Information Rates of Controlled Protein Interactions Using Terahertz Communication
11 Information Rates of Controlled Protein InteractionsUsing Terahertz Communication
Hadeel Elayan, Andrew W. Eckford, and Raviraj Adve
Abstract —In this work, we present a paradigm bridgingelectromagnetic (EM) and molecular communication througha stimuli-responsive intra-body model. It has been establishedthat protein molecules, which play a key role in governingcell behavior, can be selectively stimulated using Terahertz(THz) band frequencies. By triggering protein vibrational modesusing THz waves, we induce changes in protein conformation,resulting in the activation of a controlled cascade of biochemicaland biomechanical events. To analyze such an interaction, weformulate a communication system composed of a nanoantennatransmitter and a protein receiver. We adopt a Markov chainmodel to account for protein stochasticity with transition ratesgoverned by the nanoantenna force. Both two-state and multi-state protein models are presented to depict different biologicalconfigurations. Closed form expressions for the mutual infor-mation of each scenario is derived and maximized to find thecapacity between the input nanoantenna force and the proteinstate. The results we obtain indicate that controlled proteinsignaling provides a communication platform for informationtransmission between the nanoantenna and the protein with aclear physical significance. The analysis reported in this workshould further research into the EM-based control of proteinnetworks.
I. I
NTRODUCTION
Interest in nanoscale robotic systems has led researchersto investigate different frameworks to initiate reliable com-munication between nanomachines. One solution is molecularcommunication, which is a paradigm inspired by nature, thatentails utilizing chemical signals as carriers of information.The transmitter of this diffusion-based channel releases parti-cles into an aqueous or gaseous medium, where the particlespropagate until they arrive at the receiver; the receiver thendetects and decodes the information in these particles [1]–[3].As another solution, the emergence of plasmonic nanoantennashas paved the way towards electromagnetic (EM) communi-cation among nanodevices, where both the Terahertz (THz)band [4]–[7] and optical frequency range [8] are possiblecandidates. Specifically, in-vivo wireless nanosensor networks(iWNSNs) have emerged to provide fast and accurate diseasediagnosis and treatment. These networks are expected tooperate inside the human body in real time while establishingreliable wireless transmission among nanobiosensors [9].
We would like to acknowledge the support of the National Science andEngineering Research Council, Canada, through its Discovery Grant program.H. Elayan and R. Adve are with the Edward S. Rogers Depart-ment of Electrical and Computer Engineering, University of Toronto,Ontario, Canada, M5S 3G4 (e-mail: [email protected];[email protected]).A. Eckford is with the Department of Electrical Engineering and Com-puter Science, York University, Ontario, Canada, M3J 1P3 (e-mail: [email protected]).
One active research topic within molecular communica-tions involves establishing interfaces to connect the molecularparadigm with its external environment [10]–[13]. The authorsin [10] proposed a wearable magnetic nanoparticle detector tobe used as an interface between a molecular communicationsystem deployed inside the human body and a signal pro-cessing unit located outside. In [11], the authors presenteda biological signal conversion interface which translates anoptical signal into a chemical one by changing the pH of theenvironment. Moreover, a redox-based experimental platformhas been introduced in [12] to span the electrical and moleculardomains. This wet-lab coupling paves the way towards novelgeneration of bio-electronic components that serve as the basisof intelligent drugs, capable of biochemical and electricalcomputation and actuation. Furthermore, in a very recentwork, the authors in [13], identified genes that control cellularfunction upon responding to EM fields that penetrate deeptissue non-invasively. Their experimental results complementthe growing arsenal of technologies dedicated to the externalcontrol of cellular activity in-vivo.Among the biological structures found in the human body,protein molecules are heterogeneous chains of amino acids;they perform their biological function by coiling and foldinginto a distinct three dimensional shape as required. Changesin protein level, protein localization, protein activity, andprotein-protein interactions are critical aspects of an inter-cellular communication process collectively known as signaltransduction . One important feature associated with proteinstructures is that their vibrational modes are found in the THzfrequency range [14]. These modes provide information aboutprotein conformational change, ligand binding and oxidationstate [15]. Therefore, by triggering protein vibrational modesusing THz EM waves, we can direct mechanical signalinginside protein molecules, in turn controlling changes in theirstructure and, as a result, activating associated biochemicalevents [16].In this work, we bridge the gap between EM (specifically,THz radiation) and molecular communication; We consider acommunication link which consists of a nanoantenna trans-mitter, a protein receiver and a Markovian signal transductionchannel. We are interested especially in the process at thereceiving end of signal transduction, where a protein changesconformation due to the induced THz signal. Since thisproblem can be thought of fundamentally as an informationtransmission problem, our aim in this paper is to computethe mutual information of this communication link. In fact,gaining a detailed understanding of the input-output relation-ship in biological systems requires quantitative measures that a r X i v : . [ q - b i o . M N ] S e p capture the interdependence between components. Hence, aclosed form expression for the mutual information rate underindependent, identically distributed (IID) inputs is derivedand maximized to find the capacity for different proteininteraction scenarios. By finding the mutual information rate,experimenters are guided into the amount of information theprotein signaling pathway carries.The main contributions of the paper are as follows: • We model the stochastic protein dynamics actuatedthrough THz waves as a discrete-time, finite-state chan-nel. We