Information storage and transmission is maximized in molecular computation processes.

Information and entropy are related concepts. In fact, entropy and information can be transformed into each other [Szilard (30)]. In biological systems, which are also physical systems, entropy and information are fungible. This leads us to a principle for entropy similar to our principle for information. We use the principle of MREP (Jaynes (12); Martyushev and Seleznev (31–33) as the second general principle for biological information processing.

In an open, far-from-equilibrium (nonequilibirum) steady state (NESS), the observed conformation of the steady state is the one that produces the maximum rate of entropy production given externally applied constraints of chemical, free-energy, and information fluxes [Jaynes (12)].

The maximization of entropy and information flux in the complete system of switches requires the flux to be variable at the switch level. The concentrations and fluxes within each switch adapt to maximize the total information and entropy fluxes. For this to occur, the switches must be flexible [Engh and Bossemeyer (34); Karshikoff et al. (35); Gerstein and Echols (36); Marsh et al. (37); Jacobs et al. (38); Sauer et al. (39); Bernadó et al. (40); Teilum et al. (41); Homans (42); Teilum et al. (43)]. The flexibility of proteins within the switches implies that reaction rates among protein conformations can be variable. Storage of molecular information requires that not all conformation states are flexible, however. Some reaction rates must be rigid for information storage to be stable. This switch ensemble has been described by Schrödinger as an aperiodic crystal [Schrodinger et al. (44)].

The sequestering of reactions into flexible and rigid categories is determined by the background free energy. In systems near equilibrium, the background energy is supplied by thermal fluctuations at temperature T . If free energy is injected into the system by an external process such as phosphorylation, then the random fluctuations in the neighborhood of the injection may be much higher than thermal fluctuations. The free-energy injected by an ATP is about 12   k c a l / m o l [(45, Sec. 15.2) Qian and Reluga (29)], which is to be compared with 0.6   k c a l / m o l for thermal fluctuations. Reactions with free-energy barriers smaller than the fluctuation energy are flexible, while reactions with barriers larger than the fluctuation energy are rigid.

It should be noted that MREP is consistent with the Second Law of Thermodynamics. The Second Law describes the final equilibrium state of a system. MREP describes how quickly that equilibrium state is approached.

An ensemble of switches is a collection of M switches, R T receptors, and L ligands in which the switches present ON and OFF states. Macroscopic quantities are calculated by taking an average of switch properties over the entire ensemble. An equivalent approach is the time-average picture in which a single switch or group of switches experiences all possible allowable states over time. Macroscopic quantities are calculated by taking a time-average of the switch trajectories through their state space. The two approaches are equivalent according to the Ergodic Hypothesis [Reif (25)]. Both pictures are useful for our purposes

An ACTIVE switch maintains a finite chemical flux [Qian (46)]. Consider first an ensemble of M ACTIVE switches that support a chemical flux J i for each switch i as illustrated in Figure 1F . Here, INACTIVE switches are defined as those switches that do not support a chemical flux. The number of ACTIVE switches in the ON state is M o , while the number in the OFF state is M − M o . A given configuration x w ∈ X ( M , M o ) of M o ON states among M ACTIVE switches is, after Boltzmann, called a complexion [CERCIGNANI (47)]. As an example, if the ON state is designated by one and the OFF state by zero, then the set of complexions for M = 4 switches with M 0 = 2 of the switches being ON is (1 1 0 0), (1 0 0 1), (1 0 1 0), (0 0 1 1), (0 1 1 0), and (0 1 0 1), for a total of six complexions. We assumed the switches are distinguishable. Each switch can be found in a well-defined location.

The amount of information H ( X ) in a given ensemble of switches is an average of the logarithm of the probability Pr   ( x i ) of finding the switch in a given complexion [Shannon and Weaver (11)].

where Pr ( x w ) is the probability of the ensemble of switches being found in complexion x w . If all the switches in a probability distribution have the same probability 1 / W , then then Eq. 2 becomes

The number of configurations of M switches with M o in the ON state is

We see from Eq. 4 that the maximum information storage in Eq. 3 occurs, for a given switch number M and a given number of ON switches M o , when the number of ON switches in a spectrum of M ACTIVE switches is half the total number of ACTIVE switches M .

