BACKGROUND
a cellular pathway, whether enzymatic or signal transducing, can in a simplistic manner be described as a causal relationship between an environmental signal  and a cellular response .
cellular signals are mediated through a series of successive protein-protein interactions that bridge spatial and topological boundaries  and that allow for crosstalk between different pathways  <cit> .
this protein-based modular strategy achieves integrated cellular responses that are both specific and at the same time tuned to global environmental and cellular requirements.
this specificity is organized through the cooperativity between the members of the complex and the introduction of temporal and spatial constraints on the expression levels of the different members of the signalling pathways.
over- or under-expression, for instance, of the signalling components may have disastrous effects on the cellular phenotype, e.g.
the development of cancer.
cooperativity is a thermodynamic concept that is used in different biochemical contexts  <cit> .
here this notion refers to the formation of multi-protein complexes with non-additive free-energies of assembly, i.e.
complexes for which the stability of the final assembly is higher than the sum of all individual binary association  <cit> .
a classic way to study cooperativity is by the analysis of a thermodynamic cycle  <cit> .
consider an assembly process that involves three proteins a, b and c that together form a ternary complex abc, where b acts as an adaptor protein providing a separate binding surface for each of the two other molecules .
two alternative routes can then create the complex abc: either the binary complex ab is formed first and c binds afterwards or first the proteins b and c are joined before a is added.
every pair-wise reaction between the proteins in isolation and between individual proteins and partially formed complexes is annotated by a rate of dissociation , specifying the likelihood of that particular assembly/disassembly step.
here there are four dissociation constants: kda-b, kdb-c, kdab-c and kda-bc, where the latter dissociation constants refer to the dissociation of c from abc and a from abc respectively.
since it is known that the overall thermodynamics and free energies for both routes to construct abc is the same around the cycle , one can simply determine the cooperativity of the system by comparing the energy changes when one of the proteins is already bound or not  <cit> .
for instance, when the likelihood of binding of c is enhanced when a is already bound to b, then there is positive cooperativity, resulting in bigger gain in energy when comparing it with the same reaction without the presence of a. when the binding of c to b is inhibited by the presence of a, then there is negative or anticooperativity, producing in turn a decrease in the free energy when comparing it to the process without a. when the presence of a in the complex does not influence the binding affinity of c, then there is no cooperativity or independent binding.
even though this scheme identifies the presence and type of cooperation in the assembly process, it does not shed light on the molecular concentrations, possibly reflecting the intracellular conditions, required for efficient regulation or coordination between a pathway's active  and inactive  state.
here we provide an information theoretical method that, in the same spirit as the hill and scatchard plots  <cit> , identifies and quantifies cooperativity in macromolecular assemblies and visualizes for a spectrum of concentrations when optimal coordination is obtained for the given experimental data.
different from those established methods, our approach goes beyond multiple bindings of the same ligand to a homogeneous oligomer : we consider here the construction of heterogeneous protein assemblies mediated by multiple binding surfaces on adaptor proteins.
as such, and as far as we are aware, this method provides an original and novel approach for the analysis of the cooperativity in macromolecular complexes that are part of some signalling cascade.
RESULTS
general description of the approach
in analogy with cellular pathways, each protein in a cellular network can be considered as an element receiving an input signal  and generating an output signal .
hence, we can reinterpret the ternary protein complex abc as an instance of communication over a noisy channel  <cit> , where protein b provides the communication channel through which information is exchanged between upstream ligand a and downstream effector c .
like all noisy communication channels  <cit>  the protein complex abc is determined by a set of conditional probabilities relating input to output .
a first set of probabilities, p and p, describe the accuracy of the communication channel, i.e.
the likelihood that a given output signal corresponds to the appropriate input signal.
in other words, if no input signal is given, meaning a is not bound to b  then c should also be not bound to b , and vice versa.
a second set of probabilities, p and p, describes the intrinsic noise of the communication channel, i.e.
the likelihood that a given input signal is not correctly conveyed.
thus inappropriate transmission occurs when either c is bound to b  even though a was not bound before  or when a is bound  and c is not .
these probabilities describe a partition of the initial protein concentrations  <dig>   <dig> and  <dig> over all possible association states of the system including a, b and c in isolation, the binary complexes ab and bc and the ternary complex abc.
