Introduction

Modern molecular biology and genomics methods allow investigators to readily assay protein and mRNA expression levels and identify interactions between proteins, RNA, and other cellular components. Leveraging these data to understand the functional significance of interactions on gene expression is a key challenge in computational biology. The recent application of thermodynamic models to gene regulation is an exciting development, as each model reflects a specific, testable hypothesis regarding the physical architecture of the underlying molecular system [1]–[4]. Such models will help transform parts lists, which describe the components of regulatory systems, into models that integrate the interactions between components into accurate predictions of gene expression.

Though a gene is regulated at every step of transcription and translation, a large component of regulation operates at the level of the promoter [5]. Transcription factors bind to specific sequences and modulate transcription by influencing exposure of the polymerase binding site (chromatin remodelers [6]), chemically modifying DNA (methyltransferases [7]), and recruiting factors necessary for, or inhibitory of, polymerase complex formation [8]–[11]. These mechanisms constitute the cis-regulatory component of a gene's regulation. Understanding gene expression under a variety of cellular contexts requires a well-grounded theory for modeling cis-regulatory function.

Here we show the biochemical derivation of the thermodynamic framework used to model promoter activity. The derivation is presented in a form that can be readily coded in any programming language, allowing readers to develop cis-regulatory models specific to their own systems. We suggest how this approach can be leveraged to model virtually any cis-regulatory mechanism. We also demonstrate that modular Michaelis-like functions, another commonly used framework, are a specific subset of the thermodynamic model framework. To demonstrate this, we recast Michaelis-like functions as thermodynamic models, highlighting the physical assumptions necessary for interconversion. Viewing Michaelis functions in this form reinforces the principles of the thermodynamic framework, emphasizes the relationship between these approaches, and provides criteria for an investigator to choose an appropriate cis-regulatory model. The flexibility of the thermodynamic framework, along with its grounding in basic physical principles, makes it a powerful tool for unraveling the molecular interactions that underlie combinatorial cis-regulation.

cis-Regulatory Functions in Models of Transcription

A model of cis-regulation relates the activities of various transcription factors acting on gene to the concentration of mRNA produced by transcription of . To illustrate how cis-regulation contributes to expression, a general model of transcription, derived from physicochemical principles, is presented [12]. The concentration of any particular mRNA species, denoted as , changes over time according to the first order rate equation (Equation 1),(1)where is the degradation rate constant of the RNA transcript in units of inverse time, is the concentration of RNA transcripts generated per unit time when RNA polymerase is committed to transcription, and is the probability that a DNA template is committed to transcription. The quantity is the cis-regulatory term. The function integrates elements of the cellular milieu that affect transcription and outputs the probability that a single DNA molecule is committed to transcription. In a clonal population of cells at equilibrium, equals the fraction of those cells currently committed to transcription. Although there are numerous other discrete, continuous, and stochastic models of gene expression [12], [13], every model must contain some form of the cis-regulatory function .

Anatomy of a cis-Regulatory Function

Importantly, there is no hypothesis-independent form of the cis-regulatory function; any choice of is a hypothesis about the mechanism of gene M's transcription. Even if we choose to be a constant, we imply that gene M is constitutively transcribed at a rate unaffected by any cellular or environmental factors. There is no single correct formulation of ; investigators must formulate based on aspects of their system they know to be true, and on hypotheses they hold regarding the important features of their system.

Two approaches have been used to formulate cis-regulatory expressions: (1) Michaelis-like functions and (2) thermodynamic state ensemble models. Michaelis-like functions have been most frequently employed to study large gene regulatory networks [12], [14]–[18], owing to their modular design and limited number of free parameters. State ensemble approaches have been the model of choice for characterizing a few specific genes in great detail [1]–[3], [19]–[23]. By manipulating these two approaches analytically, we will show that the Michaelis-like models are a specific case in the thermodynamic framework, thus uniting these two approaches and also illuminating some of the subtleties of the Michaelis-like models.

