinetic theory) rather than a ∆G-based approach may be useful. We can use the probability of garnering the energy needed to disrupt an interaction to present concepts of binding specificity (selectivity) and stability. Developing an understanding of the formation and disassembly of molecular interactions builds on the same logic that Albert Einstein and Ludwig Böltzman used to demonstrate the existence of atoms and molecules and the reversibility of molecular reactions (Bernstein, 2006).
The selectivity (specificity) and stability of molecular interactions can be understood from an energetic perspective – comparing the enthalpic and entropic differences between bound and unbound states. What is often missing from such discussions, aside from the fact of their inherent complexity, particularly in terms of calculating changes in entropy and exactly what is meant by energy (Cooper and Klymkowsky, 2013) is that many students enter biology classes without a robust understanding of enthalpy, entropy, or free energy (Carson and Watson, 2002). Presenting students with a molecular collision, kinetic theory-based mechanism for the dissociation of molecular interactions, may help them better understand both the dynamics and specificity of molecular interactions. Through this approach we can gage the strength of an interaction (the sum of the forces stabilizing the interaction) based on the amount of energy (derived from collisions with other molecules) needed to disrupt it . The implication of student responses to relevant Biology Concepts Instrument (BCI) questions and beSocratic activities, as well as a number of studies in chemistry, is that few students consider the kinetic/vibrational energy delivered through collisions with other molecules (a function of temperature), as key to explaining why interactions break (see Carson and Watson, 2002 and references therein). Although this paper is 20 year old, there is little or no evidence that the situation has improved. Moreover, there is evidence that the conventional focus on mathematics-centered, free energy calculations in the absence of conceptual understanding may serve as an unnecessary barrier to the inclusion of a more diverse, socioeconomically, population of students (Ralph et al., 2022; Stowe and Cooper, 2019).
The lac operon as a context for introducing stochasticity: Studies of the E. coli lac operon hold an iconic place in the history of molecular biology and are often found in introductory courses, although typically presented in a deterministic context. The mutational analysis of the lac operon helped define key elements involved in gene regulation (Jacob and Monod, 1961; Monod et al., 1963). Booth et al (2022) use the lac operon as the context for their “modeling and simulation lesson”, Advanced Concepts in Regulation of the Lac Operon. Given its inherently stochastic regulation (Choi et al., 2008; Elowitz et al., 2002; Novick and Weiner, 1957; Vilar et al., 2003), the lac operon is a good place to start introducing students to stochastic processes. In this light, it is worth nothing that Booth et al describes the behavior of the lac operon as “leaky”, which would seem to imply a low, but continuous level of expression, much as a leaky faucet continues to drip. As this is a peer-reviewed lesson, it seems likely that it reflects a widely held understanding of how stochastic processes are introduced to, and understood by students and instructors.
E. coli cells respond to the presence of lactose in growth media in a biphasic manner, termed diauxie, due to “the inhibitory action of certain sugars, such as glucose, on adaptive enzymes (meaning an enzyme that appears only in the presence of its substrate)” (Blaiseau and Holmes, 2021). When these (preferred) sugars are depleted from the media, growth slows. If lactose is present, however, growth will resume following a delay associated with the expression of the proteins encoded by the operon that enable the cell to import and metabolize lactose. Although the term homeostatic is used repeatedly by Booth et al (2022), the lac operon is part of an adaptive, rather than a homeostatic, system. In the absence of glucose, cyclic AMP (cAMP) levels in the cell rise. cAMP binds to and activates the catabolite activator protein (CAP), encoded for by the crp gene. Activation of CAP leads to the altered expression of a number of target genes, whose products are involved in adaption to the stress associated with the absence of common and preferred metabolites. cAMP-activated CAP acts as both a transcription repressor and activator, “and has been shown to regulate hundreds of genes in the E. coli genome, earning it the status of “global” or “master” regulator” (Frendorf et al., 2019)(FIG. ↑). It is involved in the adaptation to environmental factors, rather than maintaining the cell in a particular state (homeostasis).
The lac operon is a classic polycistronic bacterial gene, encoding three distinct polypeptides: lacZ (β-galactosidase), lacY (β-galactoside permease), and lacA (galactoside acetyltransferase) . When glucose or other preferred energy sources are present, expression of the lac operon is blocked by the inactivity of CAP. The CAP protein is a homodimer and its binding to DNA is regulated by the binding of the allosteric effector cAMP. cAMP is generated from ATP by the enzyme adenylate cyclase, encoded by the cya gene. In the absence of glucose the enyzme IIAglc, encoded by the crr gene, is phosphorylated and acts to activate adenylate cyclase (Krin et al., 2002). As cAMP levels increase, cAMP binds to the CAP protein, leading to a dramatic change in its structure (↑), such that the protein’s DNA binding domain becomes available to interact with promoter sequences (↑)(Sharma et al., 2009).