next to the total concentrations  <dig>   <dig> and  <dig>  the steady state that is defined by this partitioning depends on the different dissociation constants including kda-b, kdb-c, kdab-c and kda-bc.
given the steady state concentrations, the individual and conditional probabilities may be obtained  in methods).
these probabilities are necessary to derive shannon's entropy , h, ...) and subsequently the mutual information i exchanged between a and c over channel b  in methods).
we quantify the degree of cooperativity of the system by the amount of information that is exchanged between the elements of the complex.
in terms of the protein concentrations, this mutual information expresses how well the ratio of the steady-state concentrations of the ternary complex abc and the free adaptor protein b are balanced while at the same time requiring the concentration of both binary complexes  to be as low as possible.
so, on the one hand, low information exchange corresponds to an equilibrium situation where the protein  and complex concentrations  are out of balance or where to many intermediate complexes are present making it hard for the biological system to perform its function.
on the other hand, high amounts of information exchange correspond to an optimized system where all members achieve the required coordination to switch efficiently between active and inactive states of the cellular process.
note, that the approach described here for a ternary protein complex can be further generalised to describe communication channels having multiple inputs or outputs .
in that case the mutual information between the different components needs to be deduced by a multivariate approach   <cit> .
it is also important to note that the mutual information does not change from swapping the input with the output components, i.e.
i = i.
biophysical model system
to clarify the biophysical meaning and illustrate our method we here describe the information exchange over a part of the p <dig> regulatory pathway.
the p <dig> pathway controls the degradation of the cyclin-dependent kinase  <dig>  inhibitor p <dig>  <cit>  thereby playing an important role in cell cycle progression  <cit> .
in particular, phosphorylation of p <dig> triggers cks1-mediated binding of p <dig> to skp <dig>  as skp <dig> is part of the scfskp <dig> ubiquitin ligase this results in p <dig> degradation and cell cycle progression.
in a recent study  <cit> , the assembling mechanism for part of the scfskp <dig> multiprotein complex has been analyzed in order to understand 1) how and in which order the different units assemble and 2) how the specific order of this process influences the mutual affinities between the components and intermediately formed complexes.
seeliger et al.
<cit>  showed that the skp2-cks <dig> complex increases the affinity of cks <dig> for the cdk <dig> inhibitor p <dig> a 100-fold.
additional inclusion of cdk <dig> increases the affinity for p <dig> even more.
through mutational analysis the authors also showed long-range coupling between distant functional sites in cks <dig>  making it a principal example how adaptor proteins can play a central role in tightly controlling the assembly of a critical complex.
as a consequence, it forms a biophysically meaningful case to investigate the communication between the different binding sites of the cks <dig> structure in terms of shannon's information theory .
note here that shannon's information theory can also be used to derive the communication pathway in cks <dig>  we recently demonstrated this lower-level analysis for the sh <dig> domain of fyn  <cit> .
given the appropriate structural data, the same analysis could occur which should reveal the communication between the three binding sites  <cit> .
the biophysical data obtained in  <cit> , i.e.
the dissociation constants, is used to perform the current analysis, .
the thermodynamic cycle including the adaptor protein cks <dig>   <cit> , the proteins skp <dig>  and p <dig>  produced from this data shows that both paths around the cycle are cooperative: having skp <dig> bound to cks <dig> makes it easier for p <dig> to bind and vice versa.
in a first step, we focus on the thermodynamic cycle for the formation of this ternary complex p27-cks1-skp <dig> .
since in vivo p <dig> is bound to cdk <dig>  we will in a second step consider the quaternary complex cdk2-p27-cks1-skp <dig>  in that case two signals  are integrated and conveyed over the communication channel cks <dig>  as the mutual binding affinities of this system, i.e.
kdskp2-cks <dig>  kdcks1-p <dig>  kdskp2-cks1p <dig> and kdskp2cks1-p <dig>  have been determined experimentally , we can quantify the information exchange between the input and output components of the system and study the transmission efficiency, meaning under which conditions we observe the highest degree of cooperativity, of the adaptor protein cks <dig> under a wide range of chemical potentials .
note that only one of the dissociation constants, kdskp2-cks1p <dig> or kdskp2cks1-p <dig>  is required for the derivation of the different steady-state concentrations .
dissociation constants in bold were derived experimentally.