If we assume that natural selection favors systems that maintain maximum information storage in a collection of switches, then this implies that a switch configuration that has equal numbers of ON and OFF switches tends to be favored. This is the justification for prediction (2) outlined in the abstract.

We take as an example the configurations of switches that contain M = 4 ACTIVE switches. These configurations are displayed in Figure 2 where the switches have been assembled into five groups based the number of switches that are ON. The information stored in each group is the logarithm of the number of switches in the group. Here, the group in Figure 2A has the highest information content.

Shannon and Weaver (11) defined the information I ( Y | X ) transmitted from one distribution Pr ( X ) to another distribution Pr   ( Y ) as

where H ( Y | X ) is the information stored in the conditional probability Pr ( Y | X ) . Here, X might be the switch ON/OFF spectrum and Y might be the spectrum of effector coupling, whether a given effector is activated or not. We see that maximum information transmission occurs when the storage terms H ( X ) and H ( Y ) are maximal and the conditional terms H(X | Y) and H ( Y | X ) are minimal. The conditional terms are zero if the mapping from X to Y and Y to X is unique in each direction. Every single state in X maps to a single state in Y and every single state in Y maps to a single state in X . Intuitively, the transition from X to Y is like a deck of cards that is shuffled. The amount of information stored in the cards is the same before and after shuffling, but the order of the cards is different. The maximum information transmission that is possible in a process is known as the capacity C I of the system [Shannon and Weaver (11)].

We can maximize the information stored in the receptor concentrations of each switch in Figure 1F . The total receptor concentration R T is equal to the receptor concentration R S in the switches plus the concentration Δ R of any receptors not attached to a switch.

The probability Pr ( i | j ) that receptor j that is associated with any switch is associated with switch i

where R i + R i * is the number of receptors, both ON and OFF, associated with switch i . A uniform distribution maximizes information storage.

which yields

  where R r e f is a reference concentration that is independent of the number of ACTIVE switches, the number of ON switches, and the number of OFF switches.

The constant concentration defined in Eq. 10 is a convenient set of units to measure receptor concentration. If the switch is ON, then

This normalization is experimentally accessible and is currently used in assay experiments to supply a set of units for receptor concentration. It is observed in assay experiments that the maximum response for many ligands and downstream coupling pathways have similar values [Rajagopal et al. (1)]. The maximum response to a particular ligand in this group is taken to be the reference receptor concentration R r e f . Here, we normalize all receptor concentrations to R r e f .

Proteins can have both flexible and rigid domains [Engh and Bossemeyer (34); Karshikoff et al. (35); Gerstein and Echols (36); Marsh et al. (37); Jacobs et al. (38); Sauer et al. (39); Bernadó et al. (40); Teilum et al. (43); Homans (42); Teilum et al. (41)]. The principle of MREP states that flexible internal degrees of freedom within the NESSs adjust such the naturally preferred NESS generates entropy at a maximum rate, consistent with relevant constraints such as mass and energy conservation [Jaynes (12); Dobovišek et al. (33)]. The maximum is not only the maximum rate over all possible NESSs, but also over all flexible internal states of each NESS. The receptors are flexible for bonds with binding energies below some energy determined by the distance from equilibrium and rigid for bonds with greater binding energies [Dobovišek et al. (33)].

Note that the MREP picture is quite different from a traditional mass-action treatments, such as the Operational model and TCM. [Black and Leff (7); Stephenson (8); De Lean et al. (48); Onaran et al. (10)]. The key difference results from protein flexibility and rigidity. In a mass-action simulation, the concentrations are allowed to vary, while reaction rates are held constant. The mass-action approach assumes that all reaction rates are rigid. For MREP, only rigid rates are held constant, while concentrations and flexible rates vary. This implies that fluxes J i and free-energy drops μ i ≤ 0 can vary for flexible reactions.

