The Role of Energy Cost on Accuracy, Sensitivity, Specificity, Speed and Adaptation of T Cell Foreign and Self Recognition
TThe Role of Energy Cost on Accuracy, Sensitivity, Specificity, Speed andAdaptation of T-Cell Foreign and Self Recognition
Gyubaek Shin a and Jin Wang ab ∗ The critical role of energy consumption in biological systems including T-Cell discrimination process has been investigated in variousways. The kinetic proofreading(KPR) in T-Cell recognition involving different levels of energy dissipation influences functionaloutcomes such as error rates and specificity. In this work, we study quantitatively how the energy cost influences error fractions,sensitivity, specificity, kinetic speed in terms of Mean First Passage Time(MFPT) and adaption errors. These provide the backgroundto adequately understand T-Cell dynamics. It is found that energy plays a central role in the system that aims to achieve minimum errorfractions and maximum specificity with the fastest speed under our kinetic scheme, but such an optimal condition is accomplished atsignificant amount cost of energy and sensitivity. Starting with the application of steady state approximation(SSA) to the evaluation ofthe concentration of each complex produced associated with KPR, which is used to quantify various observables, we present bothanalytical and numerical results in detail.
One of the well known biological malfunctions is the deviationfrom the normal condition of being able to maintain the abilityto efficiently differentiate foreign antigens from self-proteinsattacking living cells. It may be associated with an abnormality ofKPR processes, which prevents a bound form of “wrong” ligandsfrom being dissociated at a sufficiently high rate. The affinityratio of “correct” and “wrong” ligands with T-cell receptor istypically a measurable quantity that determines the efficiency ofsuch dissociation. Hopfield and Nino developed KPR theory inbiosynthetic processes.
Hopfield formulated error fractions forprotein synthesis. They elucidated that enzymes discriminatetwo different reaction pathways, leading to correct or incorrectproducts due to KPR. Since then, extensive researches on sen-sitivity and specificity associated with error fractions have beenperformed. Goldbeter et al found that covalent modification inprotein involving biological systems affects a sensitivity amplifica-tion using Steady State Approximation(SSA). A series of modifications after ligand binding in the KPR processinvolves extra steps which creates “time delay” τ . The extra stepsleading to signaling are critical factors that allow for reduction inerror rates, indicating high efficiency of kinetic discrimination. However, KPR also involves free energy cost for activation of an a Department of Chemistry, SUNY Stony Brook - 100 Nicolls Road, Stony Brook, NY11794, USA b Department of Physics and Astronomy, SUNY Stony Brook - 100 Nicolls Road, StonyBrook, NY 11794, USA. E-mail: [email protected] initially formed complex, which occurs in nonequilibrium states. The energy is also crucial in reducing the error rates and allowingincreased specificity. Before KPR attracted great interests, therehad been several studies focusing on the effect of energy cost forKPR in biological processes such as tRNA aminoacylation andso on.Beyond the classical studies on discrimination process for bio-logical systems such as Hopefiled, Nino, and McKeithan, therehas been a fair amount of accomplishment on T-cell recognitionwith certain modifications, which make it possible to addressseveral deficiencies found in existing models. For example, Qiancalculated an error fraction depending on both KPR steps andenergy cost using the Successive Rapid Equilibrium Approximation(SREA) by assuming that there is energy input for only the firstcycle. Chen et al. are the ones provided a the formulation ofT-Cell sensitivity and specificity in a quantitative manner, depend-ing on the number of KPR steps using a SSA. There have beensignificant contributions from Cui and Banerjee, focusing onthe detailed relationship between error rates and MFPTs. Despitetheir efforts on detailed analysis of the dynamics, their studies arebased on the kinetic scheme in terms of only energy cost, lackingcomprehensive information for which both KPR steps and energydissipation are taken into consideration. For convenience, we usethe term “KPR steps” instead of phosphoylation steps althoughtechnically a KPR process includes both phosphorylations and thedissociation of each intermediate product. Here, the followingquestions can be raised:(1) If energy consumption plays an central role in reducing1 a r X i v : . [ q - b i o . M N ] M a y KINETIC SCHEME DESCRIBING T-CELL RECOGNITION errors, how does energy influence sensitivity and specificity inT-Cell discrimination process, and what are the relationshipsamong error fraction, sensitivity and specificity under identicalconditions?(2) Although two factors, the KPR steps and energy dissipation,both of which contribute to editing process of the system haveopposite effects in terms of the time required to complete theassociated process, the retardation due to the increased KPR stepsmay be mitigated by sufficient level of energy dissipation. Canthe MFPT data provide adequate information to determine suchenergy level under the given condition?(3) What is the appropriate stimulus signal that promotes adapta-tion of the corresponding system?In order to answer the above questions, we design a kineticmodel describing T-Cell discrimination process. After introducingthe chosen model for our discussion, this paper shows a detailedprocedure leading to analytical expressions for these quantitiesbased on the SSA by imposing energy input in “every step”since a series of modifications that occur after ligand bindingrequires energy consumption and is out of equilibrium. Based onthe kinetic model, we calculated error fractions, sensitivity andspecificity in terms of both KPR steps and energy cost.We also calculated the kinetic speed in terms of MFPT, theaverage time required to complete the signaling event startingfrom an initial state, depending on energy with given KPR steps.The entire picture of the dynamics in T-Cell recognition willstill remain unclear with the only sensitivity and specificity dataavailable until the consequences of MFPT are evaluated. This isbecause the MFPT provides information on the time required forsignaling to be completed under energy dissipation. We also seehow energy influences the adaptation errors in response to theshift of a particular parameter, which is the rate constant used inour kinetic model.