the remaining values were obtained through the thermodynamic cycles  <cit> .
how much information is exchanged in the p27-cks1-skp <dig> complex?
as can be seen in figure  <dig>  only a small part of the phase space displays some or significant degree of cooperativity.
in the cooperative regime the assembly of cks <dig> to p <dig> will be conditional on the concentration of skp <dig> and vice versa.
in other words, binding of p <dig> to cks <dig> and recruitment to the scfskp <dig> ubiquitin ligase machinery will be mutually dependent events.
outside of this regime the assembly of these elements is still possible, but as under these conditions the fraction of bound protein is no longer influenced by changes in the concentration of the other, information exchange becomes very noisy.
moreover, from the difference in the extent for which each protein, skp <dig> and p <dig>  shows information exchange one can derive that it is skp <dig> that forms the natural input signal for this regulatory process: it controls the switching in an effective manner.
how does channel concentration affect robustness of the system?
interestingly, although the area of maximum cooperativity of p27-cks1-skp <dig> represents only a minor part of the phase space, it displays a relatively slow decline for increasing p <dig> concentration .
this relatively broad, although suboptimal response curve, gives a measure for the robustness of the system to extrinsic noise due to fluctuations in input or output ligand concentrations.
this is not the case for skp <dig>  as can be seen figure  <dig> , the responsive area for skp <dig> is very tightly defined .
hence variations in skp <dig> strongly influence the responsiveness of the biochemical system.
the robustness towards the skp <dig> concentration increases as the concentration of cks <dig> increases, as is shown in figure  <dig>  increasing the concentration of cks <dig> from  <dig>  μm and  <dig>  μm results in an increase of the cooperative area, particularly for the concentration of skp <dig>  this effect of cks <dig> on the concentration of skp <dig> makes sense biochemically since it is know that the expression of cks <dig> alternates in parallel with the concentration of skp <dig> when passing through the cell cycle  <cit> .
even though the area for skp <dig> increases, the concentrations of p <dig> for which cooperativity is high remains the same.
moreover, even though the actual concentration for skp <dig> changes, the maximum mutual information, or the capacity, remains always the same for this ternary system .
how much information is exchanged in the cdk2-p27-cks1-skp <dig> complex?
the right panel of figure  <dig> shows the cooperativity profile of the quaternary complex cdk2-p27-cks1-skp <dig>  in this case both cdk <dig> and p <dig>  which are associated with a 1: <dig> stochiometry can be considered as two input signals, which are combined over the cks <dig> adaptor to elicit skp <dig> binding.
the multivariate approach to mutual information  <cit>   makes it possible to analyse the different components that define the information flow in this quaternary complex.
as can be expected, the integration of the two signals gives rise to an increased cooperativity of the system, but also to a broader maximum indicating a more robust system response .
in addition, we can deconstruct the transmission within the complex cdk2-p27-cks1-skp <dig> into the mutual information between skp <dig> and p <dig> , skp <dig> and cdk <dig>  and the effect of integrating both signals, called the interaction information a  <cit>  .
even though the majority of the communication occurs between skp <dig> and p <dig>  the integration of these two signals, a , shows how the concentration of cdk <dig> affects the communication between skp <dig> and p27: first for low initial concentrations of both proteins, knowledge about cdk <dig> inhibits slightly the information exchange over the cooperative channel.
as the concentration of skp <dig> increases, the transmission is amplified resulting in the higher capacity that is shown in the right panel of the figure  <dig>  consequently, the contour plot of a gives a quantitative interpretation of how these three proteins affect the communication over the channel cks <dig>
channel capacity and noise of the p27-cks1-skp <dig> complex
the maxima of these contour diagrams represent the capacity of the system, i.e.
the maximum amount of information that can be transmitted over the channel with an arbitrary small probability of error  <cit> .
as can be seen in figure  <dig>  channel capacity is achieved for the optimal input distribution {, }* = { <dig>  μm,  <dig>  μm}  for the ternary complex p27-cks1-skp <dig>  at these concentrations ~ <dig>  bits of information is received as output for every bit of input.
as previously argued, this capacity remains the same even when the concentration of cks <dig> changes.
in terms of cooperativity, this means that even though the system is not fully cooperative, the balance between the assembled degradation system and independent adaptor protein is rather efficient while at the same time the intermediate complexes are few.
hence the capacity of the system is limited by the molecular concentrations that can be attained at steady state.
we added in each plot a blue line that marks the concentration of p <dig> where the maximum information transmission is found  for all combinations of  and  .
we immediately notice here that this vertical line lies at the intersection of the two error probabilities  as well as the 50% mark of the probability of having p <dig> bound to cks <dig> .
consequently, when the channel is symmetric the scfskp2model system studied here reaches maximal information exchange.
by adding cdk <dig> in the system, the capacity becomes ~ <dig>  bits , confirming the effect of including cdk <dig> as discussed in ref.
