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+Results {#results .unnumbered}
+=======
+
+Characterizing Transcription Factor Induction using the Monod-Wyman-Changeux (MWC) Model {#characterizing-transcription-factor-induction-using-the-monod-wyman-changeux-mwc-model .unnumbered}
+----------------------------------------------------------------------------------------
+
+We begin by considering a simple repression genetic architecture in
+which the binding of an allosteric repressor occludes the binding of RNA
+polymerase (RNAP) to the DNA [@Ackers1982; @Buchler2003]. When an
+effector (hereafter referred to as an "inducer\" for the case of
+induction) binds to the repressor, it shifts the repressor's allosteric
+equilibrium towards the inactive state as specified by the MWC model
+[@MONOD1965]. This causes the repressor to bind more weakly to the
+operator, which increases gene expression. Simple repression motifs in
+the absence of inducer have been previously characterized by an
+equilibrium model where the probability of each state of repressor and
+RNAP promoter occupancy is dictated by the Boltzmann distribution
+[@Ackers1982; @Buchler2003; @Vilar2003; @Bintu2005a; @Garcia2011; @Brewster2014]
+(we note that non-equilibrium models of simple repression have been
+shown to have the same functional form that we derive below
+[@Phillips2015a]). We extend these models to consider allostery by
+accounting for the equilibrium state of the repressor through the MWC
+model.
+
+Thermodynamic models of gene expression begin by enumerating all
+possible states of the promoter and their corresponding statistical
+weights. As shown in , the promoter can either be empty, occupied by
+RNAP, or occupied by either an active or inactive repressor. The
+probability of binding to the promoter will be affected by the protein
+copy number, which we denote as $P$ for RNAP, $R_{A}$ for active
+repressor, and $R_{I}$ for inactive repressor. We note that repressors
+fluctuate between the active and inactive conformation in thermodynamic
+equilibrium, such that $R_{A}$ and $R_{I}$ will remain constant for a
+given inducer concentration [@MONOD1965]. We assign the repressor a
+different DNA binding affinity in the active and inactive state. In
+addition to the specific binding sites at the promoter, we assume that
+there are $N_{NS}$ non-specific binding sites elsewhere (i.e. on parts
+of the genome outside the simple repression architecture) where the RNAP
+or the repressor can bind. All specific binding energies are measured
+relative to the average non-specific binding energy. Thus,
+$\Delta\varepsilon_{P}$ represents the energy difference between the
+specific and non-specific binding for RNAP to the DNA. Likewise,
+$\Delta\varepsilon_{RA}$ and $\Delta\varepsilon_{RI}$ represent the
+difference in specific and non-specific binding energies for repressor
+in the active or inactive state, respectively.
+
+![**States and weights for the simple repression motif.** RNAP (light
+blue) and a repressor compete for binding to a promoter of interest.
+There are $R_A$ repressors in the active state (red) and $R_I$
+repressors in the inactive state (purple). The difference in energy
+between a repressor bound to the promoter of interest versus another
+non-specific site elsewhere on the DNA equals $\Delta\varepsilon_{RA}$
+in the active state and $\Delta\varepsilon_{RI}$ in the inactive state;
+the $P$ RNAP have a corresponding energy difference
+$\Delta\varepsilon_{P}$ relative to non-specific binding on the DNA.