In anticipation of specializing the results to a GPCR complex, we add a GTPC to our collection of PdPCs. We consider the simple case in which we have two types of ACTIVE switches, N PdPC switches and one GTPC. Here, M = N + 1 . The collection of ACTIVE PdPCs obey a chemical reaction of the form

where R i is the receptor state in which the switch i is unphosphorylated (OFF) and R i * is the phosphorylated (ON) state. Here, k i − is the reaction rate from the ON state to the OFF state and k i + is the rate from the OFF to the ON. It is the rate at which the ON state is dephosphorylated.

The GTPC is of the form

Here, R G is the receptor concentration associated with the OFF state and R G * is the receptor concentration associated with the ON state of the GTPC, and G is the G α bound to G T P that effects the downstream effector coupling. Here, k G + and k G − are the forward and backward reaction rates, respectively. We assume the number of G α G T P is much larger than the number of free receptors.

Consider a single given receptor in a given volume V that can be in a number of internal states. The flow among the states is determined by the probability of the receptor being in a certain state and the reaction rates among the states. The total receptor concentration is 1 / V receptors per unit volume. The entropy production rate σ for all the ACTIVE switches in the receptor is

where T is the temperature of the heat bath in energy units and μ i ≤ 0 is the free-energy cost for turning ACTIVE switch i to ON and then back to OFF with similar conditions for the GTPC. Here, Pr(ON | i) is the probability that ACTIVE switch i is ON, and Pr(OFF | i) is the probability that the switch is OFF. Here, J i is the flux per receptor. The summation is over N , the total number of PdPCs.

Since, from Eq. 10, the number of receptors in each switch is constant, we can express the chemical flux J i in terms of the probability a switch is ON or OFF, Eqs. 15 and 16. The ON-state receptor concentration is

with comparable expressions for the GTPC and the OFF states.

The total free-energy drop Δ G is finite. We quantify this by setting the average drop μ 0 over switches to be constant.

The entropy production, Eq. 14, along with normalization and free-energy constraints can be maximized with the method of Lagrange multipliers (see Supplement ). We find that the chemical flux is constant J 0 in every switch

where J is the total chemical flux into the system.

Equation 21 can be used to identify the control point for turning a switch ON and OFF. From the definition of flux, Eqs. 15 and 16, the probability of a PdPC being ON is

and the probability of the GTPC being ON is

As can be seen from Eqs. 22 and 23, the ON states of the switches are completely determined by the externally applied chemical flux and the return rates k i − and k G − for the switches. These two reaction rates are the dephosphorylation rate in PdPCs and the backward rate for the reaction in Eq. 13 for the GTPC. This is prediction (1) in the abstract. The switch is ON when the reaction rate, k i − or k G − , is equal to the flux J 0 . The switch is OFF when the rate is much larger than J 0 . In the PdPCs, this rate is the rate catalyzed by the phosphatase. In the GTPC, this is the hydrolysis/association rate for GTP-bound G protein to bind to the receptor. To maintain a stable switch, the return rates k i − and k G − must be rigid, not flexible. This indicates that the return rates are the controls for turning the switches ON and OFF.

We specialize the BOIS model to a GPCR complex illustrated in Figure 3A . Again, the time-average approach is most useful and convenient. We imagine that a single GPCR complex transitions among its allowable internal states over time. The complex is like a pulsating cloud in which all flexible states of the complex are visited by R T receptors. High-energy rigid states do not fluctuate as readily as flexible states.

In most cases, the point of ligand engagement is a ligand-binding pocket in the seven-transmembrane (7TM) G protein coupled receptor (GPCR) that is accessible from the extracellular fluid. Simultaneous to the ligand binding, a ternary reaction binds guanosine di-phosphate (GDP) -bound heterotrimeric G protein to the intracellular side of the complex [Lefkowitz (49)]. Ligand binding is further allosterically stabilized by the heterotrimeric G protein complex that binds to an intracellular binding pocket in the GPCR [Lefkowitz (49); De Lean et al. (48)]. This G protein complex, at this stage, has its G α subunit bound to Guanosine Diphosphate (GDP). From the extracellular pocket, the endogenous ligand or possibly drug agonist controls signaling to downstream pathways re-conformation of the transmembrane and intracellular alpha helices of the GPCR [Wingler et al. (50)]. The conformations determine switch configurations. The switch responsible for the control of the G α pathway is a GTPC ([39]).