present both a two-state and a multi-state modelto emulate protein dynamics. In the two-state model,a change in the protein state is triggered through theapplied nanoantenna THz force. In the multi-state model,a cascade of changes in the protein configuration isstimulated, where links between different protein statesare controlled through the targeted application of THzforce. • We analytically derive the mutual information and com-pute the capacity under different constraints for thetwo-state and multi-state protein models. The achievedtheoretical rates indicate the existence of a ubiquitousmechanism for information transmission between thenanoantenna and the protein with a clear physical sig-nificance.Biological systems can be generally modelled with mi-crostates; this could refer to the covalently modified state,conformational state, cellular location state, etc. Each ofthese states defines a certain attribute related to either theprotein structure or function [17]. In our work, the biologicalmeaning of state refers to the conformational state, which weconsider as either Unfolded or Folded for the two-state model.In the case of the multi-state model, we refer to multipleintermediate states. An example is the photoactive membraneprotein,
Bacteriorhodopsin . The cycle of this protein consistsof several states including a resting state and a series ofphoto-intermediate states, each of which is associated witha conformational change [18]. The transition between proteinstates regulates biological processes, including cell signaling.Thereafter, the methodology presented in this work sheds lighton various opportunities that impact applications concerningdrug discovery, biosensing as well as disease control andprevention.The rest of the paper is organized as follows. In Sec. II,the system model of the stimulated protein signal transductionpathway is presented. In Sec. III, a communication systembased on Markov finite-states is developed to capture proteindynamics. In Sec. IV, a two-state protein model is formulated.The model is further extended and generalized to take into ac-count multi-state protein interactions in Sec. V. In Sec. VI, thenumerical results of the models are illustrated while providinga clear physical insight. Finally, we draw our conclusions inSec. VII. II. S
YSTEM M ODEL
A. The Physical Process
Living cells communicate with each other through a seriesof biochemical interactions referred to as signal transduction networks. A molecular process referred to as mechanotrans-duction, governs the transmission of mechanical signals fromthe extracellular matrix to the nucleus [19]. Proteins, whichare considered major drivers of signal transduction, displaya status change in response to mechanical stimulation. Inour work, we consider a mechanotransduction communicationchannel, composed of a nanoantenna transmitter and a proteinreceiver. We assume that the nanoantenna is tuned to a specificfrequency depending on the protein type. As such, the inter-action between the nanoantenna and the protein gives rise to amechanical response [16]. According to structural mechanics,if an external harmonic excitation has a frequency whichmatches one of the natural frequencies of the system, thenresonance occurs, and the vibrational amplitude increases [20].This is the case with protein molecules as the value of theirvibrational frequency is given as [21] f protein ≈ π (cid:114) κm . (1) κ and m are the stiffness and the mass of the protein molecule,respectively. On average, proteins have a stiffness of Nm − and a mass of − kg yielding a vibrational frequencyin the order of , thereby matching the THz nanoantennafrequencies [22].The capability to predict collective structural vibrationalmodes at THz frequencies has long attracted the researchcommunity. This interest has been fortified by the develop-ment of THz spectroscopic techniques used to investigatethe response of biomolecules [23]. In particular, vibrationscan be dipole active, and thus probed using THz dielectricspectroscopy. The detected molecular motions in the picosec-ond range correspond to collective vibrational modes or veryfast conformational changes. An extensive review by Markelzexplores measurements of the THz dielectric response onmolecules, where the author concludes that the response ishighly sensitive to hydration, temperature, binding and con-formational change [18].The investigated dielectric response of proteins includesboth a relaxational response from the amino acid side chainsalong with a vibrational response from the correlated motionsof the protein structure [15], [24]. The authors in [21] associatesuch a vibrational phenomenon with the mechanical behaviorof proteins, which act as oscillating structures in responseto THz radiation. The induced electro-chemical force allowsthe identification of relevant resonant frequencies, which mayenable a conceptual interpretation of the protein biologicalfunction. These frequencies, which range from hundreds ofGHz to tens of THz, can be mathematically captured usingmodal analysis. For instance, in lysozyme, a highly delocalizedhinge-bending mode that opens and closes the binding cleftwas found by normal mode calculations [25].In addition, measurements of chlorophyll proteins showedan increase in the THz absorbance with denaturing, whicharise due to the protein side chains’ rotational motion [26].Further, measurements reported in [14] on lysozyme proteinsshowed sharp vibrational peaks at 1.15 and 2.80 THz. Inaddition, other measurements provided in [27], showed that the Hsp70 protein, referred to as molecular chaperon, possesseddistinct spectra for protein states at sub-THz frequencies.These measurements indicate that a nanoantenna can se-lectively target the vibrational mode of the protein relatedto either folding or unfolding and induce a conformationalchange. In fact, in [28], the authors provide a descriptionof the modes of three proteins, namely, Rhodopsin, Bacteri-orhodopsin and D96N bacteriorhodopsin mutant. This givesan indication of the selectivity of these vibrational modesshowcasing the capability to single out proteins with a degreeof accuracy. In addition to initiating information flow byinducing folding behavior, stimulating proteins by EM wavesmay provide knowledge of the misfolded protein structure.This potentially makes possible future efforts to rationallydesign drugs that prevent misfolding events along with thethe evolution of certain conditions and diseases.