The following detailed kinetic scheme reflects KPR associated withthe energy consumption. This scheme is based on the McKeithan’skinetic model.
The initial complex formed by a T-cell recep-tor and equal amounts of foreign and self ligands triggers a se-ries of modifications, leading to signaling. Since the first com-plex reaches equilibrium rapidly, the values of governing rate con-stants k and k − for the corresponding forward and backward re-actions, respectively are substantially higher than the ones givenby k p and k − p for the rest of the reactions. The dissociation eventsat each intermediate complex leading to its initial state with therate of k ∗− i (i=1,2,...N) allow for a reduction in the amount ofligand bound molecules. We set the same value of the dissoci-ation constant( k disso ) for each intermediate complex for simplic-ity. There is a need to incorporate the rate constants govern-ing the direct formation process, which leads to the developmentof the complexes without passing through earlier steps into thefull “rate equation”. The direct formation constants denoted by m i ( i = , ,... N ) decrease with the KPR steps due to the higher en- ergy intermediates as KPR progresses. In addition to this, theyalso decrease with consumed energy according to the formula forenergy dissipation. We allow variation of the backward rates andthe direct formation rates so that they decrease with energy. How-ever, the reverse rate constant k − that is associated with the fastequilibrium is unchanged. The transfer rate “W” is included in theirreversible process from the final complex to the absorbing sitewhere the associated dynamics is completed. The equilibrium ATPand ADP concentrations are related to the rate constants. [ AT P ] eq [ ADP ] eq = k ◦− p [ C ] eq k ◦ p [ C ] eq = k ◦− p m k − k ◦ p k k ∗− (1)where k p and k − p are pseudo first order rate constants denoted by k p = k ◦ p [ AT P ] and k − p = k ◦− p [ ADP ] respectively.The second equality comes from the relationship between theratio of the equilibrium concentration and kinetic constants (i.e) [ C ] eq [ C ] eq = k − m k ∗− k .The free energy of ATP hydrolysis is given as ∆ G DT = ∆ G ◦ DT + RT ln (cid:18) [ AT P ][ ADP ] (cid:19) = RT ln (cid:18) k p k k ∗− k − p m k − (cid:19) (2)where ∆ G ◦ DT = − RT ln (cid:18) [ AT P ] eq [ ADP ] eq (cid:19) (3)and we can define γ as the available free energy from each ATPhydrolysis as. γ = k p k k ∗− k − p m k − ( N = ) (4) γ = k p k ∗− ( i + ) m i k − p m i + k ∗− i ( N > ) (5)where k ∗− i (i=1,2,...N) is all the same.The affinity ratio between “wrong”(self-protein) and “cor-rect”(foreign antigen) for targeting is given by θ = C (cid:48) i [ R ][ L ] C i [ R ][ L ] = m (cid:48) i k (cid:48) disso m i k disso = k disso k (cid:48) disso = k − k (cid:48) − (6)assuming m i is the same for both the correct and wrong ligands.The self-proteins bound to the receptor dissociate more quicklythan the foreign antigens indicating that the affinity ratio is lessthan 1. We set the value of θ to be 0.01. KINETIC SCHEME DESCRIBING T-CELL RECOGNITION (a) Foreign Recognition of T-Cell(b) Self Recognition of T-Cell F i g . S c h e m a t i cs o ft h ek i n e t i c m od e l f o r “ N ” k i n e t i c p r oo f r ea d i ngp r o cess ( a ):f o r e i gn li g a nd s ( b ): se l f li g a nd s F o ll o w edb y r e c ep t o r- li gandb i nd i ng , a s e r i e s o f m od i fic a t i on s t r i gge r s t he T - C e ll r e c ogn i t i on s i gna li ng . A t ea c h i n t e r m ed i a t e s t a t e , s e l f li gand s d i ss o c i a t e / θ ( θ = . i nou r c a s e ) t i m e s f a s t e r t han f o r e i gn li gand s . k and k − a r ebo t h f o r w a r dand ba ck w a r d r a t e c on s t an t s a tf a s t equ ili b r i u m . A ft e rr e c ep t o r- li gand s b i nd i ngp r o c e ss ,t hego v e r n i ng r a t e c on s t an t s o ft hephoh s pho r y l a t i on r ea c t i on s and t he r e v e r s eo ft hepho s pho y l r a t i on r ea c t i on s be t w een i n t e r m ed i a t e c o m p l e x e s a r e k p and k − p r e s pe c t i v e l y . E a c h m i deno t e s t he r a t eo ft hed i r e c tf o r m a t i on s t a r t i ng f r o m f r ee li gand s . C oup li ng t oene r g ys ou r c e i sc on s i de r ed i n e v e r ys t ep . RESULTS: ERROR FRACTIONS