<dig>  moreover, the area of optimal response increases  allowing a wider range of useful skp <dig> and p <dig> concentrations.
CONCLUSIONS
all these results show that, given the binding affinities at equilibrium and the overall concentrations of the different components, mutual information quantifies for which protein concentrations the systems' cooperativity, or more specific its coordination, is optimal.
our analysis clearly shows  that both skp <dig> and cks <dig> concentrations are crucial parameters to obtain proper regulatory behaviour in the p <dig> degradation system.
if binding is independent then no information exchange would be observed.
moreover, this narrow range for skp <dig> remains operational for a wide range of p <dig> concentrations, making the scfskp <dig> ubiquitination system robust to variations in p <dig> abundance.
when moving outside the boundaries defined for skp <dig> the coordination between the complex members is lost, leading possibly to continuous activation without proper regulation.
this observation seems to corroborate experimental studies on the over expression of skp <dig>  in relation to the development cancer  <cit> .
even though, the current analysis is performed on the steady-state information of this biochemical system, the underlying idea is that all the proteins of the macromolecular complex become expressed so that their mutual concentrations fall into the highly cooperative area .
further analysis is off course required to verify this hypothesis.
to conclude, the present result show that shannon's information theory quantifies the cooperativity of biochemical systems, making it an important tool for the current attempts to understand cooperativity in a systems perspective.

 a cellular pathway, whether enzymatic or signal transducing, can in a simplistic manner be described as a causal relationship between an environmental signal and a cellular response. cellular signals are mediated through a series of successive protein-protein interactions that bridge spatial and topological boundaries and that allow for crosstalk between different pathways. this protein-based modular strategy achieves integrated cellular responses that are both specific and at the same time tuned to global environmental and cellular requirements. this specificity is organized through the cooperativity between the members of the complex and the introduction of temporal and spatial constraints on the expression levels of the different members of the signalling pathways. over- or under-expression, for instance, of the signalling components may have disastrous effects on the cellular phenotype, e.g. the development of cancer. cooperativity is a thermodynamic concept that is used in different biochemical contexts. here this notion refers to the formation of multi-protein complexes with non-additive free-energies of assembly, i.e. complexes for which the stability of the final assembly is higher than the sum of all individual binary association. a classic way to study cooperativity is by the analysis of a thermodynamic cycle. consider an assembly process that involves three proteins a, b and c that together form a ternary complex abc, where b acts as an adaptor protein providing a separate binding surface for each of the two other molecules. two alternative routes can then create the complex abc: either the binary complex ab is formed first and c binds afterwards or first the proteins b and c are joined before a is added. every pair-wise reaction between the proteins in isolation and between individual proteins and partially formed complexes is annotated by a rate of dissociation, specifying the likelihood of that particular assembly/disassembly step. here there are four dissociation constants: kda-b, kdb-c, kdab-c and kda-bc, where the latter dissociation constants refer to the dissociation of c from abc and a from abc respectively. since it is known that the overall thermodynamics and free energies for both routes to construct abc is the same around the cycle, one can simply determine the cooperativity of the system by comparing the energy changes when one of the proteins is already bound or not. for instance, when the likelihood of binding of c is enhanced when a is already bound to b, then there is positive cooperativity, resulting in bigger gain in energy when comparing it with the same reaction without the presence of a. when the binding of c to b is inhibited by the presence of a, then there is negative or anticooperativity, producing in turn a decrease in the free energy when comparing it to the process without a. when the presence of a in the complex does not influence the binding affinity of c, then there is no cooperativity or independent binding. even though this scheme identifies the presence and type of cooperation in the assembly process, it does not shed light on the molecular concentrations, possibly reflecting the intracellular conditions, required for efficient regulation or coordination between a pathway's active and inactive state. here we provide an information theoretical method that, in the same spirit as the hill and scatchard plots, identifies and quantifies cooperativity in macromolecular assemblies and visualizes for a spectrum of concentrations when optimal coordination is obtained for the given experimental data. different from those established methods, our approach goes beyond multiple bindings of the same ligand to a homogeneous oligomer: we consider here the construction of heterogeneous protein assemblies mediated by multiple binding surfaces on adaptor proteins. as such, and as far as we are aware, this method provides an original and novel approach for the analysis of the cooperativity in macromolecular complexes that are part of some signalling cascade.  