Thermodynamic State Ensemble Approach

The “thermodynamic model” is a framework for constructing a set of states that collectively encode the rules of transcription for a particular promoter. Each state represents a particular number and arrangement of transcription factors bound to a DNA template. Some states are transcriptionally active while others remain transcriptionally dormant. All states occur at some point, but their contributions to transcription are weighted by their relative stabilities. In this formulation, is the probability of a promoter being in a transcriptionally active state. The essence of the thermodynamic framework is to compute the ratio of transcriptionally active promoter states to the sum of all states, active and inert. This ratio depends on variables including the exact cis-regulatory sequences present in the promoter, the concentrations of proteins that bind these sequences, and the affinities of the protein–DNA and protein–protein interactions that occur on the DNA. The thermodynamic formalism provides a flexible framework in which to account for molecular interactions that control cis-regulation.

Generating a model requires writing down all possible states a promoter may adopt in the form of a binding polynomial, [2], [24]. To illustrate the binding polynomial, we first consider the simple case of a basal promoter (Figure 1). Defining what is meant by basal transcription is central to the development of a model framework because activation and repression reflect changes relative to the basal level of transcription. Here, a basal promoter is a DNA template that contains a binding site for RNA polymerase (RNAP) and no other cis-regulatory sequences. Basal transcription is defined as the level of transcript produced by RNAP in the absence of regulation by transcription factors. Note that RNAP serves as a proxy for the rate limiting step of transcription, whether that be the recruitment of a particular co-factor to the RNAP holoenzyme, or binding of a specific transcription factor. The binding polynomial for the basal promoter is given in Equation 2. For reference, Box 1 contains definitions relevant for the derivation.(2)

Box 1. Definitions

Basal promoter: a promoter in which the sequence codes only for binding of RNA Polymerase.

Basal transcription: the RNA expression level attained by driving a gene with a basal promoter.

Binding polynomial: a mathematical expression calculated by summing the concencentrations of all states of a particular macromolecule (in this case, DNA).

cis-regulatory site: a specific sequence recognized and capable of being bound by a transcription factor or polymerase.

Cooperativity: a binding modality in which the occupancy of a state where two or more factors are bound to DNA is not equal to the occupancy expected if each factor were to bind independently. In terms of energy, which is additive: if factors A and B bind independently, then the energy of the state where both are bound is . If the actual energy, , is not equal to , then there exists some interaction between A and B such that , where is the cooperative energy term of this interaction. reflects positive cooperativity, or an adhesive interaction between A and B, while indicates negative cooperativity, or a repulsive interaction between A and B.

Equilibrium: when either the time average or population average of all relevant concentrations of biochemical species are not changing.

Equilibrium binding constant: in an interaction between biochemical species, the equilibrium binding constant is the equilibrium concentration of the product divided by the product of the equilibrium concentrations of all reactants. In a cellular equilibrium, as defined above, these are actually “apparent” equilibrium constants.

Macroscopic binding constant: an analog to the total energy required to bind all species in a state from an unbound state. If only two species are interacting, the macroscopic binding constant equals the equilibrium binding constant. In all other situations, the macroscopic binding constant is equal to the product of all equilbrium binding constants neccesary to convert two or more species from the free to bound state.

Partition function: the binding polynomial normalized by the concentration of a reference state (in this case, free DNA, [DNA]). The probability of observing a particular state may be calculated by dividing a state contained in the partition function by the total partition function.

Promoter: the sequence adjacent to the coding region of a gene containing RNA polymerase binding sequence and any other cis-regulatory binding sequences.

RNA polymerase (RNAP): the biochemical machinery needed for basal expression. In the context of an experiment it can also be thought of as the aspect of the experiment not being altered.

State: a specific arrangement of transcription factors and/or RNAP bound to DNA.

Transcription factor (TF): any protein capable of both binding a promoter and affecting expression by influencing the polymerase's ability to bind DNA and/or transcribe.

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Figure 1. States of a basal promoter.

A basal promoter is composed of two states, one where DNA is bound with RNAP and is transcriptionally active, and another where DNA is free and inactive.

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This DNA-centric binding polynomial enumerates the two mutually exclusive states of a basal promoter; either DNA is free or bound by RNAP. From P, we can determine the fraction of DNA bound with RNAP, . At equilibrium, this is the concentration of bound DNA divided by the total concentration of DNA, (Equation 3).(3)

Equation 3 is a basic cis-regulatory function () for a basal promoter where concentration of bound polymerase is the only determinant of transcription. The primary assumption of the thermodynamic model, originally introduced by Shea and Ackers [2], is that binding of the polymerase complex is the key event leading to production of a transcript, and that other proteins affecting expression operate by recruiting or inhibiting the polymerase complex. Thus, the fraction of polymerase complex bound is directly proportional to the number of transcripts produced.