The G protein is composed of three sub-units, G α and a G β γ dimer. Nucleotide exchange occurs and the G-protein bound GDP is replaced with a bound guanosine tri-phosphate (GTP). This destabilizes the complex leading to the release of G α GTP into the intracellular fluid where it activates G-protein effector coupling. This is the GTPase switch. The G β γ dimer is involved with other effector coupling, including β arrestin ( β arr) pathways. The C tail of the GPCR contains a number of phosphorylation sites [Latorraca et al. (5)]. These sites can be ACTIVATED and act as PdPCs, switches that leads to effector coupling of β arr pathways.

The GPCR acts as a Guanine-nucleotide Exchange Factor (GEF) that facilitates the exchange of GDP bound to the G α subunit of the G protein complex for GTP [Coleman et al. (51); Sprang (52)]. The heterotrimeric G protein separates into two subunits, the G α subunit bound to GTP and a β γ dimer [Liang et al. (53)] Both the G α subunit and the released dimer transduce the signal initiated by ligand binding to the GPCR.

The other transducer, β arrestin ( β arr) is controlled by multiple PdPCs that can desensitize the G α pathway or activate pathways downstream of β arr ([38,40]). Here, β arr can bind to the receptor core and sterically desensitize the G α pathway. The transducer β arr can also form a megaplex by binding through phosphorylation sites to the C tail of the GCPR leading to possible continued activation of G α pathways [Thomsen et al. (54)]. Downstream β -arrestin pathways can also be activated by the phosphorylation configuration on the C tail [Latorraca et al. (5)]. In this study, we focus on the information flow through the megaplex rather than G α desensitization.

The C terminus of the G protein-coupled receptor binds to the β γ dimer (55]) and facilitates the phosphorylation of the C terminus and intracellular loops of the GPCR. After phosphorylation, β arrestin binds to the phosphorylated barcode and intracellular loops formed by the placement of phosphates on the intracellular portions of the GPCR. The β arrestin provides a scaffold for downstream pathways. [Smith et al. (56)] The β arrestin protein takes on many conformations with each independent conformation associated with a downstream pathway ([30]). Each conformation corresponds to a distinct configuration of PdPC switches. The GPCR is dephosphorylated by a phosphatase. Free energy is cycled into and out of the process by phosphorylation-dephosphorylation [Beard and Qian (57); Qian and Reluga (27, 29, 46)] and nucleotide exchange-hydrolysis [Coleman et al. (51); Sprang (52)].

This biological model is greatly simplified by the BOIS picture illustrated in Figure 3B . Here, the OFF states of the switches are pooled into a common state C , that is ACTIVATED by ligand L binding to free receptors R F . In this picture, the OFF state for a given switch is characterized by an absence of receptor in the ON state R i * . The arguments stemming from MIST are unchanged and still hold in the specialized case. Moreover, the equations for MREP, Eqs. 14 through 14, remain unchanged. The key properties of the general BOIS model are preserved in this specialized configuration, including a common value for the flux in each ACTIVE switch and the constancy of the ON-state receptor concentration for each switch.

Most of the complication of the biological model is included in a single state C, which acts as a flux source J and sink for the switch outputs which are the phosphorylation sites on the C-tail and intracellular loops of the GPCR complex and the release of GTP-bound G α . State C represents all of the ligand-bound GPCR complex other than the switch outputs: other than the concentrations of the phosphorylation site outputs and the concentration of the GTP-bound free G α subunit. Free energy is cycled into and out of the process by phosphorylation-dephosphorylation [Beard and Qian (57); Qian and Reluga (29); 27, 46)] and nucleotide exchange-hydrolysis [Coleman et al. (51); Sprang (52)]. In the BOIS model, the ON/OFF state of the switches is determined by dephosphorylation in the PdPCs and by hydrolysis of GTP in the GTPC.