general description of the approach in analogy with cellular pathways, each protein in a cellular network can be considered as an element receiving an input signal and generating an output signal. hence, we can reinterpret the ternary protein complex abc as an instance of communication over a noisy channel, where protein b provides the communication channel through which information is exchanged between upstream ligand a and downstream effector c. like all noisy communication channels the protein complex abc is determined by a set of conditional probabilities relating input to output. a first set of probabilities, p and p, describe the accuracy of the communication channel, i.e. the likelihood that a given output signal corresponds to the appropriate input signal. in other words, if no input signal is given, meaning a is not bound to b then c should also be not bound to b, and vice versa. a second set of probabilities, p and p, describes the intrinsic noise of the communication channel, i.e. the likelihood that a given input signal is not correctly conveyed. thus inappropriate transmission occurs when either c is bound to b even though a was not bound before or when a is bound and c is not. these probabilities describe a partition of the initial protein concentrations and over all possible association states of the system including a, b and c in isolation, the binary complexes ab and bc and the ternary complex abc. next to the total concentrations and the steady state that is defined by this partitioning depends on the different dissociation constants including kda-b, kdb-c, kdab-c and kda-bc. given the steady state concentrations, the individual and conditional probabilities may be obtained in methods). these probabilities are necessary to derive shannon's entropy, h,...) and subsequently the mutual information i exchanged between a and c over channel b in methods). we quantify the degree of cooperativity of the system by the amount of information that is exchanged between the elements of the complex. in terms of the protein concentrations, this mutual information expresses how well the ratio of the steady-state concentrations of the ternary complex abc and the free adaptor protein b are balanced while at the same time requiring the concentration of both binary complexes to be as low as possible. so, on the one hand, low information exchange corresponds to an equilibrium situation where the protein and complex concentrations are out of balance or where to many intermediate complexes are present making it hard for the biological system to perform its function. on the other hand, high amounts of information exchange correspond to an optimized system where all members achieve the required coordination to switch efficiently between active and inactive states of the cellular process. note, that the approach described here for a ternary protein complex can be further generalised to describe communication channels having multiple inputs or outputs. in that case the mutual information between the different components needs to be deduced by a multivariate approach. it is also important to note that the mutual information does not change from swapping the input with the output components, i.e. i = i. biophysical model system to clarify the biophysical meaning and illustrate our method we here describe the information exchange over a part of the p regulatory pathway. the p pathway controls the degradation of the cyclin-dependent kinase inhibitor p thereby playing an important role in cell cycle progression. in particular, phosphorylation of p triggers cks1-mediated binding of p to skp as skp is part of the scfskp ubiquitin ligase this results in p degradation and cell cycle progression. in a recent study, the assembling mechanism for part of the scfskp multiprotein complex has been analyzed in order to understand 1) how and in which order the different units assemble and 2) how the specific order of this process influences the mutual affinities between the components and intermediately formed complexes. seeliger et al. showed that the skp2-cks complex increases the affinity of cks for the cdk inhibitor p a 100-fold. additional inclusion of cdk increases the affinity for p even more. through mutational analysis the authors also showed long-range coupling between distant functional sites in cks making it a principal example how adaptor proteins can play a central role in tightly controlling the assembly of a critical complex. as a consequence, it forms a biophysically meaningful case to investigate the communication between the different binding sites of the cks structure in terms of shannon's information theory. note here that shannon's information theory can also be used to derive the communication pathway in cks we recently demonstrated this lower-level analysis for the sh domain of fyn. given the appropriate structural data, the same analysis could occur which should reveal the communication between the three binding sites. the biophysical data obtained in, i.e. the dissociation constants, is used to perform the current analysis,. the thermodynamic cycle including the adaptor protein cks, the proteins skp and p produced from this data shows that both paths around the cycle are cooperative: having skp bound to cks makes it easier for p to bind and vice versa. in a first step, we focus on the thermodynamic cycle for the formation of this ternary complex p27-cks1-skp. since in vivo p is bound to cdk we will in a second step consider the quaternary complex cdk2-p27-cks1-skp in that case two signals are