We apply the mass action law to express the time derivative ofconcentration for each bound state, which is given by dC dt = k [ R ][ L ] − ( k − + k p ) C + k − p C dC dt = k p C − ( k − p + k p + k disso ) C + k − p C dC dt = k p C − ( k − p + k p + k disso ) C + k − p C + m [ R ][ L ] ... dC N − dt = k p C N − − ( k − p + k p + k disso ) C N − + k − p C N + m N − [ R ][ L ] dC N dt = k p C N − − ( k − p + k disso + W ) C N + m N [ R ][ L ] (7)Here, [R] and [L] denote the concentrations of unbound TCRand ligands, respectively.Applying the SSA to each intermediate including the final com-plex that contributes to signaling, we get C = k [ R ][ L ] + k − p C k − + k p (8) C = k p C + k − p C + m [ R ][ L ] k − p + k p + k disso (9) C = k p C + k − p C + m [ R ][ L ] k − p + k p + k disso (10)The general expression for C N − just before the formation of afinal complex is as follows. C N − = k p C N − + k − p C N + m N − [ R ][ L ] k − p + k p + k disso (11)The initial concentration C given by above can be replaced by C = k [ R ][ L ] k − assuming k − >> k p and k [ R ][ L ] >> k − p C .The concentration at the final state is given by C N = k p C + m [ R ][ L ] k − p + k disso + W ( N = ) ; (12) C N = k p C N − ( k − p + k disso + W )( k − p + k p + k disso ) − k p k − p + m N [ T ]( k − p + k p + k disso ) ( N > ) (13)Note that each series of C N depends on the number of KPR steps,whose expression for N > C N − term, generating additional terms succes-sively (i.e) C N − , C N − , and so on ending with C for n=even and C for n=odd. On the other hand, C N − = k p C N − + k − p C N − + m N − [ T ] k − p + k p + k disso with substitution of the expression for C N − and C N − respectively.Solving for C N − by igonoring the term k − p C N generated accord-ingly due to its negligibility compared to the other terms for takingthe advantage of numerical calculation without producing signifi-cant errors, we get C N − = k p C N − + k p m N − [ R ][ L ]+ k − p m N − [ R ][ L ]( k − p + k p + k disso ) − k − p k p + m N − [ R ][ L ]( k − p + k p + k disso ) (14)Applying the same trick to the rate expressions containing timederivative terms, dC N − dt , dC N − dt and so on, the initial concentrationthat controls the recursion relationship for C N − is C = k p C ( k − p + k p + k disso ) + k − p m [ R ][ L ]( k − p + k p + k disso )( k − p + k p + k disso ) − k p k − p ( N = odd ) (15) C = k [ R ][ L ]( k − p + k p + k disso ) + k − p m [ R ][ L ]( k − + k p )( k − p + k p + k disso ) − k p k − p ( N = even ) (16)The error fraction f is defined as the ratio of the rate of “wrong”product formation to the rate of “correct” product formation (i.e) C N , sel f C N , foreign for T-Cell targeting. Therefore, its full expression forour N-cycle kinetic proofreading model is given by f = (cid:0) k p C , sel f + m [ R ][ L ] (cid:1)(cid:0) k − p + k disso + W (cid:1)(cid:0) k p C , f oreign + m [ R ][ L ] (cid:1)(cid:16) k − p + k disso θ + W (cid:17) ( N = ) (17) f = ( k pCN − , sel f + kpmN − [ R ][ L ] ) (cid:20)(cid:18) k − p + kdisso θ + W (cid:19)(cid:18) k − p + kp + kdisso θ (cid:19) − kpk − p (cid:21) + m N [ R ][ L ] (cid:16) k − p + k p + kdisso θ (cid:17) ( k pCN − , foreign + kpmN − [ R ][ L ] ) [( k − p + kdisso + W )( k − p + kp + kdisso ) − kpk − p ] + m N [ R ][ L ]( k − p + k p + k disso ) ( N > ) (18)where C N − , f oreign = k p C N − , f oreign + k p m N − [ R ][ L ] + k − p m N − [ R ][ L ]( k − p + k