We can reformulate Equation 3 in terms of its component free species and their association constants. The apparent association constant for the binding of RNAP to DNA is .(4)

This simplification presumes that the concentrations of all cofactors required to form the RNAP complex are invariant. Solving for in Equation 4 and substituting it into Equation 3 results in Equation 5. The [DNA] factor is present in all terms and is subsequently dropped.(5)

The denominator of the right-hand-side of Equation 5 is called the biochemical partition function () for our system, and is exactly equal to . Dividing any state or sum of states listed in by results in the probability of observing . The reference state, where DNA is unbound, is represented by the in ; consequently, the probability of finding DNA unbound is . Equation 5 is perhaps the most intuitive form of the thermodynamic model as it shows clearly the origin of each state. Each state is a summand, and the elements within a summand serve as a sort of recipe for how to make that state. For example, the numerator term in Equation 5 can be read as “binding of RNAP to DNA has an equilibrium binding constant of ”. This form is particularly useful because it expresses the model in terms that are more accessible to experiment. While in vivo binding constants and concentrations of free species are difficult to determine, reasonable proxies for these quantities can often be obtained experimentally [20], [25].

Several other manipulations of these equations are employed in the literature. In addition to writing states in terms of free species concentrations, Shea and Ackers substitute association constants with Boltzmann weights [2]. Others course-grain the product of association constants and concentrations into single parameters [1], [20], reducing computational complexity. These manipulations to the free species form described above are discussed in the supplement (Text S1, Alternate cis-Regulatory Function Forms).

Building a Thermodynamic State Ensemble Model

The framework suggested by Shea and Ackers allows great flexibility for assembling models to reflect a wide variety of mechanisms and behavior.

For any particular system, construction of a thermodynamic cis-regulatory function requires three components: (1) a list of all states, (2) the macroscopic equilibrium constant for each state, and (3) a boolean for whether each state is capable of transcribing or not. We will illustrate the formalism using a promoter with a single binding site for a transcription factor and a binding site for RNA polymerase (Figure 2). We have introduced linear algebra to showcase the building blocks of the framework while demonstrating how to code a specific model. We will encode the list of states in a position matrix L, which we will then convert into the functional state vector s.States are written as a function of position with concentrations representing what can bind each position in each state. A “1” denotes nothing is bound in that particular position and state. The product of all terms in each state are used to generate the state vector . Unique states in may result in degenerate states in . This example with two sites requires two columns; other, more complicated systems with multiple sites are modeled by adding new columns.State 1, the first row of s, corresponds to the reference state where DNA has nothing bound. State 2 has RNAP bound by itself, state 3 has TF bound by itself, and state 4 has both TF and RNAP bound. Simply by writing these states we are already specifying the architecture of our system. For example, if RNAP were to require TF to be present before it binds, then state 2, where RNAP is bound by itself, would not exist and would not be included among the list of possible states.

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Figure 2. Thermodynamic state ensemble model example.

(A) Four states are allowed in this example, two where transcription is inactive (states 1 and 3) and two states where transcription is active (states 2 and 4). (B) The function is composed of the concentrations of transcriptionally active states summed in the numerator divided by the sum of the concentrations of all possible states.

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Figure 3. Graphical representations of thermodynamic cis-regulatory functions.

Proteins/complexes are represented as ovals, binding sites as rectangles. (A) Repressor-RNAP competition with activator release model, see Box 2. The ovals represent RNAP (blue), repressor (red), and activator (green). Note that the repressor and RNAP binding sites are overlapping to reflect competition between sites. (B) Sequential binding model, see Box 2. The ovals correspond to RNAP (blue), activator A1 (dark green), and activator A2 (light green).

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Vector b contains the macroscopic equilibrium constants for each state ; as such, it will be the same length as s. Macroscopic equilibrium constants reflect the energy difference between that state and the reference (unbound) state, and comprise the product of the stepwise equilibrium constants in the intervening steps.The macroscopic binding constant for the reference state is always , representing free [DNA]; thus, (see Equation 5).