+$N_{NS}$ represents the number of non-specific binding sites for both
+RNAP and repressor. A repressor has an active conformation (red, left
+column) and an inactive conformation (purple, right column), with the
+energy difference between these two states given by $\Delta
+    \varepsilon_{AI}$. The inducer (blue circle) at concentration $c$ is
+capable of binding to the repressor with dissociation constants $K_A$ in
+the active state and $K_I$ in the inactive state. The eight states for a
+dimer with $n=2$ inducer binding sites are shown along with the sums of
+the active and inactive
+states.](main_figs/fig2.pdf){#fig_polymerase_repressor_states}
+
+Thermodynamic models of transcription
+[@Ackers1982; @Buchler2003; @Vilar2003; @Bintu2005; @Bintu2005a; @Kuhlman2007; @Daber2011a; @Garcia2011; @Brewster2014; @Weinert2014]
+posit that gene expression is proportional to the probability that the
+RNAP is bound to the promoter $p_{\text{bound}}$, which is given by
+$$\label{eq_p_bound_definition}
+p_\text{bound}=\frac{\frac{P}{N_{NS}}e^{-\beta \Delta\varepsilon_{P}}}{1+\frac{R_A}{N_{NS}}e^{-\beta \Delta\varepsilon_{RA}}+\frac{R_I}{N_{NS}}e^{-\beta \Delta\varepsilon_{RI}}+\frac{P}{N_{NS}}e^{-\beta\Delta\varepsilon_{P}}},$$
+with $\beta = \frac{1}{k_BT}$ where $k_B$ is the Boltzmann constant and
+$T$ is the temperature of the system. As $k_BT$ is the natural unit of
+energy at the molecular length scale, we treat the products
+$\beta \Delta\varepsilon_{j}$ as single parameters within our model.
+Measuring $p_{\text{bound}}$ directly is fraught with experimental
+difficulties, as determining the exact proportionality between
+expression and $p_{\text{bound}}$ is not straightforward. Instead, we
+measure the fold-change in gene expression due to the presence of the
+repressor. We define fold-change as the ratio of gene expression in the
+presence of repressor relative to expression in the absence of repressor
+(i.e. constitutive expression), namely,
+$$\label{eq_fold_change_definition}
+\foldchange \equiv \frac{p_\text{bound}(R > 0)}{p_\text{bound}(R = 0)}.$$
+We can simplify this expression using two well-justified approximations:
+(1) $\frac{P}{N_{NS}}e^{-\beta\Delta\varepsilon_{P}}\ll 1$ implying that
+the RNAP binds weakly to the promoter ($N_{NS} = 4.6 \times 10^6$,
+$P \approx 10^3$ [@Klumpp2008],
+$\Delta\varepsilon_{P} \approx -2 \,\, \text{to} \, -5~k_B
+T$ [@Brewster2012], so that
+$\frac{P}{N_{NS}}e^{-\beta\Delta\varepsilon_{P}}
+\approx 0.01$) and (2)
+$\frac{R_I}{N_{NS}}e^{-\beta \Delta\varepsilon_{RI}} \ll
+1 + \frac{R_A}{N_{NS}} e^{-\beta\Delta\varepsilon_{RA}}$ which reflects
+our assumption that the inactive repressor binds weakly to the promoter
+of interest. Using these approximations, the fold-change reduces to the
+form $$\label{eq_fold_change_approx}
+\foldchange \approx \left(1+\frac{R_A}{N_{NS}}e^{-\beta \Delta\varepsilon_{RA}}\right)^{-1} \equiv \left( 1+p_A(c) \frac{R}{N_{NS}}e^{-\beta
+    \Delta\varepsilon_{RA}} \right)^{-1},$$ where in the last step we
+have introduced the fraction $p_A(c)$ of repressors in the active state
+given a concentration $c$ of inducer, such that $R_A(c)=p_A(c) R$. Since
+inducer binding shifts the repressors from the active to the inactive
+state, $p_A(c)$ grows smaller as $c$ increases [@Marzen2013].