p + k disso ) − ( k − p k p )+ m N − [ R ][ L ]( k − p + k p + k disso ) (19) C N − , sel f = k p C N − , sel f + k p m N − [ R ][ L ] + k − p m N − [ R ][ L ]( k − p + k p + k disso θ ) − ( k − p k p )+ m N − [ R ][ L ]( k − p + k p + k disso θ ) (20)Again, the expression for C N − can be given in terms of either C for even n or C for odd n.We obtained the numerical results for error fractions dependingon both number of KPR steps and energy dissipation, featuringtheir decrease with both factors. The error fractions graduallydecline until energy γ reaches without giving a significantdifference in the numerical values for any KPR steps. However,drastic drops in error fractions are observed in higher energycost regime above the branch point of the energy cost measuredin γ and its salient feature is pronounced at higher KPR steps.It is also found that the error fraction converges to Hopfieldlimit( θ i + where i=1,2,...N) which is a minimum at large γ when RESULTS: SENSITIVITY AND SPECIFICITY (a)(b)
Fig. 2 (a) The red and blue lines are the minimum error rates for the KPRsteps of 3 and 6 respectively with energy γ . They have converged valuesclose to Hopefield limit at large energy around of γ . (b) 3D plot of theerror rates dependent on both KPR steps and energy γ the absorption rate W approaches zero as suggested by Hopfield. T-Cell reduces error rates by recognizing foreign antigens with thehelp of multiple phosphorylation steps and energy expenditureeven though the misrecognition of self-proteins as foreign peptidescommonly occurs. Experimentally, it is well known that the typicalerror fraction is less than − . Another study estimating theerror rate based on a simple kinetic proofreading model suggeststhat the rate is approximately − at the affinity ratio of 0.01 forN=4. Both sensitivity and specificity based on the kinetic model werecomputed. Sensitivity is defined as the probability of having thenumber of foreign antigens sufficient to generate major signalingout of the total complex. On the other hand, specificity is definedas a factor to determine the ability to discriminate the correct ligands (foreign antigens) from the wrong ones (self-proteins) intheir active states which contribute to major signaling. Chan et al. provided a simple expression for these quantitiesin kinetic proofreading in the context of T-Cell recognition inthe following manner. We directly follow the procedures theypresent.This implies the definition of sensitivity and specificity can beexpressed as follows:Sensitivity=TP/(TP+FN)Specificity=TP/(TP+FP)whereTP = The number of signaling events for a “correct” ligandFN = The number of zero signaling events for a “correct” ligandTN = The number of zero signaling events for a “wrong” ligandFP = The number of signaling events for a “wrong” ligandIf we simply use the fraction of the active complexes, taking C N C total as α N , thenTP= C total , f oreign α Ncorrect
FP = C total , sel f α Nwrong
FN= C total , f oreign (cid:0) − α Ncorrect (cid:1)
TN = C total , sel f (cid:0) − α Nwrong (cid:1)
Therefore,Sensitivity = C total , f oreign α Ncorrect C total , f oreign α Ncorrect + C total , sel f (cid:0) − α Ncorrect (cid:1) (21)Sepcificity = C total , f oreign α Ncorrect C total , f oreign α Ncorrect + C total , sel f α Nwrong (22)Here, C total can be achieved by adding the concentrations of allintermediates including the ligand-receptor complex at the finalstate, which is taken from both foreign and self-ligands, sorted bydifferent “N”. The associated concentrations of foreign ligands forthe purpose of numerical calculation were taken from the equation(8) to (16).Chan and et al. shows the feature of decrease in sensitivitydepending on the number of KPR steps based on their idealizedkinetic scheme for which reverse reactions between intermediatestates are not taken into account without using energy γ . Theyalso obtained the result through increased specificity, reaching to1.0 depending on the number of KPR steps. The trade-off betweensensitivity and specificity is also observed in our model. RESULTS: SENSITIVITY AND SPECIFICITY (a)(b)