Lastly, we define vector t, which contains boolean values for whether a state is capable of transcribing. For example, we might assume that transcription occurs any time RNAP is bound, as assumed by Shea et al. [2]. Changes in the values of the t vector can accommodate situations where this assumption proves to be false.The cis-regulatory function is the sum of states capable of transcribing divided by the sum of all possible states. The denominator of is the biochemical partition function , which can be expressed as the dot product of the transpose of s with b. Taking the pairwise element product of b and t results in vector .Then the dot product of the transpose of s with yields the sum of transcriptionally active states.

Generally, for any architecture L written as a vector s that contains the concentrations of all relevant species, vector b containing the macroscopic equilibrium constants for each state, and vector t relating whether a state is capable of transcribing, the cis-regulatory function is:(6)

For our example,(7)

The terms reflect the most general treatment of this system, but can also be written as functions of their stepwise equilibrium constants. In the scenario above, the macroscopic equilibrium constant is exactly equal to the equilibrium constant for binding of RNAP to DNA, denoted as . Similarly, is exactly equal to the equilibrium constant for association of TF to DNA, denoted as . can be a number of different expressions depending on the system. For example, setting implies completely independent non-cooperative binding of TF and polymerase; that is, binding of one does not influence binding of the other. In this case,(8)

Completely independent binding of transcription factor and RNAP implies that the presence of TF has no bearing on the probability of RNAP being bound, a scenario reflected in the equation by factoring and canceling out the TF terms, revealing our basal promoter function:(9)

In order for the TF to affect binding of the polymerase we must introduce a cooperative binding term . Then and as before, but . The new no longer simplifies to the trivial case.(10)

The cooperative term reflects the energy associated with the interaction of the polymerase with the TF. If  = 1, we recover the case above where binding of the TF has no bearing on the binding of the polymerase. If , the TF acts like an activator; if the TF is bound, it stabilizes the state where polymerase is also bound. Conversely, if , the TF acts like a repressor; TF binding decreases the stability of the state where polymerase is also bound. See Box 2 for additional examples.

Box 2. Example Model Implementations

Repressor-RNAP competition with activator release model. In this example, RNAP is blocked from binding by a repressor, R, bound to the same site. Activator A binds to an adjacent site and, through negative cooperativity, ejects the repressor from DNA, thus freeing the RNAP binding site. Neither activator nor repressor interacts directly with polymerase. The b vector element contains the cooperative term , which needs to be less than one for the activator and repressor to repel each other. Note that the repressor and polymerase need not have exactly the same binding site, as long as the presence of one excludes binding of the other (see Figure 3A). This general principle of allowing or disallowing states can be expanded to account for promoters with overlapping binding sites [2], [30].

Sequential binding model. Several known regulatory mechanisms involve sequential binding of activators. In this model, activator A1 permits binding of activator A2, which in turn recruits polymerase. This model subscribes to Michaelis-like model logic where all activators are required for binding, but the sequential aspect can only be captured using a state ensemble approach. We have engineered sequential binding by disallowing activator A2 to bind without activator A1, and disallowing polymerase to bind without activator A2 (see Figure 3B).

When constructing a thermodynamic model, an investigator explicitly selects the number of binding sites, decides which proteins bind to each site, determines whether a state is transcriptionally active, and assigns cooperative interactions between binding partners. The resulting cis-regulatory function's numerator contains transcriptionally active states while the denominator encodes all binding states. These traits confer considerable versatility to the thermodynamic modeling approach, making it a powerful tool for exploring cis-regulatory control of gene expression.

Modular Michaelis Functions as Partition Functions

What is the physical interpretation of the Michaelis function architecture? By converting the Michaelis model formulations above (Equation 13) into thermodynamic functions we will reveal assumptions underlying Michaelis-like models that are not obvious in their original formulation. The steps involved in converting one model to the other also highlight the similarity between these models, and demonstrate that the Michaelis formulation is simply a thermodynamic model with specific cis-regulatory rules.

