Control of Voltage-Dependent Biomolecules Via Nonequilibrium Kinetic Focusing

Transcription

1 Control of VoltageDependent Biomolecules Via Nonequilibrium Kinetic Focusing Mark M. Millonas Dante R. Chialvo SFI WORKING PAPER: SFI Working Papers contain accounts of scientific work of the author(s) and do not necessarily represent the views of the Santa Fe Institute. We accept papers intended for publication in peerreviewed journals or proceedings volumes, but not papers that have already appeared in print. Except for papers by our external faculty, papers must be based on work done at SFI, inspired by an invited visit to or collaboration at SFI, or funded by an SFI grant. NOTICE: This working paper is included by permission of the contributing author(s) as a means to ensure timely distribution of the scholarly and technical work on a noncommercial basis. Copyright and all rights therein are maintained by the author(s). It is understood that all persons copying this information will adhere to the terms and constraints invoked by each author's copyright. These works may be reposted only with the explicit permission of the copyright holder. SANTA FE INSTITUTE

2 Control of VoltageDependent Biomolecules Via Nonequilibrium Kinetic Focusing Mark M. Millonas 2 3 and Dante R. Chialvo 2 3 Theoretical Division, and CNLS, MS B258, Los Alamos National Laboratory, Los Alamos, NM 87545, USA. 2 Division of Neural Systems, Memory and Aging, University of Arizona, Tucson, AZ Fluctuations and Biophysics Group, Santa Fe Institute, 399 Hyde Park Rd., Santa Fe, NM We show thatvoltagesensitive macromolecules can, by the application of the right type of external voltage uctuations, be focused with great probabilityinto a desired kinetic substate. As an illustration we consider an eightstate model of the Shaker K + channel driven by dichotomous voltage uctuations. This technique may provide a powerful new tool for the study of the kinetics of voltage sensitive ion channels. Ion channels are semipermeable macromolecular \pores" in the membranes of eukaryotic cells through which various ionic substances are selectively allowed to pass. [] To a large degree, they determine the properties of excitable cells in nerve andmuscle tissue. In voltage sensitive ion channels the coupling between membrane voltage and channel gating involves charges or electrical dipoles which act as sensors of the potential across the membrane. When the channel changes state these charges move, and produce capacitive currents called gating currents. [2] Information about the conformational states of channel molecules can be obtained by measuring the components of these gating currents, and indicates that voltagedependent channels dwell inanumber of states, most of them corresponding to closed conformations, and that the transition between these states are inuenced by the membrane potential. [3] One of the principal goals of studying gating currents is to discover detailed information about the conformational substates of ion channels. It is hoped that this will lead to a knowledge of the corresponding molecular mechanisms, and eventually to a comprehensive understanding of the physiological role each channel plays. Here we propose a method we callnonequilibrium kinetic focusing in which a single kinetic substate of an ion channel can be selectively enhanced and its properties studied by the application of the right type of external voltage uctuations. As a particular example of the application of nonequilibrium kinetic focusing we will concentrate on one particular model system, the Shaker K + channel, which has been studied extensively in the last few years. [4] In a recent publication, Bezanilla, Perozo and Stefani (BPS) report on a 8;state linear kinetic model for this channel C 0! C! C! C 2! C 2! C 3! C 4! O which is able to reproduce many of the experimental observed gating currents. The \C" states all represent closed conformations of the channel, and the \O" state is the open conformation. The arrows indicate allowed kinetic transitions between substates. One possible interpretation of this model is pictured in Fig.. The features in Fig. are meant to represent various structural elements in the ion channel protein, and this is not the only conceivable arrangement. [] The mobile voltage sensor can be in one of eight possible states, only one of which (the open state) allows for the passage of K + ions. The equilibrium position of the spring is determined by the membrane potential through the coupling to the mobile charged element. The