Fig. 3 (a) The sensitivities for different number of KPR steps, N = (red)and N = ( blue ) with given energy γ . They decrease with energy dissipa-tion and reach converged minima. (b) 3D plot of the sensitivities depen-dent on both KPR steps and energy γ Our results show that the sensitivity decreases and convergesto a certain minimum as the energy cost γ increases with agiven number of KPR steps. It is also found that the sensitivitydecreases with the number of KPR steps for given energy asexpected from the equation (21). A rapid drop in the sensitivityis observed in low energy regime, especially γ < γ , while the concentrations of allintermediates including the final products formed by both foreignantigens and self-proteins also approach a converged minimumrapidly for any N, resulting in the obtained minimum sensitivityat large γ . This observation can be interpreted as that a largeamount of energy input which drives the forward reactions alsoinvolves the immediate dissociation at each intermediate complex,yielding lower value of sensitivity with elevation of energy. At thesame time, the decreasing trend of the final concentration canbe accelerated by reducing the rate of the direct formation withenergy consumption. For this reason, the successive increment of the forward rate relative to the backward rate with the growth ofKPR steps results in a sharp drop of in the sensitivity.The specificity obtained from our model using SSA has thefeature approaching a maximum value rapidly as the energy cost γ increases. In addition to this, it is noticeable that the number ofKPR steps does not affect the specificity in a significant mannerwith given energy γ , showing marginal growth of the quantity as Nincreases. We observe the rapid increase of specificity convergingto the approximate value of 1.0 (Exact value of 1.0 found at N > The estimated sensitivity is notin agreement with our numerical results, which are O( − )measured at the detailed balance condition. This is mainly dueto the fact that the nature of these quantities are not robust,exhibiting large variations depending on several factors such asoverall kinetic scheme and the values of specific parameters. (a)(b) Fig. 4 (a) The specificities for different number of KPR steps, N = ( red ) ,and N = ( blue ) with energy. They approach maximum values quickly withenergy supply. It is found that the values are rarely affected by KPR steps,having any noticeable distinction especially at low energy regime. (b) 3Dplot of the specifities dependent on both KPR steps and energy γ RESULTS: MEAN FIRST PASSAGE TIME
The speed of KPR cascade associated with Mean First PassageTime(MFPT) provides information on how rapidly the immunesystem responds to the foreign ligand. More precisely speaking,it is the average time taken to produce the final product thatcontributes signaling immediately from foreign antigens
We find that the energy input and the number of KPR steps arethe major factors that determine the MFPT in KPR model. Westart with the construction of the transition matrix governing thekinetic model in Laplace domain, followed by the calculation ofthe corresponding probability of each state. Our work in thispart is directly towards the evaluation of MFPT depending onthe energy consumption when the foreign ligands are involvedin KPR. Similar works have been done by Banerjee et al. forthe calculation of MFPT of DNA replication process. However,it is based on a different style of biological network that takesseparate mechanisms relying on the type of ligands, both correctand incorrect ones forming associated complexes.