The GTPC switch may be more complicated than the PdPCs. Here, association and dissociation of G α adds some complication as does the presence of hydrolysis and its catalyst RGS. The value of the free G α concentration also has an effect. This added complication does not affect the results and conclusions here.

The number of receptors bound to ligands depends on the free-ligand concentration. If the receptors have not been exposed to ligands and the current free-ligand concentration is zero, then, obviously, the number of ligand-bound receptors is zero. As the free-ligand concentration increases, ligands become bound to receptors. This is a dynamic process with ligands associating and dissociating with the receptors. If the addition of free ligands is slow enough, the ligand-bound receptor concentration achieves a steady state that depends only on the ligand concentration. In other words, the bound receptor concentration changes adiabatically with ligand concentration. When the ligand concentration becomes very large, all receptors become bound.

No information is transmitted by the ligands when the number of ligand-bound receptors is zero or when all receptors are bound. In these two cases, the information H stored in the bound-receptor probability distribution is zero. The maximum storage of one bit occurs when half the receptors are ligand-bound. Any information transmission rate greater than one bit must be supplied by another mechanism than the simple ligand binding described here.

It should be noted that reversing the ligand population from a high value to zero does not necessarily reproduce the same ligand dependence on the number of bound receptors as in the forward case. This is due to the fact that once a receptor is bound to a ligand, it can change into other states such as, for instance, a ligand-bound phosphorylated state, which may not have the same ligand dissociation constant as the unphosphorylated state. One then expects hysteresis in the forward and reverse trajectories of the bound receptor concentration. We are not aware of any experiments that have examined the removal of free-ligand concentration from a population of fully bound receptors. We do not believe that the hysteresis prediction has been tested.

The kinetics of the model is described by the following ordinary differential equations (ODE) where we designate the state concentrations by the name of the state

and

where N is the number of ACTIVE phosphorylation switches and the k notation indicates reaction rates. All receptor concentrations are normalized to R r e f from Eq. 10. Here, R i * is the ON-state concentration of the of the PdPC and is equal to the probability that the PdPC is ON, while R G * is the same quantity for the GTPC.

Simulations were performed in which the ligand concentration was initially set to zero. The concentration was increased until all receptors were bound. At that point, all ON switch concentrations were at their common maximum value.

For a finite value of the chemical flux J 0 , some initial conditions lead to non-physical results. For instance, in the relevant case in which all receptors are initially in the free state R F and none are in any ligand-bound state, Eqs. 26 and 27 immediately drive the ON-state concentrations P i and R G to negative values, which is non-physical and not permitted. This implies that in the state in which the complex encounters the ligand for the first time, all switches are INACTIVE and contain no chemical flux ( J 0 = 0 ). Simulations indicate that no physical solutions are possible until the ligand concentration increases to the level at which the complex concentration C is equal to the number of receptors that will turn on one switch ( C = 1 ).

At that point one switch can be ACTIVATED and turn ON. The concentration goes to zero at this point. As the ligand concentration increases more, the receptor concentration C once again increases to a value of one at which a second pair of ON/OFF switches are ACTIVATED. This process continues until the total number of INACTIVE switches is ACTIVATED or the ACTIVE switches have consumed the total number of receptors. This is illustrated in the simulation runs in Figure 4 .

This argument highlights the important transition from inactivated switches to activated switches as free-ligand concentration increases in the presence of receptors that are not initially bound to ligands. At low free-ligand concentrations, the total flux into the system is zero. No switches are ACTIVE. The complex is not capable of sending information to downstream processes. As free-ligand concentration increases, however, a point is reached when a phase transition occurs and the chemical flux becomes finite. At this point, the complex is able to transmit information. This supports prediction (3) in the abstract.

The sharp transition between ON and OFF states illustrated in Figure 4 is an artifact of the theory. In physical phase transitions and in all theories based on the mean-field-theory approximation, as is the BOIS model, fluctuations spread sharp transitions. When fluctuations are included in the theory, the transitions become smooth [Uzunov (58)].