integrated and conveyed over the communication channel cks as the mutual binding affinities of this system, i.e. kdskp2-cks kdcks1-p kdskp2-cks1p and kdskp2cks1-p have been determined experimentally, we can quantify the information exchange between the input and output components of the system and study the transmission efficiency, meaning under which conditions we observe the highest degree of cooperativity, of the adaptor protein cks under a wide range of chemical potentials. note that only one of the dissociation constants, kdskp2-cks1p or kdskp2cks1-p is required for the derivation of the different steady-state concentrations. dissociation constants in bold were derived experimentally. the remaining values were obtained through the thermodynamic cycles. how much information is exchanged in the p27-cks1-skp complex? as can be seen in figure only a small part of the phase space displays some or significant degree of cooperativity. in the cooperative regime the assembly of cks to p will be conditional on the concentration of skp and vice versa. in other words, binding of p to cks and recruitment to the scfskp ubiquitin ligase machinery will be mutually dependent events. outside of this regime the assembly of these elements is still possible, but as under these conditions the fraction of bound protein is no longer influenced by changes in the concentration of the other, information exchange becomes very noisy. moreover, from the difference in the extent for which each protein, skp and p shows information exchange one can derive that it is skp that forms the natural input signal for this regulatory process: it controls the switching in an effective manner. how does channel concentration affect robustness of the system? interestingly, although the area of maximum cooperativity of p27-cks1-skp represents only a minor part of the phase space, it displays a relatively slow decline for increasing p concentration. this relatively broad, although suboptimal response curve, gives a measure for the robustness of the system to extrinsic noise due to fluctuations in input or output ligand concentrations. this is not the case for skp as can be seen figure, the responsive area for skp is very tightly defined. hence variations in skp strongly influence the responsiveness of the biochemical system. the robustness towards the skp concentration increases as the concentration of cks increases, as is shown in figure increasing the concentration of cks from μm and μm results in an increase of the cooperative area, particularly for the concentration of skp this effect of cks on the concentration of skp makes sense biochemically since it is know that the expression of cks alternates in parallel with the concentration of skp when passing through the cell cycle. even though the area for skp increases, the concentrations of p for which cooperativity is high remains the same. moreover, even though the actual concentration for skp changes, the maximum mutual information, or the capacity, remains always the same for this ternary system. how much information is exchanged in the cdk2-p27-cks1-skp complex? the right panel of figure shows the cooperativity profile of the quaternary complex cdk2-p27-cks1-skp in this case both cdk and p which are associated with a 1: stochiometry can be considered as two input signals, which are combined over the cks adaptor to elicit skp binding. the multivariate approach to mutual information makes it possible to analyse the different components that define the information flow in this quaternary complex. as can be expected, the integration of the two signals gives rise to an increased cooperativity of the system, but also to a broader maximum indicating a more robust system response. in addition, we can deconstruct the transmission within the complex cdk2-p27-cks1-skp into the mutual information between skp and p, skp and cdk and the effect of integrating both signals, called the interaction information a. even though the majority of the communication occurs between skp and p the integration of these two signals, a, shows how the concentration of cdk affects the communication between skp and p27: first for low initial concentrations of both proteins, knowledge about cdk inhibits slightly the information exchange over the cooperative channel. as the concentration of skp increases, the transmission is amplified resulting in the higher capacity that is shown in the right panel of the figure consequently, the contour plot of a gives a quantitative interpretation of how these three proteins affect the communication over the channel cks channel capacity and noise of the p27-cks1-skp complex the maxima of these contour diagrams represent the capacity of the system, i.e. the maximum amount of information that can be transmitted over the channel with an arbitrary small probability of error. as can be seen in figure channel capacity is achieved for the optimal input distribution {, }* = { μm, μm} for the ternary complex p27-cks1-skp at these concentrations ~ bits of information is received as output for every bit of input. as previously argued, this capacity remains the same even when the concentration of cks changes. in terms of cooperativity, this means that even though the system is not fully cooperative, the balance between the assembled degradation system and independent adaptor protein is rather efficient while at the same time the intermediate complexes are few. hence the capacity of the system is limited by the molecular concentrations that can be attained at steady state. we added in each plot a blue line that marks the concentration of p where the maximum information transmission is found for all combinations of and. we immediately notice here that this vertical line lies at the intersection of the two error probabilities as well as the 50% mark of the probability of having p bound to cks. consequently, when the channel is symmetric the scfskp2model system studied here reaches maximal information exchange. by adding cdk in the system, the capacity becomes ~ bits, confirming the effect of including cdk as discussed in ref. moreover, the area of optimal response increases allowing a wider range of useful skp and p concentrations.  