In otherwords, the machinery completed by Banerjee et al. can be utilizedto extract information such as first passage probability density of“correct” products among the coexistence of two types of ligands,which is different from our case. Basically, we follow the recipefrom Bel et al. for the evaluation of the MFPT. The detailedprocedure to obtain the MFPT is given in the Appendix.However, when the algebraic equations used for obtaining theparameters λ and λ are applied for each case, N = > >
1) since the dynamics at the product C still affects therest of intermediate complexes.(B) Each equation (31) in Appendix which increases its degree asthe number of the KPR steps grows has imaginary roots, whichmakes it cumbersome to find and collect appropriate real rootsthat determine the solution of the corresponding equation inLaplace domain.How can we address the problems? Based on the numericalresult, we have found that the MFPT for N=1 where the networkis governed by the first ATP hydrolysis is much higher than theMFPT for N > k , k p and the direct formationrate m , while both the immediate dissociation events at theproduct and the direct formation rate that drastically decreaseswith the growth of energy make the signal to escape the loopdifficult at large energy input. However, as the number of KPRsteps increases, the forward rates, especially pronounced in highenergy regime allow the system to make a completion in a muchshorter time.Based on this observation, we made an approximation to dealwith the intractable situation by only imposing the quadraticequation (25), which allows two real roots to control the entiresystems such that we can simply ignore the influence of otherseries of equations (31) because the rate determining process is associated with the kinetics within the first ATP hydrolysis due tothe “ trapping effect” discussed below.Collecting all quantities including the expression for E and E to get the probability density at the absorbing state, which is givenby F ( s ) = W P N + ( s ) in the Laplace domain, the mean first passagetime T whose expression, in general, is (cid:82) ∞ t f ( t ) dt = ( − ) dF ( s ) ds | s = can be computed for our model. The expression for the MFPTprobability density is given by F ( s ) = W (cid:16) E N + + E N + (cid:17) . (a)(b) Fig. 5 (a) The mean first passage time(MFPT) depending on energy input γ with given the number of kinetic proofreading steps, N = (red) and N = (blue) respectively. (b) 3D plot of the MFPT dependent on both KPRsteps and energy γ The numerical results reveal that in general, the MFPT increaseswith the formation of more phosphorylated complexes, butdecreases with the energy input. There is a steep drop in theescape time until the energy γ reaches 100, but above the value,its variation is negligible for any KPR steps. However, as shownin the figure, such features are found to have deviations for smallKPR steps(N =
3) having long escape time at low energy comparedto the case of higher KPR steps(N = RESULTS: ADAPTATION ERRORS
As discussed earlier, the consequence is directly associated withour kinetic scheme that allows the rate of direct formation todecrease with consumed energy, and the backward rate constant k − still having a constantly large value regardless of the energyinput, which makes it difficult for the signals to escape the firstPdPC hydrolysis cycle as energy increases. Such a trap in the firsthydrolysis affects the escape time for larger KPR steps, yieldinghigher values of MFPT for N =
3, but other factors that facilitatethe transport of signals such as successive forward rates becomedominant in determining the escape time for N >
3. Moreover, asindicated in the figure featuring the unusually drastic drop of theescape time for N =
3, energy expenditure can be used to acceleratethe rate of signal transduction. The change in the MFPT untilthe energy cost γ approaches 100 for N = = − ) sec.A very simple kinetic model without considering energy ex-penditure for TCR activation indicates that the estimated waitingtime using a phosphorylation rate of . s − with 10 KPR steps isaround 10 sec. Compared to the particular estimation, the lowervalue of the MFPT for our case is mainly attributed to the dras-tic drop of the backward rate with the increase of the energy input.
In biological systems, a stimulus signal generates correspondingoutcomes. The change in output in response to the perturbationallows the systems to return to the original one whose outputis measured without a signal input. For T-Cell recognition, asudden shift of a given parameter leads to change in an outputactivity to some extent despite its eventual recovery. It is meaning-ful to find out how accurately a perturbed system returns to theunperturbed one depending on KPR steps and energy dissipation.We take a slight change of forward rate constant k p associatedwith the phosphorylation process in PdPC as a signal input tomonitor its response which is the concentration of all intermediatecomplexes for foreign antigens and self-proteins(total concen-trations of foreign antigens and self-proteins!). The adaptaionerror is defined as | a − aa | , where a =the amount of change inoutput activity without perturbation and a=the amount of changein output activity due to perturbation, and the error is expectedto decline with energy cost. Among several candidates as anoutput activity in response to a given input in order to measure theadaptation errors featuring decline, we have found that the totalconcentration of foreign antigens and self-proteins is the only onethat displays a gradual drop in error with both KPR