Binding of activated (cAMP-bound) CAP is not, by itself sufficient to activate expression of the lac operon because of the presence of the constitutively expressed lac repressor protein, encoded for by the lacI gene. The active repressor is a tetramer, present at very low levels (~10 molecules) per cell. The lac operon contains three repressor (or “operator”) binding sites. In the absence of lactose, but in the presence of cAMP-activated CAP, the operon is expressed in discrete “bursts” (Novick and Weiner, 1957; Vilar et al., 2003). Choi et al (2008) found that these burst come in two types, short and long, with the size of the burst referring to the number of mRNA molecules synthesized (↑). The difference between burst sizes arises from the length of time that the operon’s repressor binding sites are unoccupied by repressor. The tetravalent repressor protein can bind to two operator sites at the same time. When released from one site, polymerase binding and initiation produces a small number of mRNA molecules; persistent binding to the second site means that the repressor concentration remains locally high, favoring rapid rebinding to the operator and the cessation of transcription (RNA synthesis). When the repressor releases from both operator sites simultaneously (a rarer event), it is free to diffuse away and interact (non-specifically, i.e. with low affinity) with other DNA sites in the cell, leaving the lac operator sites unoccupied for a longer period of time. The number of such non-specific binding sites greatly exceeds the number (three) of specific binding sites in the operon. The result is the synthesis of a larger “burst” (number) of mRNA molecules. The average length of time that the operator sites remain unoccupied is a function of the small number of repressor molecules present and the repressor’s low but measurable non-sequence specific binding to DNA.
The expression of the lac operon leads to the appearance of β-galactosidase and β-galactoside permease. An integral membrane protein, β-galactoside permease enables extracellular lactose to enter the cell while cytoplasmic β-galactosidase catalyzes its breakdown and the generation of allolactone, which binds to the lac repressor protein, inhibiting its binding to operator sites, and so removing repression of transcription. In the absence of lactose, there are few if any of the proteins (β-galactosidase and β-galactoside permease) needed to activate the expression of the lac operon, so the obvious question is how, when lactose does appear in the extracellular media, does the lac operon turn on? Booth et al and the Wikipedia entry on the lac operon (accessed 29 June 2022) describe the turn on of the lac operon as “leaky” (see above) . The molecular modeling studies of Vilar et al (2003) and Choi et al (2008)(which, together with Novick and Weiner, are not cited by Booth et al) indicate that the system displays distinct threshold and maintenance concentrations of lactose needed for stable lac gene expression. The term “threshold” does not occur in the Booth et al article. More importantly, when cultures are examined at the single cell level, what is observed is not a uniform increase in lac expression in all cells, as might be expected in the context of leaky expression, but more sporadic (noisy) behaviors. Increasing numbers of cells are “full on” in terms of lac operon expression over time when cultured in lactose concentrations above the operon’s activation threshold. This illustrates the distinctly different implications of a leaky versus a stochastic process in terms of their impacts on gene expression. While a leak is a macroscopic metaphor that produces a continuous, dependable, regular flow (drips), the occurrence of “bursts” of gene expression implies a stochastic process.
As the roles and effects of stochastic processes in biological systems becomes increasingly apparent and impactful, e.g. in the prediction of phenotypes from genotypes (Karavani et al., 2019; Mostafavi et al., 2020), it becomes increasingly important that students understand the un-predictable, that is stochastic, aspects of biological systems. As exampled, revealed particularly through the application of single cell RNA sequencing methods, the variation in gene expression has become increasing appreciated as an important influence on organism development and behaviors. For example, in diploid eukaryotic cells is now apparent that in many cells, and for many genes, only one of the two alleles present is expressed; such “monoallelic” expression can impact a range of processes (Gendrel et al., 2014). While stochastic processes are directly relevant to biological systems, they are often not well convey through chemistry courses (Williams et al., 2015) or effectively integrated into, and built upon in molecular (and other) biology curricula.
Summary: There are a number of ways that do not involve complex mathematical (or chemical energy and entropy) calculations through which introductory level biology students can be introduced into the stochastic features of biological systems, including the complex relationship between genotype and phenotype  These approaches can help students appreciate (and be immunized against) the increasingly popular (apparently) illusion of genetic determinism, as illustrated by Harden’s Genetic Lottery (Harden, 2021), reviewed by Feldman & Riskin (2022) and Coop & Przeworski (2022).
It may also help make sense of discussions of whether human (and other organisms) have “free will”. Clearly the situation is complex. From a scientific perspective we are analyzing systems without recourse to non-natural processes. At the same time, “Humans typically experience freely selecting between alternative courses of action” (Maoz et al., 2019). It seems possible that recognizing the intrinsically unpredictable nature of many biological processes (including those of the central nervous system) may lead us to conclude that whether or not free will exists may seen as a non-scientific question, unanswerable by science alone approaches.
 For this discussion I will ignore entropy, a factor that figures in whether a particular reaction in favorable or unfavorable, that is whether, and the extent to which it occurs.
 The same low affinity binding to DNA allows proteins to undergo 1-dimensional diffusion, a process that can greatly speed up the time it takes for a DNA binding protein to “find” a high affinity binding site (Stanford et al., 2000; von Hippel and Berg, 1989).
 Similar molecular collisions are relevant to covalent (chemical) reactions, in that they supply the energy to overcome the activation energy barrier.
 The function(s) of the lacA polypeptide remain unclear but “may assist cellular detoxification by acetylating nonmetabolizable pyranosides, thereby preventing their reentry into the cell” (https://www.rcsb.org/structure/1krv)
 In the context of genetics and evolutionary mechanisms, the process of genetic drift provides a useful context within which to introduce these ideas; students can be directed to the predictability of the system when using different population sizes (see (https://youtu.be/B5M_C8gBvYo). Recently we have discovered a new genetic drift applet here.
Thanks to Melanie Cooper and Nick Galati for taking a look and Chhavinder Singh for getting it started.
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