We can reconcile the thermodynamic model with the Michaelis framework by treating polymerase as an activator. Since polymerase is required for transcription, we incorporate the basal thermodynamic function (Equation 5) into the Michaelis-like formulation, Equation 13, as an activator function (Equations 16 and 17).(16)(17)

Comparing Equations 16 and 17 with Equation 13 illustrates that the original Michaelis-like function requires the assumption that , such that the activator function for polymerase . In other words, the Michaelis approach assumes that the polymerase site is saturated, or always occupied.

The asymmetry in the way Michaelis functions treat RNAP becomes clear when they are recast in the thermodynamic framework. Consider the following Michaelis-like models: activator only (Equation 18), repressor only (Equation 19), and one activator and one repressor (Equation 20).(18)(19)(20)

Adding in the polymerase function as in Equation 16 and multiplying out the terms, we generate the following expressions.(21)(22)(23)

Comparing the resulting models shows that the Michaelis-like activator and repressor functions treat the state in which only RNAP is bound very differently. A one activator promoter (Equation 21) transcribes only when both RNAP and activator are present, as represented by the sole numerator term. The presence of the term indicates that polymerase can bind DNA without activator, but because this state is only in the denominator, binding does not result in transcription. In contrast, the repressor model (Equation 22) only transcribes when RNAP is bound and repressor R is not bound, as reflected by the state being the sole numerator term. Thus, the presence of repressor inhibits expression absolutely. In order to appropriately model their own systems with Michaelis-like functions, investigators should be aware of the different interpretation of the RNAP-only state in the activator and repressor functions.

Recasting the original Michaelis-like functions as a thermodynamic ensemble model also highlights its implicit AND-circuitry. The inclusion of both an activator and repressor in the Michaelis-like formulation results in a model with only a single term in the numerator (Equation 23). This means that transcripts are generated only when activator is bound and repressor is not bound. Higher numbers of transcription factors continue these patterns. For example, a two or more activator model requires that all activators are bound for transcription, and a two or more repressor model requires that none of the repressors are bound. In a mixed system with multiple activators and repressors, the trend set by the one activator and one repressor model (Equation 23) prevails; transcripts are produced only when all activators accompany polymerase with no repressors present. Investigators must decide on the validity of this constraint when employing Michaelis-like functions.

The implicit AND logic associated with Michaelis-like functions leads to a seeming paradox. The more activators a promoter contains, the lower its expression. This is because the probability of having all activators bound at the same time decreases with the number of activator binding sites in a promoter. This seeming paradox and the general AND-circuitry associated with this formalism led some groups to produce an OR-logic function for activators (Equation 24) and repressors (Equation 25) [16], [17]:(24)(25)

The activator function involves addition rather than multiplication of individual transcription factor effects. Following the same steps outlined above, one can show that the OR-logic model here no longer produces zero expression when any single activator concentration (or affinity) goes to zero. However, if all activator concentrations are zero, transcription is abolished, implying that some activator (of either type) is required to produce transcripts.

To allow basal expression even in the absence of transcription factors, some groups [14], [16], [17] introduce an empirical basal leak term to the Michaelis function formulation. Leak functions can also be reformulated as thermodynamic models, revealing a similar set of implicit physical assumptions (see Text S1, Michaelis-like Functions with Basal Leak).

These are reasonable models provided that the mechanisms described appropriately reflect the logic of the system being modeled. Michaelis-like functions can be a simple and powerful framework for modeling many types of regulatory logic. The purpose of reformulating these models in the thermodynamic framework was to demonstrate that Michaelis-like functions are simply one type of thermodynamic model. The assumptions that underlie these particular models, which are easy to see in the thermodynamic framework, are likely to be valid for many, but not all types of cis-regulatory logic.

Some regulatory mechanisms require the use of the more general thermodynamic framework. For example, a repressor might function by directly blocking polymerase binding, so that simultaneous binding of polymerase and repressor does not occur [2]. Or, an activator might boost expression, but transcription continues even in the absence of activator [5]. Michaelis-like functions can be applied in these situations, but cannot distinguish between various mechanisms. Box 2 illustrates two examples of cis-regulatory architectures that can only be represented using the more general thermodynamic approach.