steps and theamount of energy cost when the forward rate is slightly enhanced.The error becomes somewhat lower as more phosphorylatedproducts are created with given energy, but conversely, we finda sharp growth of the error with KPR steps at equilibrium(zeroenergy input).Such an opposite situation is attributed to the increase inconcentration of the final products with the growth of KPRsteps when the systems is perturbed at equilibrium, which is incontrast to the general decline of the concentration under thesame condition when there is available energy cost. The increased forward rate as stimulation, combined with the relatively largerdirect formation and backward rates at equilibrium compared toout of equilibrium is mainly responsible for the growth of eachintermediate product, having a cumulative effect on the concen-tration of total complexes. This reveals its sharp increase as KPRproceeds, resulting in the salient feature of the adaptation errorunder the detailed balance condition. Our numerical results alsoshow that there is a significant drop in the adaptation error in lowenergy regime( γ < γ > .On the other hand, when the backward rate is slightly enhancedas a stimulus signal, there is an increase of the adaptation errorwith KPR steps for all range of energy consumption although theerror is reduced as more energy is involved for a given KPR step.In this case, the adaptation error decreases and converges to acertain minimum for N ≥
5, but for N <
5, the error simply dropsand reaches zero with increased energy. (a)(b)
Fig. 6 (a) The adaptation error that measures the concentration of allintermediate complexes in response to perturbation of the forward rate k p displays its decrease, but featuring plateau with energy cost. (b) 3D plotof the corresponding adaptation errors dependent on both KPR steps andenergy γ DISCUSSION
The general trend of the decreasing adaptation error with en-ergy can be interpreted as a trade-off between two factors: Thereis a serial increment of each forward rates between intermediatecomplexes as perturbation, which increases the concentration ofthe final products drastically, yielding large errors. This becomesnoticeable as KPR steps grows due to the additional elevationof the forward rate. However, the total concentration may bemoderated by the successive decline of the direct formation rateswith energy consumption, producing small errors in the regime oflarge energy cost.When T-Cell dynamics, initially influenced by perturbationis under the condition where the lowest adaptation error isachieved, it means that the system has recovered most of thefeatures of physical outcomes including error fractions, sensitivityand specificity. Such a feature can be pronounced when sufficientamount of energy is involved in T-Cell recognition process basedon the model we design.
It is difficult to predict the consequences of T-Cell dynamicswithout numerical calculation due to the complexity of our T-Cellscheme. For example, the dissociation event at each intermediateproduct and the direct process forming a phosphorylated complexwithout passing through previous intermediate stage are necessaryelements to understand T-Cell recognition, as well as forward andbackward rates between two products. Moreover, consideringthe nonequilibrium nature of living organisms, interacting withenvironments constantly, we had to incorporate energy sourceassociated with ATP hydrolysis into our system. As used by Qian, the energy dissipation is expressed in terms of several kinetic rateconstants, and it indicates that most of the rates governing ourT-Cell system depend on the consumed energy, which makes therelated dynamics more complex. Hence, it is important to take allthe information into account to set up an appropriate model forunderstanding T-Cell recognition.Despite existing studies on kinetic proofreading in T-Cellrecognition, the lack of simultaneous comparisons of physicaloutcomes has prevented us from fully understanding the dynamicsof the process in terms of energy input. As part of addressingsuch a problem, we present all the results regarding error rates,sensitivity, specificity, speed and adaptation errors in terms ofenergy dissipation with given KPR steps.It has been found that the error fractions decrease with energydissipation and KPR steps, and they have asymptotic behaviors,converging to certain minimum values when a sufficient amountof energy is supplied, which is consistent with the consequence ofHopfield’s work. Compared to the numerical results of specificity,we have also found that the error rates determined at largeamount of energy consumption lead to maximized specificity.Trade-off between sensitivity and specificity featured as thenumber of KPR steps grows is also observed when energycost increases. In addition to this, the energy supply plays ancentral role in reducing the escape time accelerating the speedof signal transduction by minimizing the time-delay caused bythe growth of KPR steps under the given kinetic scheme. Finally, the only measurable quantity where adaptation error graduallydecreases with both KPR steps and energy cost when the system isperturbed is the total concentration of all intermediates.Despite our efforts, the optimal condition that allows for theT-Cell discrimination process to work such that it is being madeunder maximum allowed efficiency is still not completely revealed.Nevertheless, it is found that such a condition characterized by theminimum error fractions, the minimum MFPT and the maximumspecificity is obtained at high energy with loss of the sensitivity forour particular model.