B. Boltzmann Distribution
Signaling inside proteins results in a spring-like effectwhich shifts their minimum energy [29]. Protein structuresare therefore investigated using energy functions where theyobey statistical laws based on the Boltzmann distribution. Onthe one hand, the energy levels of EM waves in the THzfrequency band are very low, corresponding to 1-12 meV [30],[31]. These values match energies in the range of − Joules. Since the energy expended = force × distance, andwe deal with protein conformational changes, measured innanometers [32], this will yield forces in the piconewton range.On the other hand, this energy scale conform with energiesrequired for ATP hydrolysis, ranging from k b T to k b T (here, k b is Boltzmann’s constant and T temperature in Kelvin; 1 k b T at Kelvin ≈ × − ) [32]. Thereby, utilizing aTHz force to drive a protein activity and a controlled molecularresponse is compatible with intra-body energetics.The protein conformational change from one state to anothermimics a stretch activated channel. Based on statistical me-chanics, the Boltzmann distribution provides probability thata system will be in a certain state as a function of the state’senergy and system temperature. The probability of the proteinexisting in a certain state i is P i = 1 Z exp (cid:20) − E i k b T (cid:21) , (2)where E i is the Gibbs free energy of the state and Z is anormalization factor which results from the constraint that theprobabilities of all accessible states must add up to one, i.e.,the normalization factor is given by Z = M (cid:88) i =1 exp (cid:20) − E i k b T (cid:21) , (3)where M is the number of states accessible to the proteinnetwork.In our model, the Boltzmann distribution is altered to takeinto account the nanoantenna THz force. By applying an exter-nal force, F , the average position of the mechanotransductionchannel is shifted, thereby impacting the state probability of the protein. This relation can be seen when finding the energydifference between states given as ∆ E = ∆ E ij − F ∆ (cid:96), (4)where ∆ E ij = E i − E j is the difference in Gibbs freeenergy between initial state i and final state j . ∆ (cid:96) denotesthe change in the protein length, which corresponds to aconformational change in the protein structure requiring work φ ( F ) = F ∆ (cid:96) . Gibbs free energy expresses the thermodynamicenergy reflecting the chemical potential between interactingproteins [33]. In fact, upon the change of concentration ofone molecular species, the reactions in which these molecularspecies participate are affected. Hence, a change in one proteinconcentration will percolate through the network changing itsenergy. The final result represents perturbation in the networkleading to changes in the energetic landscape, or Gibbs energyof the molecule [34]. If the protein is subject to a force, anatural reaction coordinate is the length of the protein in thedirection of the force, and the total energy difference is givenin (4). C. Stochastic Model of Protein Folding
To model the stochasticity of proteins involved upon trigger-ing them by a THz force, we use the kinetic master equation atthe single protein level since it captures the chemical kineticsof the receptor [35]. Such approach is similar to the onespresented in [36]–[38]. A transition rate matrix R describesthe rate at which a continuous time Markov chain movesbetween states. Elements r ij (for i (cid:54) = j ) of matrix R denotethe rate departing from state i and arriving in state j . Diagonalelements r ii are defined such that r ii = (cid:88) j (cid:54) = i r ij . (5)In addition, the probability vector, p ( t ) , as a function of time t satisfies the transition rates via the differential equation d p ( t ) dt = p ( t ) R. (6)To represent the protein change of state as a discrete-timeMarkov chain, we discretize the time into steps of length ∆ t .As such, the master equation provided in (6) becomes d p ( t ) dt = p ( t ) R = p ( t + ∆ t ) − p ( t )∆ t + o (∆ t ) . (7)We neglect the terms of order o (∆ t ) and manipulate (7) tohave p ( t + ∆ t ) = ∆ t p ( t ) R + p ( t ) = p ( t )( I + ∆ tR ) , (8)where I is the identity matrix. If we denote p i = p ( i ∆ t ) , wearrive at a discrete time approximation to (8) as, p i +1 = p i ( I + ∆ tR ) . (9)Thus, we obtain a discrete-time Markov chain with a transitionprobability matrix Q given as Q = I + ∆ tR. (10) III. P
ROTEIN C ONFORMATIONAL I NTERACTION AS A C OMMUNICATION S YSTEM
We now discuss how induced protein interactions can bedescribed as information-theoretic communication systems:that is, in terms of input, output, and conditional input-output probability mass function (PMF). The channel inputis the nanoantenna force transmitted to the protein receptor:at the interface between the receptor and the environment, thereceptor is sensitive to the induced force, undergoing changesin configuration as force is applied. The channel output is thestate of the protein. A Markov transition PMF dictates theinput-output relationship since the protein state depends onboth the current input and the previous state. This relationshipis given as p Y | X ( y | x ) = n (cid:89) i =1 p Y i | X i , Y i − ( y i | x i , y i − ) , (11)where p Y i | X i , Y i − ( y i | x i , y i − ) is provided according to theappropriate entry in matrix Q given in (10) and n is the fixedchannel length.For any communication system with inputs x and outputs y , the mutual information, I ( X ; Y ) , provides the maximuminformation rate that may be transmitted reliably over thechannel for a given input distribution. Maximizing