Hill Cooperativity in the Context of a cis-Regulatory Function

Cooperativity is a repulsion or attraction between proteins on the surface of DNA such that the sum of the free energies of proteins binding independently differs from the energy of the proteins binding together. We discussed cooperativity in the thermodynamic framework using Equation 10. Another commonly used method to capture cooperativity is the addition of Hill coefficients () to the Michaelis-like functions [14], [16], [18], [27]. For example, the modified one activator and one repressor models correspond to:(26)(27)

These functions are known as Hill functions [24]. Hill functions measure cooperativity by quantifying deviation from independent binding in a traditional biochemical binding titration experiment. Used as cis-regulatory functions, Goutsias and Kim point out that these functions imply extreme cooperativity; for two proteins, n = 2 implies that the proteins can only bind simultaneously, never independently [12]. We will show the origin of this assumption again using the thermodynamic formalism.

Discussion

Using expression-profiling methods, investigators routinely collect large quantities of gene expression data. A mature and robust quantitative framework would draw meaningful conclusions from these rich but complex datasets. Here we derived a thermodynamic state ensemble framework for capturing cis-regulatory architectures. Our intention here was to clarify the assumptions of the thermodynamic framework, to provide a step-by-step guide for constructing such a model, and to impart guidance in interpreting the physical meaning of the parameters of these models. Different investigators will collect different types and amounts of data, in turn requiring pre- and post-processing steps specific to their respective systems. This includes data filtering and fitting routines for parameter estimation that we could not address here and must be dealt with on a case-by-case basis. What we did attempt to address were the aspects of thermodynamic modeling that will be common to all investigators; namely the construction and interpretation of such models.

The flexibility of the thermodynamic formalism makes it simple to model different promoter architectures and regulatory mechanisms. Discrete promoter states determine the overall architecture of the model, with individual states constructed from the product of activities of DNA-bound molecules. The balance between productive and silent states determines the probability of transcription (), a term mathematically composed of a denominator comprising the sum of all states and a numerator containing the sum of transcriptionally active states. Selecting whether a state is transcriptionally active, and even whether a state exists at all, allows a large number of possible models to be constructed. With this versatility comes both a warning and a virtue; any architecture devised reflects a specific hypothesis about the physical system being modeled.

Michaelis-like models are simplified forms of the thermodynamic framework. Each type of Michaelis-like cis-regulatory function can be derived from the thermodynamic model framework by making a few key assumptions. Understanding these assumptions will help investigators to choose appropriate models for their systems. Michaelis models generally assume that polymerase is present in excess and that each transcription factor included acts at an independent site. Products of Michaelis-like functions represent the hypothesis that all activators, and no repressors, must be bound to initiate transcription. Sums of Michaelis functions correspond to situations in which at least one activator must be bound for transcription to occur (basal transcription is disallowed). A thermodynamic reformulation of Hill functions reflect a specific type of cooperativity in which either a site is free, or bound by proteins, for a Hill coefficient of . Alternatively, a Hill coefficient of can imply binding of an -mer to the promoter. These two situations imply two distinct interpretations for the parameters. Michaelis and Hill-like functions are valid simplifications of the thermodynamic framework. It is up to individual investigators to decide when the assumptions underlying these simplifications are appropriate.

In some cases, investigators must employ the more general form of the thermodynamic framework. For example, repressors might inhibit transcription by binding directly to the RNAP binding site, a mode of repression that cannot be specifically represented using the Michaelis formulation. Such a mechanism can be captured by a thermodynamic state ensemble model in which one disallows the state in which both RNAP and repressor are simultaneously bound (for examples, see Box 2). In general, it may be wise to first cast any system under study in the thermodynamic framework before simplifying to the corresponding Michaelis model so that the underlying assumptions about the system are clear.

With the exception of a few well-characterized systems like lac and the lysis-lysogeny operator of -bacteriophage, combinatorial cis-regulation of genes is not understood to the point where one can predict levels of transcription from the cis-regulatory content of a gene. The parts list of cis-regulatory components is growing rapidly; soon we will know the binding preferences of all transcription factors and their activating or repressing activities [28], [29]. Even with this catalog in hand, we will not understand gene regulation until we understand how the interactions between cis-regulatory components generate specific patterns of transcription. We are optimistic that the thoughtful application of state ensemble models will provide mechanistic insight into the physical interactions that underlie combinatorial cis-regulation.

Supporting Information

Acknowledgments

The authors thank members of the Cohen lab for critical reading of the manuscript.