An assumption of the BOIS model in Eqs. 24 - 27 is that the only state in which a receptor can dissociate from a ligand is C . This is consistent with many models, including the Operational Model. The dissociation constant from phosphorylated states, for instance, is zero. An important implication is that, in the BOIS model, once a switch has been activated it does not become deactivated if the free-ligand concentration decreases to zero. The ligands bound to the receptors in the switches remain bound. A process from outside the model must deactivate the switches. It must be terminated by an external process such as arrestin blockage. This is consistent with [Tran et al. (59); Gurevich and Gurevich (60); Wilden et al. (61); Lohse et al. (62)].

Both the Operational model and the BOIS model treat much of the receptor complex as a black box represented by state C in Eq. 24. The Operational model differs from the BOIS model in that the constant flux J 0 in Eqs. 26 and 27 is replaced by fluxes k i + C and k G + C , respectively. The specialized BOIS model makes a very different set of predictions from the Operational model. There is no concept of ACTIVE and INACTIVE switches in the Operational model as there is in TCM. The flux in a switch can take on any values given by k i + C and k G + C . In the scenario in which all ligands are free and the ligand concentration is increased from zero, a single switch becomes saturated when it has absorbed the entire number of receptors R T . If multiple switches are present in the system, the response is partitioned equally among the switches if they are identical. This is illustrated by the green curve in Figure 4 . If the number of switches changes, then so does the maximum response.

Importantly, in the Operational model, the saturated outputs drop to zero if the free-ligand concentration drops back to zero. We can see this quantitatively by noting that the Operational model for the system describes the NESS by [Black and Leff (7)].

Here, R * is the ON concentration of the switch and K S are given by

for a switch. Note that the Operational model, Eq. 30, depends continuously on ligand concentration and linearly on the total receptor concentration R T . This is quite different from the BOIS model, which is step-wise dependent on the ligand concentration. The Operational model increases the ON concentration for all switches simultaneously, while the BOIS model turns on one switch at a time. From Eq. 30 we see that the output concentration R * goes to zero as the free-ligand concentration L goes to zero.

Molecular switches exist within a milieu of energy fluctuations, which cause the molecular conformations to rapidly change. On the other hand, for information transmission to be useful, the conformations associated with information storage and transmission must be long-lived. [Schrodinger et al. (44)], otherwise, the information rapidly disappears. Therefore, bonds associated with the switches, whether covalent or conformational, must have lifetimes comparable to the time needed for information storage and transfer. The bonds associated with switch configurations must be rigid. In the BOIS model these rigid bonds are represented by the reaction rates k i − and k G − in Eqs. 26 and 27.

MIST requires that the number of ON states is equal to the number of OFF states, at least statistically. Some indications of this have been observed experimentally [Latorraca et al. (5)]. This is a global requirement involving the entire portion of the GPCR megaplex associated with switches. The mechanism to implement this requirement is unclear but we can gain some insight with a simple simulation illustrated in Figure 5 .

As was pointed out in the discussion of Figure 1 , biological processes, in order to repeat actions, must exist as cycles [Kauffman (63)]. We can perform a very simple numerical experiment to see how these cycles might form from a very simple set of physical rules. We imagine a collection of four switches with states as illustrated in Figure 2 . We then imagine that each state s transitions to another state at times t = 1 , 2 , 3 , ⋯ with a certain transition probability p [ s ( t ) → s ( t + 1 ) ] . We then run a Monte Carlo simulation of state transitions with a single requirement: if the simulation transitions from s ( t ) to s ( t + 1 ) , then the transition probability for those two states is slightly increased. This type of modification to the transition probability is known as Hebbian learning and is important in the study of neural systems [Munakata and Pfaffly (64); Keysers and Perrett (65); Gerstner and Kistler (66)].

Each simulation starts with a random initial state. The requirement states that a transition from one state to another affects the conformation of the flexible protein matrix such the conformation has a physical memory, a lasting impression, of the transition. We see in Figure 5A that the simulation spontaneously forms cycles. Precisely which cycles are formed varies from simulation to simulation and is determined by the random number seed. We see from Figure 5B that the probability of a state being in a cycle in which the average number of ON states among the cycle switches clusters around two, which is half the total number of switches M = 4 . We found this to be true for all the approximately 100 simulations we ran.