this mutualinformation over the input distribution provides the channelcapacity. This analysis is important in order for us to identifythe maximum rate by which a protein can receive informationand, thereby, we assess the impact of THz force on commu-nication. For tractability, we restrict inputs to the set of IIDinput distributions, where p X ( x ) = (cid:81) ni =1 p X ( x i ) . The authorsin [39] showed that the IID input distribution was capacityachieving (i.e., max achievable rate) for two-state intensity-driven Markov chains. T he protein state y forms a time-homogeneous Markov chain given as p Y ( y ) = n (cid:89) i =1 p Y i | Y i − ( y i | y i − ) , (12)where y is null and p Y i | Y i − ( y i | y i − ) = (cid:88) x i p Y i | X i , Y i − ( y i | x i , y i − ) p X ( x i ) . (13)The mutual information can be written as I ( X ; Y ) = n (cid:88) i =1 (cid:88) y i (cid:88) y i − (cid:88) x i p Y i , X i , Y i − ( y i , x i , y i − )log p Y i | X i , Y i − ( y i | x i , y i − ) p Y i | Y i − ( y i | y i − ) . (14)Thereafter, the channel capacity is given as C = max p X ( x ) I ( X ; Y ) . (15)In our analysis, we deal with the input, x , as either a discreteor continuous parameter. We use the bisection method tocompute the capacity for the discrete case and deploy theBlahut-Arimoto (BA) algorithm to find the capacity for thecontinuous scenario. In fact, given an input-output transition Fig. 1. Two-state protein model represented by unfolded ( U ) and folded ( F )Markov states. matrix, the classical BA algorithm is a general numericalmethod for computing the capacity channel [40]. The max-imization of the mutual information is attained through analternating maximization procedure to the global maximum. Avariation of the BA algorithm is the constrained BA method,which incorporates an average power constraint on the channelinputs.We provide several capacity measures with different con-straints for the EM-triggered protein communication channel.Specifically, we derive the capacity per channel use and withaverage energy constraint. Capacity per channel use is asuitable measure in applications involving targeted therapyor targeted drug delivery. The capacity with an average en-ergy constraint is a useful measure for efficient intra-bodycommunication, where both medium compatibility and safetymetrics are practical constraints accounted for. In each case,the optimum input distribution and the resulting maximizedcapacity measures are attained.IV. T WO -S TATE P ROTEIN M ODEL
A. Mathematical Model
In our two-state model, the protein resembles a binarybiological switch, represented using a finite-state Markovchain. The states of the protein depicted are the folded, F ,and unfolded, U , as those govern the activation of biolog-ical processes and chemical interactions. The input to ourmechanotransduction channel is the force induced by thenanoantenna, while the output is the state of the protein. Incontinuous time, the protein folding can be represented asa Poisson process, transitioning between F and U . We let p Y ( t ) = [ p F ( t ) , p U ( t )] denote the time-varying vector of stateoccupancy probabilities.As demonstrated in Fig. 1, in this system, the transitionrate from unfolded, U , to folded, F , is α , while the transitionrate from F to U is β . The latter transition is considered arelaxation process which returns the protein to the unfoldedstate. Such process is independent of the excitation signal sinceprotein folding is entropically unfavorable [41]. The mainreason for protein to get folded is to acquire its function. Thefunction implies a general architecture of the protein whichhas to be stable in time and flexible enough to allow thebiological process to occur. Therefore native state of a proteinis not necessarily the most stable one. To model the two-state conformational change which captures the behavior ofa protein, the normalization factor, provided in (3), is givenby Z = exp (cid:20) − E U k b T (cid:21) + exp (cid:20) − E F k b T (cid:21) , (16) where E U and E F denote the Gibbs free energies associatedwith the unfolding and folding states, respectively. As such,the steady-state probability of the protein being in one state,the folded for example, can be found from (2) and (16) as p Y ( y = F ) = 11 + exp (cid:104) ∆ Ek b T (cid:105) . (17)The transition rates controlling such two-state interaction aregiven by the rate matrix R as R = (cid:20) − α αβ − β (cid:21) . (18)From (10), the transition probability matrix yields Q = (cid:20) − α ∆ t α ∆ tβ ∆ t − β ∆ t (cid:21) . (19) B. Kinetic Detailed Balance
The steady state probability is the eigenvector of thestochastic matrix, which can be found using the followingrelation p Y ( y ) Q = p Y ( y ) . (20)Hence, for our two-state Markov model the steady-states yield p Y ( y ) = (cid:40) αα + β , y = F βα + β , y = U . (21)The relationship between α and β can therefore be found byequating (17) and (21) for y = F , resulting in β = α exp (cid:18) ∆ Ek b T (cid:19) . (22)(22) satisfies the detailed balance theory, which has beenformulated for kinetic systems [42]. Detailed balance ensuresthe compatibility of kinetic equations with the conditions forthermodynamic equilibrium. The rate constants pulling againstan applied force resembles a biased random walk that allowsthe protein to perform work per unit step, i.e., φ ( F ) = F ∆ (cid:96) ,in agreement with the second law of thermodynamics and asshown in (4).Since the value of the energy, ∆ E , gets altered whenthe system is subject to an external force, the value of α (the probability of