+
+We use the MWC model to compute the probability $p_A(c)$ that a
+repressor with $n$ inducer binding sites will be active. The value of
+$p_A(c)$ is given by the sum of the weights of the active repressor
+states divided by the sum of the weights of all possible repressor
+states (see ), namely, $$\label{eq_p_active}
+p_A(c)=\frac{\left(1+\frac{c}{K_A}\right)^n}{\left(1+\frac{c}{K_A}\right)^n+e^{-\beta \Delta \varepsilon_{AI} }\left(1+\frac{c}{K_I}\right)^n},$$
+where $K_A$ and $K_I$ represent the dissociation constant between the
+inducer and repressor in the active and inactive states, respectively,
+and $\Delta
+\varepsilon_{AI} = \varepsilon_{I} - \varepsilon_{A}$ is the free energy
+difference between a repressor in the inactive and active state (the
+quantity $e^{-\Delta \varepsilon_{AI}}$ is sometimes denoted by $L$
+[@MONOD1965; @Marzen2013] or $K_{\text{RR}*}$ [@Daber2011a]). In this
+equation, $\frac{c}{K_A}$ and $\frac{c}{K_I}$ represent the change in
+free energy when an inducer binds to a repressor in the active or
+inactive state, respectively, while
+$e^{-\beta \Delta \varepsilon_{AI} }$ represents the change in free
+energy when the repressor changes from the active to inactive state in
+the absence of inducer. Thus, a repressor which favors the active state
+in the absence of inducer ($\Delta \varepsilon_{AI} > 0$) will be driven
+towards the inactive state upon inducer binding when $K_I < K_A$. The
+specific case of a repressor dimer with $n=2$ inducer binding sites is
+shown in .
+
+Substituting $p_A(c)$ from into yields the general formula for induction
+of a simple repression regulatory architecture [@Phillips2015a], namely,
+$$\label{eq_fold_change_full}
+\foldchange = \left(
+1+\frac{\left(1+\frac{c}{K_A}\right)^n}{\left(1+\frac{c}{K_A}\right)^n+e^{-\beta \Delta \varepsilon_{AI} }\left(1+\frac{c}{K_I}\right)^n}\frac{R}{N_{NS}}e^{-\beta \Delta\varepsilon_{RA}} \right)^{-1}.$$
+While we have used the specific case of simple repression with induction
+to craft this model, the same mathematics describe the case of
+corepression in which binding of an allosteric effector stabilizes the
+active state of the repressor and decreases gene expression (see ).
+Interestingly, we shift from induction (governed by $K_I < K_A$) to
+corepression ($K_I > K_A$) as the ligand transitions from preferentially
+binding to the inactive repressor state to stabilizing the active state.
+Furthermore, this general approach can be used to describe a variety of
+other motifs such as activation, multiple repressor binding sites, and
+combinations of activator and repressor binding sites
+[@Bintu2005; @Brewster2014; @Weinert2014].
+
+The formula presented in enables us to make precise quantitative
+statements about induction profiles. Motivated by the broad range of
+predictions implied by , we designed a series of experiments using the
+*lac* system in *E. coli* to tune the control parameters for a simple
+repression genetic circuit. As discussed in , previous studies from our
+lab have provided well-characterized values for many of the parameters
+in our experimental system, leaving only the values of the the MWC
+parameters ($K_A$, $K_I$, and $\Delta \varepsilon_{AI}$) to be
+determined. We note that while previous studies have obtained values for
+$K_A$, $K_I$, and $L=e^{-\beta \Delta \varepsilon_{AI}}$
+[@OGorman1980; @Daber2011a], they were either based upon biochemical
+experiments or *in vivo* conditions involving poorly characterized
+transcription factor copy numbers and gene copy numbers. These
+differences relative to our experimental conditions and fitting
+techniques led us to believe that it was important to perform our own
+analysis of these parameters. After inferring these three MWC parameters
+(see , Section "" for details regarding the inference of $\Delta
+\varepsilon_{AI}$, which was fitted separately from $K_A$ and $K_I$), we
+were able to predict the input/output response of the system under a
+broad range of experimental conditions. For example, this framework can
+predict the response of the system at different repressor copy numbers
+$R$, repressor-operator affinities $\Delta\varepsilon_{RA}$, inducer
+concentrations $c$, and gene copy numbers (see Appendix
+[\[AppendixFugacity\]](#AppendixFugacity){reference-type="ref"
+reference="AppendixFugacity"}).