mobile element moves in a stepping ratchet potential and is found the vast majority of the time near one of the the potential energy minima, which represent the eight distinct kinetic substates observed experimentally. [5] Stepping Ratchet + Mobile Voltage Sensor Extracellular Intracellular FIG.. Cartoon of one possible arrangement of the voltage sensor of the Shaker K + channel, shown here in the open state. Occasionally the mobile element makes thermally activated jumps to neighboring states, and over time these jumps lead to an equilibrium occupation probabilityover the eight substates. The distribution of states of a system in thermal equilibrium at temperature T is very well understood. The probability of state i of a molecule with n states is approximately P i exp(;e i =kt )= K + Aqueous Pore in Channel nx j= exp(;e j =kt ): ()

3 We derived the energy function picture in Fig. 2 by tting the kinetic data in [6]. The information provided in [6] is not sucient to ll in a very detailed model of the conformational dynamics. [7] However, by making use of some mild auxiliary assumptions we were able to combine this information into a continuous model which is consistent with the observation of BPS, and which contains the basic details. Let us then consider an alternative to this situation where a external time dependent voltage V 0 +V (t) (where <V(t) >= 0) is applied across the membrane. The gating of the channel is modeled as the motion of a mobile subunit with charge valence z and position x in an internal potential u(x) subject to thermal uctuations, _x(t) =;u 0 (x)+z[v 0 + V (t)]= + p 2kT(t) U/kT C 0 C C C 2 C 2 C 3 where is the friction coecient (inverse mobility) of the mobile unit, and we have assumed that its motion is overdamped. The length scale is the membrane thickness, and (t) is white noise with < (t) >= 0, and <(t)(0) >= (t). After converting to dimensionless units X = zx=, = (z 2 kt= 2 )t, v 0 = V 0 =kt and v(t) =V ()=kt, and U(X) =u(x)=kt which isinfact the denition of the energy function in Fig. 2, we have 5 _X() =;U 0 (X)+v 0 + v()+ p 2() (2) FIG. 2. Conformational energy function for the Shaker K + channel (at 0 mv membrane potential). The gating states of the channel are described by a single dimensionless reaction coordinate X = zx= where z is the eective charge valence and x is the coordinate of the mobile unit. Thus, large distances between the stable states represent large gating currents. The dimensionless energy function U(X) = E(x)=kT has a complicated form with several minima, each ofwhich corresponds to one of the states of the ion channel. In BPS the kinetic features of particular regions of substates of Shaker K + channels are deduced by measuring the gating current during relaxation after applying a 20 ms pulse from a hyperpolarized holding potential of 90 mv. The time of the pulse is long enough for an equilibrium state to form, and by adjusting its voltage a certain limited degree of control over the system is achieved. An external voltage V across the membrane changes the energies in Eq. in a linear way, E i! E i +VX i, where X i are the relative positions of the states. Fig. 3 shows the equilibrium probabilities of the eight states of the Shaker K + channel as the external voltage is varied. As observed experimentally in BPS there are major charge motions around 44 mv and 63 mv. To observe the relaxation kinetics in the region of a particular substate one should enhance that substate as much as possible by pulsing to the voltage at which the probability is a maximum. For instance, the voltage which produces the largest probability for the state C 3 is around 45 mv. This situation represents a fundamental constraint which limits the experimentalist's ability to study a particular substate and its kinetics. C 4 O X where <() >= 0,and <()(0) >= (t). Equilibrium Probability C C 0 C C 2 C 2 C 3 C Static Voltage (mv) FIG. 3. Equilibrium probabilities for the eight states of the Shaker K + channel. Although there are an unlimited number of types of nonequilibrium uctuations which can be applied, for the purpose of illustration we will conne ourselves here to what is known as dicotomous noise, or telegraph noise. This is noise which has two states V 0 + V. Transitions between states occur at random moments with probabilities W out of the plus and minus state respectively. For this reason this noise is known as Markovian colored noise. This noise is very suitable as a rst choice because it is easy to apply experimentally and because together with the sawtooth form of the eective potential picture in Fig. 3, it allows for a particularly simple analysis. We have already considered this noise in another context. [8] It is also important to note that while we use this particular noise as an illustration, the basic principles behind our results will also apply to a host of other types of noise. While more detailed work needs to be done to determine O 2