the forward transition rate) will alsovary accordingly. As such, α can be divided into α NF , thenatural transition rate when no force is applied, and α AF , thetransition rate when a force is applied, resulting in an averagefolding probability. The values of α NF and β for differentproteins can be found from experimental studies available inthe literature since protein folding is a naturally occurringphenomenon driven by the change in Gibbs energy [43].Therefore, (22) can take two different forms depending onwhether the system is being subject to an external force ornot as follows β = α NF exp (cid:18) ∆ Ek b T (cid:19) , ∆ E = ∆ E ij (23) α AF exp (cid:18) ∆ Ek b T (cid:19) , ∆ E = ∆ E ij + φ ( F ) (24)Here, NF and AF correspond to No Force and Applied Force,respectively. C. Capacity of Two-State Protein Conformation1) Discrete Case:
Based on our developed model, we let x denote a binary input which stimulates the protein. Thisinput is induced either due to intra-body interactions with noexternal force or could be triggered due to an applied THznanoantenna force, in which x ∈ { NF , AF } . The channeloutput is the state of the protein given as either unfolded orfolded, where y ∈ { U , F } . We have, as a result, a discretechannel, where the inputs and outputs form vectors. In order tofind the capacity, we follow the formulation presented in Sec.III. Assuming the previous state of the protein, y i − = U , wehave p Y i | Y i − ( F | U ) = (cid:88) x i p Y i | X i , Y i − ( F | x i , U ) p X ( x i )= p NF α NF + p AF α AF = ¯ α, (25)and p Y i | Y i − ( U | U ) = 1 − ¯ α . Here, ¯ α represents the averagefolding probability. On the other hand, if y i − = F , p Y i | Y i − ( U | F ) = (cid:88) x i p Y i | X i , Y i − ( U | x i , F ) p X ( x i )= β, (26)and p Y i | Y i − ( F | F ) = 1 − β . The transition probability matrixprovided in (19) can now be written as ¯ Q = (cid:20) − ¯ α ∆ t ¯ α ∆ tβ ∆ t − β ∆ t (cid:21) . (27)In addition, the steady state probabilities given in (21) areadjusted to take into account the average folding probability, ¯ α . The mutual information, I ( X ; Y ) , which was given in (14),can also be represented as I ( X ; Y ) = H ( Y i | Y i − ) − H ( Y i | X i , Y i − ) , (28)for i ∈ { , , ..., n } . To compute (28), we use the binaryentropy function as follows H ( p ) = − p log p − (1 − p ) log(1 − p ) . (29)Then, each term in the right hand side of (28), is dealt withseparately. H ( Y i | Y i − ) yields = p Y ( U ) H ( Y i | Y i − = U ) + p Y ( F ) H ( Y i | Y i − = F )= β ¯ α + β H (¯ α ) + ¯ α ¯ α + β H ( β ) . (30)In a similar manner, H ( Y i | X i , Y i − ) results in = (cid:88) x i p X ( x i ) p Y ( U ) H ( Y i | X i = x i , Y i − = U )+ (cid:88) x i p X ( x i ) p Y ( F ) H ( Y i | X i = x i , Y i − = F )= β ¯ α + β ( p NF H ( α NF ) + p AF H ( α AF )) + ¯ α ¯ α + β H ( β ) . (31)By substituting back into (28), the mutual information yields I ( X ; Y ) = β ¯ α + β ( H (¯ α ) − p NF H ( α NF ) − p AF H ( α AF )) . = H ( p NF α NF + p AF α AF ) − p NF H ( α NF ) − p AF H ( α AF )1 + ( p NF α NF + p AF α AF ) /β . (32) Finally, the capacity of the two-state model is found bymaximizing (32) with respect to the nanoantenna applied forceas C = max p AF H ( p NF α NF + p AF α AF )1 + ( p NF α NF + p AF α AF ) /β + − p NF H ( α NF ) − p AF H ( α AF )1 + ( p NF α NF + p AF α AF ) /β . (33)It is sufficient to maximize over p AF since p NF = 1 − p AF .
2) Continuous Case:
In the previous part, we developed themodel as a discrete case given a binary input binary outputsystem. Nonetheless, an in-depth picture for the capacityassociated with protein conformational transitions is attainedby applying a continuous input. By having the nanoantennaforce transmit continuously, the capacity versus applied forcecan be studied over a range of values. This is achieved byexpanding ¯ α in (25) to become ¯ α = α NF p NF + N − (cid:88) i =1 α AF ( f i ) p AF ( f i ) , (34)where p AF ( f i ) denotes the probability of applying a force, f i , towards the protein. The dependency of α AF on the forcefactor has been demonstrated in (24).We find the capacity for the two-state model under theconstraint of a maximum applied force per channel use asmax p AF I ( X ; Y ) subject to ≤ F applied ≤ F max . (35) F max in this case is the maximum amount of nanoantennaapplied force and p AF is the probability vector of appliedforces. The objective function in (35) is concave with respectto the input probability vector and the constraint is linear;hence, the optimization problem is concave. Therefore, the so-lution of the problem can be obtained using the BA algorithm.The algorithm begins with the transition probability matrix,initially defined in (27), but extended to take into accountthe N maximum force samples along with an arbitrary butvalid, choice for p AF . Since the mutual information in (35)is concave in terms of the input probability, the output of thealgorithm is the optimal, capacity-achieving, input probabilitydistribution, ˆ p AF .V. M ULTI -S TATE P ROTEIN M ODEL
A. Mathematical Model