Acknowledgements
We thank supports from National Science Foundation(NSF-Phys.76066 and NSF-CHE-1808474. Discussions with BotondAntal for plotting 3D graphs are appreciated. DISCUSSION (a) (b)(c) (d)
Fig. 7
3D plot displaying the interplay among the measured quantities for N=6
Appendix
The governing equation expressed as ˙ p(t) = Ap(t) gives the seriesof probabilistic outcomes denoted by p , p , ... and p N + due tothe stochastic nature of the system. The direct transition matrixfor our Markovian model is given by ˙ p T ( t ) ˙ p ( t ) ˙ p ( t ) ˙ p ( t ) ... ˙ p N − ( t ) ˙ p N ( t ) = − k − N ∑ i = m i k − k disso ... k disso k disso k − ( k − + k p ) k − p ... m k p − ( k − p + k p + k disso ) ... m k p ... ... m N − ... − ( k − p + k p + k disso ) k − p m N ... k p − ( k − p + k disso + W ) (23)Performing Laplace transform, we get ( s − A ) P ( s ) = s + k + N ∑ i = m i − k − − k disso − k disso ... − k disso − k disso − k s + k − + k p − k − p ... − m − k p s + k − p + k p + k disso − k − p ... − m − k p s + k − p + k p + k disso ... ... − m N − ... s + k − p + k p + k disso − k − p − m N ... − k p s + k − p + k disso + W (24) Each row in the above matrix has the value of (1,0,0,0,...0,0)because of the relation ( s − A ) P ( s ) = p ( t = ) . Putting the generalsolution for the equation, given by P i ( s ) = E λ i + E λ i (i=1,2,...N)where E , are constants determined at boundaries into the “sys-tem controlling equation(s)(SCEs)”, we get algebraic equations ineach λ for two different number of kinetic proofreading cases(eqn(25) and (31) in Apendix for N=1, N > respectively). Note thatthe SCEs are the series of equations that do not include the initialand final algebraic equations solely used for boundary conditions.For N=1, k − p s + k − + k p λ µ − λ µ + k s + k − + k p = (25)Using the fact E , must satisfy the equations determined atboundaries (i=0 and i=2), the following relations are obtained. (cid:0) s + k + m − k − λ − k disso λ (cid:1) E + (cid:0) s + k + m − k − λ − k disso λ (cid:1) E = (26)and EFERENCES REFERENCES (cid:2) k p λ − ( s + k − p + k disso + W ) λ + m (cid:3) E = (cid:2) − k p λ +( s + k − p + k disso + W ) λ − m (cid:3) E (27)Combining these two equations to solve for E and E , we get E = s + k + m − k − λ − kdisso λ + ( s + k + m − k − λ − kdisso λ )[ kp λ − ( s + k − p + kdisso + W ) λ + m ] − kp λ +( s + k − p + kdisso + W ) λ − m (28) E = E (cid:34) k p λ − ( s + k − p + k disso + W ) λ + m − k p λ + ( s + k − p + k disso + W ) λ − m (cid:35) (29)where λ and λ are given by λ , = ( s + k − + k p ) ± (cid:113) ( s + k − + k p ) − k k − p k − p . (30)For N > , kps + k − p + kp + kdisso λ i − µ , i k − ps + k − p + kp + kdisso λ i + µ , i − λ i µ , i + mi − s + k − p + kp + kdisso = ( < i < N ) (31)In the same manner, the other equations determined at bound-aries (i=0 and i=N+1) where E , must obey can be expressed asfollows. ( s + k + N ∑ i = m i )( E λ + E λ ) − k − ( E λ + E λ ) − k disso (cid:34) E λ (cid:16) − λ N (cid:17) − λ + E λ (cid:16) − λ N (cid:17) − λ (cid:35) = (32) m N k p ( E + E ) + E λ N + E λ N = s + k − p + k disso + Wk p (cid:104) E λ N + + E λ N + (cid:105) (33)The expressions for E and E can be obtained as follows: E = (cid:104) λ N (cid:16) s + k − p + k disso + Wk p λ − (cid:17) − m N k p (cid:105) E (cid:104) λ N (cid:16) − s + k − p + k disso + Wk p λ (cid:17) + m N k p (cid:105) (34) E = A (cid:34) λ N (cid:16) s + k − p + kdisso + Wkp λ − (cid:17) − mNkp λ N (cid:16) − s + k − p + kdisso + Wkp λ (cid:17) + mNkp (cid:35) + B (35)where A = s + k + N ∑ i = m i − k − λ − k disso λ (cid:0) − λ N (cid:1) − λ (36) B = s + k + N ∑ i = m i − k − λ − k disso λ (cid:0) − λ N (cid:1) − λ (37) References
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