4 if other types of easily applied voltage uctuations (e.g. Gaussian noise [9]) will work better in this or other situations, the basic principles should apply regardless of the specic type of nonequilibrium voltage uctuations applied across the membrane. We start by putting the external voltage uctuation in dimensionless form v = V =kt where we chose v + = s D v + ; V ; = ; s D v ; + : (3) The transition probabilities w + from the plus to the minus state and w ; from the minus to the plus state are w + = + 2 v w ; = ; 2 v : (4) This noise has mean zero <v(t) >= 0 and correlation function <(t)(0) >= (D= v ) exp (;= v ). The dicotomous noise thus has three characteristic dimensionless parameters, the \amplitude" D, the correlation time v, and an asymmetry parameter. This noise goes over to white noise with strength D as v! 0and! 0. A probabilistic treatement of Eq. 24 leads to the set of coupled equations [0] (X ) = X ~U 0 (X) ; v + x w + + (X ) w ; ; (X ) (X ) where (X ) are the conditional probability densities for the ion to be at X when the noise is in the plus or minus state v() =v. The total probability density is just P (X ) = + (X ) + ; (X ). For simplicity we have introduced U(X) ~ =U(X) ; v0 X After introducing the auxiliary variable Q = w + + (X ) ; w ; ; (X ), and rewriting the above set of equations in terms of P (X ) and Q(X ) we obtain, Q(X ) P(X ) = X ~U 0 + X P ; A X Q = ~U 0 ; X + Q ; Q ; X v A P X : where = p v =D, = 2= p ; 2, and A = 2 v = p ; 2. We wish to nd the stationary probability distribution P 0 (X) Q 0 (X) of the above equations, where P 0 = =0 and Q 0 = =0. Ifwe further note that t P (X ) = ; x J(X ) wherej is the probability current, and that the stationary current must vanish in this closed system, we can obtain the stationary state equations 0= ~ U 0 P 0 + P 0 X ; A Q 0 (5) 0= ~U 0 ; X + Q 0 ; Q 0 ; P 0 X v A X : (6) We will consider only the case where the amplitude of the driving is large with respect p to the thermal uctuations, and where T! 0, and D= > sup ju 0 j. Substituting Eq. 5 into Eq. 6 and integrating, and solving for the stationary distribution for the potential in Fig. 2, we obtain the result we are after Z X U P 0 (X) Ne ;(X) (X) = 0 (y) ; v 0 dy W (y) h i W (y) =+D ; 2 [ U ~ 0 (y)] 2 + U ~ 0 (y) : (7) The stationary probabilities of the substates are then given approximately by P i exp(;(x i )=kt )= X j exp(;(x j )=kt ): (8) This expression can be evaluated on the computer for the potential shown in Fig.2. More detailed theoretical and numerical studies with a wider range of applicabilty will appear elsewhere. [] Fig. 4 shows the focusing eect we are after. We have chosen to illustrate the enhancement ofthec 3 substate because it is the the state with the smallest probability under optimal equilibrium conditions, and because the eects are particularly dramatic. Qualitatively similar results can be obtained for any state. The correct noise parameters are found by trial and error by evaluating Eqs. 7 and 8. The probabilities of any state can be signicantly enhanced from the optimal equilibrium values. For the case pictured in Fig. 4, the optimal probability under static conditions for C 3 is about After nonequilibrium kinetic focusing the probability of this state has been enhanced to 0.96, some 25 times the optimal static value. Perhaps more importantly, under optimal static conditions, most of the other states are present ineven greater proportions, and the state C 3 is smothered by signals from the other states. After focusing with nonequilibrium uctuations the state C 3 is by far the predominant state and under these circumstances it is no exaggeration to talk about focusing the channels into a specic state. The physical eect which makes this unusual focusing possible is the existence of an additional nonequilibrium source of transport due to the interaction of the voltage uctuations with the complex underlying dynamics of the channel voltage sensor. Many of the conceptual roots can be found in the pioneering work of Landauer. [2] A certain amount of generic intuition about this behavior can be gleaned from more recent work on ucutation induced transport. [8,9,3,4] For dicotomous noise there are three principal factors to consider. The rst is amplitude of the uctuations. If the amplitude is too small 3