Successive events occur inside a living cell through asequence of protein activation in which signaling cascades areoften illustrated by kinetic schemes. Although a node in anetwork is represented by a single protein, the protein itselfcan have multiple gene products with many conformations.Each node of the protein can slightly differ in sequence. Suchdifferences allow a node to bind with hundreds of partners atdifferent times and perform many essential biological func-tions [44].In this section, we further extend the two-state protein con-formation model to consider the transition between different protein configurations in order to more accurately resemble theprotein signaling pathway especially when there are multiplefolding routes from different starting points [45]. As such,we generalize the two-state model presented previously totake into account multiple-states. The selectivity attained byusing THz signals allows us to target specific links in agiven network in order to create controlled interactions. Thesemacroscopic interactions resemble the creation or removal ofedges between nodes in a graph [46]. By targeting the THzforce on specific locations of the protein molecule, distinctresponses can be induced.We let p Y ( t ) = (cid:2) p y ( t ) , p y ( t ) , ...., p y m +1 ( t ) (cid:3) be the prob-ability vector accounting for n = m + 1 states and m links.In this case, the generalized rate matrix yields R = − α α .... ....β − ( β + α ) α .... .... β − ( β + α ) α .... .... : : : : : :: : : : β m − β m . (36)Following the same formulation presented in (10), the gen-eralized probability matrix is given in (37). We note thatthroughout the analysis, we will use ¯ Q rather than Q , whereeach α j is replaced by ¯ α j , indicating an average state changeprobability.To compute the mutual information, I ( X ; Y ) , for the multi-state conformational model, we follow the same approach asin the previous section, where we provide a generalizationof the formulation. First, following (28), we first compute H ( Y i | Y i − ) as = p Y ( y ) H (¯ α ) + m (cid:88) j =2 p Y ( y j ) (cid:18) H ( β j − ) + H (¯ α j ) (cid:19) + p Y ( y m +1 ) H ( β m ) . (38)Then, we find H ( Y i | X i , Y i − ) as = p Y ( y ) (cid:18) p AF H ( α AF ) + p NF H ( α NF ) (cid:19) + m (cid:88) j =2 p Y ( y j ) (cid:18) H ( β j − ) + (cid:18) p AF j H ( α AF j ) + p NF j H ( α NF j ) (cid:19)(cid:19) + p Y ( y m +1 ) H ( β m ) . (39)Substituting back in (28) we get I ( X ; Y ) = m (cid:88) j =1 p Y ( y j ) H (¯ α j ) − m (cid:88) j =1 p Y ( y j ) (cid:18) p AF j H ( α AF j ) + p NF j H ( α NF j ) (cid:19) . (40)The capacity of the multi-state protein model is found bymaximizing (40) with respect to the nanoantenna applied force ¯ Q = − ¯ α ∆ t ¯ α ∆ t ... ...β ∆ t − ( β + ¯ α )∆ t ¯ α ∆ t ... ... β ∆ t − ( β + ¯ α )∆ t ¯ α ∆ t ... ... : : : : : :: : : : β m ∆ t − β m ∆ t . (37)as C = max p AF (cid:20) m (cid:88) j =1 p Y ( y j ) H (¯ α j ) − m (cid:88) j =1 p Y ( y j ) (cid:18) p AF j H ( α AF j ) + p NF j H ( α NF j (cid:19)(cid:21) . (41)In this case, p AF is a vector constituting the probability offorce applied to the m links. B. Example: Four State Protein Model
Fig. 2. Multi-state protein model with several transitions.
To show the applicability of the protein multi-state model,we apply it to a 4 state protein chain. We have the probabilityoccupancy vector as, p ( t ) = [ p A ( t ) , p B ( t ) , p C ( t ) , p D ( t )] . Therelationship between the states is formulated using a Markovtransition PMF, which is previously given in (11) and (13).Hence, based on Fig. 2, if the previous state, y i − = A , wehave p Y i | Y i − ( B | A ) = (cid:88) x i p Y i | X i , Y i − ( B | x i , A ) p X ( x i )= p NF α NF + p AF α AF = ¯ α , (42)and p Y i | Y i − ( A | A ) = 1 − ¯ α . On the other hand, if y i − = B , p Y i | Y i − ( A | B ) = (cid:88) x i p Y i | X i , Y i − ( A | x i , B ) p X ( x i )= β , (43)and p Y i | Y i − ( B | B ) = 1 − ( β + ¯ α ). The relationship betweenthe remaining states follows accordingly.Using (20), the steady state probabilities are found as p Y ( y ) = β β β β β β +¯ α β β +¯ α ¯ α β +¯ α ¯ α ¯ α , y = A ¯ α β β β β β +¯ α β β +¯ α ¯ α β +¯ α ¯ α ¯ α , y = B ¯ α ¯ α β β β β +¯ α β β +¯ α ¯ α β +¯ α ¯ α ¯ α , y = C ¯ α ¯ α ¯ α β β β +¯ α β β +¯ α ¯ α β +¯ α ¯ α ¯ α , y = D (44) In (44), we have considered the steady states after a forcehas been applied to the system, i.e., each α j is replaced by ¯ α j . We note also that the same relationship between α and β holds as (22) in Sec. III. Finally, both the mutual informationand capacity are found by substituting the given states in (40)and (41) accordingly. C. Capacity with Average Energy Constraint
A variation on the optimization in (35) is when the averageenergy of applied nanoantenna force per channel use is alsoconstrained. In this case, the constrained BA algorithm isdeployed to find the capacity of the multi-state protein model.The resulting optimization problem is given asmax p AF I ( X ; Y ) subject to (cid:88) i p AF i E i (cid:54) E max , ≤ p AF i ≤ . (45) E i is the energy applied to link i . The capacity with averageenergy constraint E max is defined as C = max p AF (cid:34)(cid:88) i p AF i ¯ Q log ¯ Q (cid:80) i p AF i ¯ Q − λ ( (cid:88) i p AF i E i − E max ) (cid:35) . (46)Here, ¯ Q is the transition probability matrix defined in (37). Thecost function in (46) is parametrized using Lagrange multiplier λ . The procedure followed to optimize the input distributionis similar to that without the average energy constraint. Theadditional step involves obtaining a value for λ after updatingthe distribution vector p AF . This can be obtained using asimple bisection search.VI. N UMERICAL R ESULTS
In this section, we demonstrate the results of numericallysimulating our developed models. The aim of the presentedwork is to find the information rates by which proteinmolecules convey information when triggered by THz nanoan-tennas. Several scenarios are presented to take into accountdifferent protein configurations undergoing either single ormultiple signaling interactions.