These results are easy to understand. If we pick a state with uniform probability from among the 16 possible choices, then the probability that the state will be associated with a particular cycle is greater the more states are in the cycle. Larger cycles, on the other hand, sample more of the total number of states than smaller cycles. As the cycles become larger, the average number of ON switches in the cycle approaches 2, which is the average number for all 16 states. This leads to the tight clustering around the average of 2 seen in Figure 5B .

It is well known that balance of this nature, excitatory-inhibitory balance, occurs at the cellular level [Froemke (67); Ganguly et al. (68)]. At least in some periods of neural development, the number of inhibitory and excitatory neurons are statistically equal in neuron switches. The balance in the cellular case is determined by Cl   − concentration in the extra- and intra- cellular fluid [Ganguly et al. (68)], which, as we propose here, is a system-wide control of ON/OFF balance.

One important implication of this ON/OFF balance is that if a switch is flipped, then, at least statistically, another switch is flipped oppositely. The hypotheses generated by the arguments of this section are theoretical predictions that merit experimental tests.

We compare the assay data from [Rajagopal et al. (1)] with simulations based on the BOIS model in Figure 6 . Assay results on two different receptors, β 2 adrenergic ( β 2 A R ) and angiotensin II type 1A ( A T 1 A R ), for which IP and cAMP, respectively, were measured as an indirect proxies for these switch ON states, are displayed. The G proteins were in the G q class for β 2 A R and in the G s class for A T 1 A R . The study also measured recruitment of β arrestin 2.

Two different downstream pathway concentrations were measured for each receptor, the G α pathway and one β -arr pathway. Other pathways may have been present, but they were not measured. The estimated reference receptor concentration R e x p was taken to be the maximum Formoterol concentration for the adrenergic receptor and angiotensin II maximum concentration for the angiotensin II receptor. The maximum downstream concentrations normalized to R e x p for each pathway are given in columns marked G α and β arr in Table 1 . The E C 50 concentrations in the table are measured in moles per liter. Several distinct ligands, listed in Table 1 , were applied to each receptor.

Experimental dose-response and bias curves from [Rajagopal et al. (1)] are displayed in Figure 6 . All concentrations are normalized to the maximum observed concentrations R e x p for each receptor. Here, R e x p is the maximum concentration of Formoterol for the adrenergic receptor; it is the maximum concentration of angiotensin II for the angiotensin II receptor. Figure A displays the G α and β arr recruitment responses to Formoterol and angiotensin II for the adrenergic and angiotensin II receptors, respectively.

Figure 6A illustrates the two dose response curves for the ligand Formoterol binding to the adrenergic receptor. The G α response is indicated by gold markers and the β arr response is indicated by cyan markers. The maximum normalized Formoterol response concentration is one since R e x p is equal to the maximum Formoterol concentration.

The BOIS model is indicated in red. We assume that four switches are involved in the process. As the ligand concentration L increases from zero in the presence of a receptor that has not been exposed to the ligand before, the theoretical response is zero. The chemical flux through all the switches is zero. When the normalized bound-ligand concentration in the receptor (state C) reaches one, the first switch turns on (dashed curve). We set the E C 50 for the first switch of the BOIS model to be the E C 50 of the first switch measured in the experiment. The receptor becomes activated. A chemical flux flows through the receptor states. The ligand concentration in receptor states associated with the body of the GPCR, state C, approaches zero and all the receptor states are associated with the first switch that turns ON. As the ligand concentration increases further, the concentration in the body of the GPCR complex increases once more until the second switch (dotted) turns ON. The E C 50 of this switch in the BOIS model is determined by the E C 50 of the first switch.