all these results show that, given the binding affinities at equilibrium and the overall concentrations of the different components, mutual information quantifies for which protein concentrations the systems' cooperativity, or more specific its coordination, is optimal. our analysis clearly shows that both skp and cks concentrations are crucial parameters to obtain proper regulatory behaviour in the p degradation system. if binding is independent then no information exchange would be observed. moreover, this narrow range for skp remains operational for a wide range of p concentrations, making the scfskp ubiquitination system robust to variations in p abundance. when moving outside the boundaries defined for skp the coordination between the complex members is lost, leading possibly to continuous activation without proper regulation. this observation seems to corroborate experimental studies on the over expression of skp in relation to the development cancer. even though, the current analysis is performed on the steady-state information of this biochemical system, the underlying idea is that all the proteins of the macromolecular complex become expressed so that their mutual concentrations fall into the highly cooperative area. further analysis is off course required to verify this hypothesis. to conclude, the present result show that shannon's information theory quantifies the cooperativity of biochemical systems, making it an important tool for the current attempts to understand cooperativity in a systems perspective. 
we show here that shannon's mutual information quantifies information in biological system and more specifically the cooperativity inherent to the assembly of macromolecular complexes. the proposed framework provides a new and original approach to analyse the effects of cooperativity in the assembly of macromolecular complexes. as such, this information measure identifies and quantifies those protein concentrations that render the biochemical system most effective in switching between the active and inactive state of the intracellular process. it shows the conditions, provided by the protein concentrations, for which a particular system acts most effectively, i.e. further we show, using a portion of the p regulatory pathway, how classical equilibrium thermodynamic quantities such as binding affinities and chemical potentials can be used to quantify information exchange but also to determine engineering properties such as channel noise and channel capacity. intra-cellular information exchange, propelled by cascades of interacting signalling proteins, is essential for the proper functioning and survival of cells. as such this framework opens the possibility of grasping biological qualities such as system sensitivity, robustness or plasticity directly in terms of their effect on information exchange. although these parameters might also be derived using classical thermodynamic parameters, a recasting of biological signalling in terms of information exchange offers an alternative framework for visualising network cooperativity that might in some cases be more intuitive.
 intra-cellular information exchange, propelled by cascades of interacting signalling proteins, is essential for the proper functioning and survival of cells. now that the interactome of several organisms is being mapped and several structural mechanisms of cooperativity at the molecular level in proteins have been elucidated, the formalization of this fundamental quantity, i.e. information, in these very diverse biological contexts becomes feasible.  we show here that shannon's mutual information quantifies information in biological system and more specifically the cooperativity inherent to the assembly of macromolecular complexes. we show how protein complexes can be considered as particular instances of noisy communication channels. further we show, using a portion of the p regulatory pathway, how classical equilibrium thermodynamic quantities such as binding affinities and chemical potentials can be used to quantify information exchange but also to determine engineering properties such as channel noise and channel capacity. as such, this information measure identifies and quantifies those protein concentrations that render the biochemical system most effective in switching between the active and inactive state of the intracellular process.  the proposed framework provides a new and original approach to analyse the effects of cooperativity in the assembly of macromolecular complexes. it shows the conditions, provided by the protein concentrations, for which a particular system acts most effectively, i.e. exchanges the most information. as such this framework opens the possibility of grasping biological qualities such as system sensitivity, robustness or plasticity directly in terms of their effect on information exchange. although these parameters might also be derived using classical thermodynamic parameters, a recasting of biological signalling in terms of information exchange offers an alternative framework for visualising network cooperativity that might in some cases be more intuitive. 