A. Discrete Case Result
In our discrete scenario, the system is binary, where thenanoantenna force is either present or absent as mathematically formulated in Sec. IV. The mutual information is calculatedfrom the analytically derived model and the capacity is com-puted using a bisection search. This method is guaranteed toconverge to a root, which is the value of p AF that maximizesthe capacity in our case. The discrete scenario proves theexistence of a communication channel, where information canbe transmitted upon triggering the protein by THz EM waves.Figs. 3 and 4 illustrate the mutual information curves for β = 0 . and β = 0 . , respectively. The value of α NF isfixed to . while the values of α AF vary for both cases. Asexpected, the higher the value of α AF , the higher the capacitysince the value of α AF corresponds to the probability offolding. In addition, we notice that higher values of β indicatea higher capacity. This observation can be deduced from (32),where an increased value of β corresponds to a higher value of I ( X ; Y ) . The values of p AF which maximize the capacity areclearly indicated using circles on the demonstrated 2D plotsof the mutual information curves. p AF α AF M u t ua l I n f o r m a t i on ( b i t s / c hanne l u s e ) (a) p AF M u t ua l I n f o r m a t i on ( b i t s / c hanne l u s e ) (b)Fig. 3. (a) 3D contour plot of the mutual information curve where p AF and α AF are varied. (b) 2D plot showing the maximizing values of p AF bycircles. α NF = 0 . and β = 0 . , while α AF varies from the bottom from . to . with a . increment. B. Capacity Per Channel Use Result
For the case of a continuous force, the BA algorithm isdeployed to find the capacity. The attained result furtherfortifies the discrete case by providing a more detailed analysisof how the capacity varies as a function of force. We utilize the p AF α AF M u t ua l I n f o r m a t i on ( b i t s / c hanne l u s e ) (a) p AF M u t ua l I n f o r m a t i on ( b i t s / c hanne l u s e ) (b)Fig. 4. (a) 3D contour plot of the mutual information curve where p AF and α AF are varied. (b) 2D plot showing the maximizing values of p AF bycircles. α NF = 0 . and β = 0 . , while α AF varies from the bottom from . to . with . increment. relationships given in (34) and (35) to simulate this scenario.Protein conformational changes are measured in nanometers(nm) and forces are given on the scale of piconewtons(pN) [47]. The value for the protein conformational distancewas fixed at ∆ (cid:96) = 2 nm for maximum forces ranging between − pN. The selected force range of the nanoantennareflects THz transmissions based on intra-body link budgetanalysis [4] and force sensitivity at the cellular level [16].Fig. 5 demonstrates the capacity as a function of the appliednanoantenna force. We observe that given a fixed value of β and α NF , the value of the capacity increases upon increasingthe nanoantenna applied force. In addition, the higher the valueof α NF , the higher the achieved capacity for the value of β = 0 . . In order to understand such behavior, the change inGibbs free energy, ∆ E ij , must be examined. In fact, ∆ E ij iscomputed using the relationship presented in (23), which isrearranged to yield ∆ E ij = k b T ln (cid:20) α NF β (cid:21) . (47)By increasing the value of α NF , ∆ E ij witnesses incrementsuntil it approaches equilibrium ( ∆ E ij = 0 ) at α NF = 0 . . Theequilibrium state indicates a chemical balance, where no workshould be done on the system as it is currently in a stable state. As such, the amount of force directed from the nanoantennawill be solely dedicated towards increasing the capacity atwhich the protein receives information. Hence, no force will belost in order to first stabilize the system and then contribute tothe capacity. Even for low values of α NF , a capacity-achievingchannel is attained upon applying a force. This indicates thatthe presented EM-molecular interface allows transmission ofinformation under different biological scenarios, where the EMforce can be regarded as a powerful tool that controls theenergy pathways of proteins. Maximum Force (pN) C hanne l C apa c i t y ( b i t s / c hanne l u s e ) α NF =0.3 α NF =0.5 α NF =0.7 α NF =0.9 Fig. 5. The channel capacity as a function of the nanoantenna applied force.The value of β is fixed to . while the value of α NF varies. C. Capacity Result with Average Energy Constraint
For the multi-state protein model formulated in Sec. V,we opt to find the capacity by which a cascade of proteinconfigurations transduce information and carries out inter-actions upon THz stimulation. This scenario sparks a re-semblance of enzymes and receptors that are activated viaprotein phosphorylation. In addition, the selectivity providedby using a THz nanoantenna allows us to control α AF bygoverning p AF applied to each link and therefore bias ournetwork in a specific direction. The constrained BA algorithmis deployed, where an average energy constraint is applied tothe capacity as formulated in Sec. V-C. For simulations, wewill use the model illustrated in Fig. 2, constituting of 4 proteinstates. We examine different values of α NF while assuming α NF = α NF = α NF . The value of β is studied when it iseither fixed or varied for the three links. By selecting differentvalues of β , we can analyze how forward transition rates areimpacted as nanoantenna force is being applied to the system.
1) Fixed β : Since protein interaction reflects a biologicalphenomenon, a protein network will favor the condition whichachieves equilibrium. As such, at equilibrium, the system willalways have the highest capacity as indicated by Figs. 6and 7. The results match the conclusion achieved in Sec. VI-B,indicated by (47). When the system is out of equilibrium, heatdissipation occurs and work should be done to bring the systemback to equilibrium, therefore reducing the attained capacity.It can be also noticed that the maximum achieved capacity ofFigs. 6 and 7 is lower compared to Fig. 5. This is attributed tothe energy constraint set by E max in (46). The chosen E max value corresponds to the typical energy consumed by a motorprotein [32]. Maximum Force (pN) C hanne l C apa c i t y ( b i t s / c hanne l u s e ) α NF =0.3 α NF =0.5 α NF =0.7 α NF =0.9 Fig. 6. The channel capacity for the multi-state protein model as a functionof the nanoantenna applied force. The value of β is fixed to . for the threelinks while the value of α NF varies. Maximum Force (pN) C hanne l C apa c i t y ( b i t s / c hanne l u s e ) α NF =0.1 α NF =0.3 α NF =0.5 α NF =0.7 Fig. 7. The channel capacity for the multi-state protein model as a functionof the nanoantenna applied force. The value of β is fixed to . for the threelinks while the value of α NF varies.