For Formoterol applied to the adrenergic receptor, the experimental observations indicate that the first switch that turns ON is the G α (gold) response. The second switch is the observed β arr recruitment (cyan) switch. This can be seen also to be true for all ligands tested with the adrenergic receptor in Figure 6C . Here, we have reproduced the bias curves (blue) from Rajagopal et al. (1). Each curve represents a unique ligand given in Table 1 . The markers indicate the maximum output for each ligand and receptor. The horizontal axis measures the output to the G α or GTPC pathways, while the vertical axis measures the output to the β arr PdPC pathway. We see that all responses start as zero as would be expected. All measured ligands turn the G α switch ON first and then some of the ligands also turn on the β arr recruitment switch. It is important to note that the maximum responses all lie roughly around the points (0,0), (0,1), and (1,1). This is support for the BOIS prediction that all responses saturate at a common concentration. The biological interpretation is that the ligand-bound receptor first binds to the G protein. Then, the receptor can stay in that state with the β arr downstream pathway untriggered or the ligand/G protein-bound receptor can facilitate the recruitment of β arr.

Figure 6B displays the response curve for Angiotensin II ligand on the angiotensin II receptor. As was the case with the adrenergic receptor, the first switch to turn ON for the angiotensin II receptor is a G α switch as seen in Figure 6B . The second switch is associated with β arr. The interpretation of the bias curves Figure 6D is not as simple as the interpretation of the bias curves for the adrenergic receptor, Figure 6C . We note that if a receptor triggers the G α response it also triggers a β arr response at a higher ligand concentration. We see that the right blue curve in the figure first passes near the (0,1) state before proceeding to the balanced (1,1) state. This indicates, within the context of the BOIS model, the G α switch is turned on first and then all receptors that have the G α switch turned ON transition into the β arr recruitment state. In more biological language, this means that the ligand-bound receptor first binds with G protein, which facilitates the recruitment of β arr. The left blue curve in Figure 6D indicates that the β arr can occur without the intermediary of the measured G α pathway. In other words, the measured G α pathway does not facilitate β arr recruitment. The β arr recruitment occurs with an

E C 50 of the second switch. Within the context of the BOIS model, this indicates that an unmeasured switch was the first to turn ON and that this unmeasured switch may have facilitated the recruitment of β arr.

We see that the saturated responses, the ON states, for the experimental assays are clustered about the points (0,1), (1,1), and (1,0) in the bias curves. One ineffective measured ligand had a response of (0,0). This point is not displayed. This indicates that the common saturated-concentration value of R_exp is the common concentration for all the ligands, for multiple numbers of ON switches. The BOIS predictions indicate the this number is R r e f . We can conclude that the measured value R e x p and the BOIS value R r e f are equal.

The fits to data in Figure 6 rely on just three free parameters, R r e f , the E C 50 for one switch, and Δ R (see Eq. 7). Each of the three parameters has an intuitive meaning.

The Operational model and TCM predict different maximum response for different numbers of ON switches. This is not observed in the observations in Figure 6 . The Operational model and TCM predict that all switches turn on simultaneously. This also, is not observed. This defect of the Operational model can be corrected in a more elaborate Mass-Action model that introduces additional degrees of freedom for reaction rates that vary the E C 50 for each switch. The BOIS model, however, makes the prediction with a paucity of free parameters.

Recent single-cell experiments on the muscarinic acetylcholine receptor M3R [Keshelava et al. (3)] indicate that more than two bits of information is transmitted through the receptor complex. In other words, the channel capacity C I was measured to be greater than two bits. This is significantly higher than previous population-level experiments that measured transmission at about one bit (see Keshelava et al. (3) and Zielińska and Katanaev (2) for references) The difference is attributed to extrinsic noise, that is inter-cellular noise, in the cell population. Increased information flow is observed when measurements are made at the cellular level as opposed to the population level. Accurate measurement of information flow seems to be sensitive to the resolution of the experiments.

The authors point out that a capacity slightly greater than two allows for four possible downstream effector couplings, which allows for ligand bias. Bias is measured in Rajagopal et al. (1). The BOIS model predicts that for four switches, the capacity is C I = log   ( 6 ) = 2.58 bits, which is consistent with both the single-cell capacity measurements and the bias experiments.