5 there is little or no eect while if it is too large the response of the voltage sensors will be so large as to smooth out the probability distribution. Secondly, if there is some spatial asymmetry about a particular substate the addition of time correlated voltage uctuation will cause an enhancement of the activation rates out of the state in one direction over another. This will eectively shift the probability distribution over the substates in the favored direction. It is interesting to note that in a sense the degree of spatial asymmetry is articially altered by varying the external net part of the voltage V 0. Lastly there is temporal asymmetry (in the present case parameterized by ) which is an asymmetry in the voltage noise itself due to a nonvanishing of the higher order odd correlation functions of the uctuations. Temporal asymmetry can bias the transport in either direction, and can compete with the eects due to a spatial asymmetry. [8,4] Stationary Probability Stationary Probability Optimal Static Voltage for C 3 V 0 = 45 mv C 0 C C C 2 C 2 C 3 C 4 O Focusing of C 3 with Voltage Fluctuations V + V = 87 mv 0 + V + V = 35 mv 0 ε = τ v = 0.00 C 0 C C C 2 C 2 C 3 C 4 O FIG. 4. C3 Focusing results for a model of the Shaker K + channel. For Gaussian noise, it is the shape of the power spectrum of the noise that controls the behavior. [9] Generally speaking it can be shown that \narrow" states are made more stable relative to \wide" states by Gaussian uctuations whose power spectrum has positive curvature at zero frequency, and that the opposite is true when the power spectrum has negative curvature at zero frequency. All of these eects act in consert, and focusing is the consequence of tuning the noise parameters in such away as to enhance transport out of the competitors of the chosen state, and to suppress transport out of the chosen state. When this is done to as great a degree as is possible the results are often quite dramatic, as is illustrated above. The only restriction seems to be that there exists enough variation between the substates for a competition between them to be set up. The complexity and nonlinearity of these eects are only magnied when one considers higher dimensional systems of substates, such as those where inactivation is taken into account. The authors wish to acknowledge very useful conversations with Francisco Bezanilla, Enrico Stefani and Tim Elston. This work is partially supported by the NIMH. [] B. Hille, Ionic Channels of Excitable Membranes (2nd Ed.) Sinauer Associates, Sunderland Mass. (992). [2] C. M. Armstrong and F. Bezanilla, Nature(Lond.) 242, 459 (973). [3] W. N. Zagotta and R. W. Aldrich, J. Gen. Physiol. 95, 29 (990). [4]W.Stuhmer, F. Conti, M. Stoker, O. Pongs, and S. H. Heinemann, Pfugers Arch. 48, 423 (99) F. Bezanilla, E. Perozo, D. M. Papazian, and E. Stefani, Science (Wash. D.C.) 254, 679 (99) E. Perozo, D. M. Papazian, F. Bezanilla, and E. Stefani, Biophys. J. 62, 60 (992) N. E. Schoppa, K. MacCormack, M. A. Tanouye, and F. J. Sigworth, Science (Wash. D.C.) 255, 72 (992) D. Sigg, E. Stefani, and Francisco Bezanilla, Science (Wash. D.C.) 264, 578 (994) [5] C. M. Armstrong, Physiol. Rev. 6, 644 (98). [6] F. Bezanilla, E. Perozo and E. Stefani, Biophys. J. 66, 0 (994). [7] The experiments proposed here might be used to ll in some of the details since the precise eects will depend on more subtle details than can be deduced from standard techniques. [8] M. M. Millonas and D. R. Chialvo, Phys. Rev. Lett. (submitted). [9] For examples see M. I. Dykman, Phys. Rev. A 42, 2020 (990) M. M. Millonas and D. I. Dykman, Phys. Lett. A 83, 65 (994) M. M. Millonas, Phys. Rev. Lett., 74, 0 (995) M. M. Millonas and C. Ray, Phys. Rev. Lett. (in press). [0] W. Horsthemke and R. Lefever, Noise Induced Transitions Springer, Heidelberg, 984. [] K. Anderson, D.R. Chialvo, T.Elston and M. M. Millonas, in preparation. [2] R. Landauer, J. Appl. Phys. 33, 2209 (962) J. Stat. Phys. 9, 35 (973), 525 (974) 3, (975). [3] A. Ajdari and J.Prost, C. R. Acad. Sci. Paris 35, 635 (992) M. Magnasco, Phys. Rev. Lett. 7, 477 (993) J. Prost, J.F. Chauwin, L. Peliti, and A. Ajdari, Phys. Rev. Lett. 72, 2652 (994) C. Doering, W. Horsthemke and J. Riordan, Phys. Rev. Lett. 72, 2984 (994). [4] A.Ajdari,D.Mukamel,L.Peliti and J.Prost, Journ. Phys. France 4, 55 (994) D. R. Chialvo and M. M. Millonas, SFI Report ,(994). 4