2) Different β : Figs. 8 and 9 show the channel capacity forthe multi-state protein model as a function of the nanoantennaforce when the value of β is set different for each link.The capacity of the system depends on the combination of β and α NF for the three links as reflected from the mutualinformation formula. The maximum capacity is achieved whenthe overall free energy values of the system, composed inour case of the three links, is closest to equilibrium. Thisrelationship is deduced from (47) and is given as ∆ E ij = k b T m (cid:88) k =1 ln (cid:20) α NF k β k (cid:21) . (48)This case resembles a more realistic intra-body scenariobecause unfolding rates between protein intermediates are notnecessarily equal. Our results match the fact that physical sys-tems in equilibrium have a statistical tendency to reach statesof maximum entropy or minimum Gibbs free energy [33].VII. C ONCLUSION AND D ISCUSSION
In this paper, we present a communication system whichbridges the link between EM nanonetworks and molecular Maximum Force (pN) C hanne l C apa c i t y ( b i t s / c hanne l u s e ) α NF =0.3 α NF =0.5 α NF =0.7 α NF =0.9 Fig. 8. The channel capacity for the multi-state protein model as a functionof the nanoantenna applied force. The value of β is different for each linkwhere β = 0 . , β = 0 . , β = 0 . . Maximum Force (pN) C hanne l C apa c i t y ( b i t s / c hanne l u s e ) α NF =0.3 α NF =0.5 α NF =0.7 α NF =0.9 Fig. 9. The channel capacity for the multi-state protein model as a functionof the nanoantenna applied force. The value of β is different for each linkwhere β = 0 . , β = 0 . , β = 0 . . paradigms. The developed stimuli-responsive system consti-tuting of a nanoantenna transmitter and a protein receiver,paves the way towards controlled intra-body interactions ata molecular level. The key idea relies on stimulating theprotein vibrational modes to induce a change in their state.Protein conformational changes activate biochemical eventsthat transduce through intra-body pathways.The presented mathematical model uses the Boltzmanndistribution to represent the system states. For the commu-nication channel, a Markov chain finite-state model is used torepresent the system inputs and outputs. Both a two-state and amulti-state protein model are developed. In the former model,the focus is on a single folding and unfolding interactionwhich results in a controlled biological change in the mediumfollowed by a cascade of reactions. Such a model is inspiredfrom mechanosensitive channels that adopt two fundamentalconformational channel states separated by an energy barrier.In the latter model, we investigate a series of interactionsrepresenting a protein undergoing intermediate changes inconfiguration, where we generalize the presented two-statemodel. Expressions for the mutual information are derived forboth cases, indicating the possible information rates achievedby stimulating proteins by THz nanoantennas. Several capacity constraints are also introduced to make sure the system iscompatible with the intra-body medium.The results attained indicate a feasible communication plat-form for information transmission between the nanoantennaand the protein. It also expresses a fundamental link betweenkinetics and thermodynamics since protein interactions favorconditions of equilibrium even when an external force isapplied to the system, which shows that the results adhereto the second law of thermodynamics. The results agree withthe fact that a time-homogeneous Markov chain converges tothe Gibbs equilibrium measure, i.e., thermal equilibrium. Inessence, the concept of mutual information introduced in thiswork not only indicates the amount of information the proteinsignaling pathway carries but can also be further interpretedin terms of molecular disorder, where the highest capacity isobtained when minimum energy is lost. Such a conclusionwill result in various medical opportunities where proteins arecontrolled and directed towards certain favorable interactions.As a future direction, we aim to present a mathematicalmodel that captures the interaction between THz waves andprotein dynamics from a mechanical perspective. This involvesstudying the resonance response associated with protein con-formational changes by modeling the protein as a large set ofcoupled harmonic oscillators. The mechanical model must beintegrated with the current work in order to have a completesystem that relates the triggered natural frequencies of proteinsto the probability of folding. In addition, the authors wouldlike to further study the relationship between THz waves andmisfolded proteins associated with neurodegenerative diseases.This involves understanding how THz waves may alter thepathological mechanisms and how this knowledge can bereflected to develop disease-modifying therapeutic strategies.R EFERENCES[1] N. Farsad, H. B. Yilmaz, A. Eckford, C.-B. Chae, and W. Guo, “AComprehensive Survey of Recent Advancements in Molecular Commu-nication,”
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Hadeel Elayan (S’12) is currently a PhD Candi-date in the Electrical and Computer Engineeringdepartment at the University of Toronto, Canada.Her research interests include Nanonetworks, Ter-ahertz Intra-body Communication as well as Molec-ular Communication. Hadeel completed a researchinternship at the Ultra-broadband NanonetworkingLab, University at Buffalo, USA during summer2016. She worked as a Research Associate in theHealthcare Engineering Innovation Center, KhalifaUniversity until August 2018. Hadeel received sev-eral awards for her research and academic excellence including the 2016 IEEEPre-doctoral Research Grant Award, the 2017 Photonics School InternshipAward from KAUST and the 2019 Ontario Graduate Scholarship. Andrew Eckford is an Associate Professor in theDepartment of Electrical Engineering and ComputerScience at York University, Toronto, Ontario. Hisresearch interests include the application of infor-mation theory to biology, and the design of com-munication systems using molecular and biologicaltechniques. His research has been covered in mediaincluding The Economist, The Wall Street Journal,and IEEE Spectrum. His research received the 2015IET Communications Innovation Award, and was afinalist for the 2014 Bell Labs Prize. He is also a co-author of the textbook Molecular Communication, published by CambridgeUniversity Press.