Direction dependent mechanical unfolding and Green Fluorescent Protein as a force sensor
aa r X i v : . [ c ond - m a t . s o f t ] J u l Direction dependent mechanical unfolding and Green Fluorescent Protein as a forcesensor
M. Caraglio ∗ Dipartimento di Fisica and CNISM, Politecnico di Torino,c. Duca degli Abruzzi 24, Torino, ItalyINFN, Sezione di Torino, Torino, Italy
A. Imparato † Department of Physics and Astronomy, University of Aarhus,Ny Munkegade, Building 1520, DK–8000 Aarhus C, Denmark
A. Pelizzola ‡ Dipartimento di Fisica,CNISM and Center for Computational Studies,Politecnico di Torino,c. Duca degli Abruzzi 24, Torino, ItalyINFN, Sezione di Torino, Torino, ItalyHuGeF Torino, Via Nizza 52, I-10126 Torino, Italy
An Ising–like model of proteins is used to investigate the mechanical unfolding of the Green Flu-orescent Protein along different directions. When the protein is pulled from its ends, we recover themajor and minor unfolding pathways observed in experiments. Upon varying the pulling direction,we find the correct order of magnitude and ranking of the unfolding forces. Exploiting the directiondependence of the unfolding force at equilibrium, we propose a force sensor whose luminescencedepends on the applied force.
PACS numbers: 87.15.A-; 87.15.Cc; 87.15.La
INTRODUCTION
In recent years many efforts have been devoted to thestudy of the mechanical properties of biopolymers un-der mechanical loading. Many experimental groups havestudied the unfolding and refolding trajectories of pro-teins and nucleic acids by applying a controlled forcewith AFM or optical tweezers techniques [1–6]. Theseworks have triggered a number of numerical investiga-tions, where the same molecules have been studied, underconditions otherwise not accessible to the experimentaltechniques [6–17].One of the most interesting proteins studied with forcespectroscopy is the Green Fluorescent protein (GFP),which exhibits bright green fluorescence when exposedto light with a suitable wavelength. GFP has many ap-plications in biotechnology, from localization of proteinsin a living cell, to metal ion or pH sensor [18]. Further-more, such a molecule has been the subject of mechanicalexperiments and numerical simulations [3, 4, 8] aimedat characterizing its response to external force and thestructure of its intermediate states. The final goal of suchstudies is a full characterization of the GFP response tomechanical stress, so as to pave the way to its use as amolecular force sensor. Indeed, it is commonly believedthat GFP fluoresces only when its structure is almostintact [19, 20]. This represents a restriction for the useof the GFP to probe forces in vivo, since a fluorescent (non-fluorescent) protein indicates that the applied forceis below (above) some typical rupture force, but one can-not obtain an estimate of the actual value of the force bymeasuring the fluorescence. Thus in the present letter wepropose a practical method to circumvent this limitation,by exploiting the fact that if pulled along different direc-tions, the GFP exhibits different mechanical properties,and thus different rupture forces, as already observed inexperiments [4]. We will first introduce a model for theGFP that has already been used to evaluate the phasediagram, the free energy landscape [9] and the unfold-ing pathways [13, 16] of widely studied proteins and ofRNA molecules [15]. We will compare the response ofthe model protein to experimental outcomes, and thenwe will study the rupture force of the GFP when pulledalong different directions. Finally, we will introduce amodel polyprotein made up of different GFP modules,and we will show how such a molecule can easily providethe value of the applied force in a wide range of values,and thus be used as a force probe. To the best of ourknowledge, it is the first time such a molecular device isproposed in the literature.
THE MODEL
A native–centric, Ising–like model of protein folding[23] has been generalized in previous works [9] to dealwith the case of mechanical unfolding. In such a modelthe state of a N residues long protein is determinedby two sets of binary variables: the variables m k areassociated to each residue being 1 or 0 according towhether the residue is native–like or not, and the vari-ables σ ij = ± i and j > i in a non–native state, such that S ij = (1 − m i )(1 − m j ) Q j − k = i +1 m k . A negative interactionenergy h ij is associated to two residues i and j if theyare in contact in the native structure and if they are inthe same native stretch. The corresponding hamiltonianis H ( { m } , { σ } ) = N − X i =1 N X j = i +1 h ij j Y k = i m k + U ( L ( { m } , { σ } )) , (1)where L = b − X i = a b X j = i +1 l ij σ ij S ij is the molecule elongation(distance between the C α atoms of the two residues a and b where force is applied, 1 ≤ a < b ≤ N ), l ij is thenative distance between C α atoms of residues i and j , and U ( L ( { m } , { σ } )) is the term describing the coupling tothe external force, which depends on the pulling protocol.In the case of a constant force the energy reads U = − f L .In the case of a constant velocity protocol the force isapplied through a harmonic potential whose center movesat constant velocity, and the corresponding energy is U = k/ L − vt ) .Under a constant force the equilibrium thermodynam-ics of the model is exactly solvable [9], while for the studyof the kinetics we resort to Monte Carlo simulations.Before studying the molecule response to externalforces, it is interesting to consider the free energy pro-file at zero force. More specifically, we have computedthe free energy profile as a function of the fraction of na-tive residues M (Fig. 1) and contacts Q (Fig. 2) at thedenaturation temperature. Inspection of these plots in-dicates that at this temperature: (i) when the protein isin its native state, all the native contacts are formed, andalmost all the residues are in the native configuration; (ii)in the unfolded state, no native contacts are formed, and1/3 of the residues are in the native configuration; (iii)the transition state corresponds to Q ∼ . ÷ .
4, whileat Q ∼ . ÷ . k B T in both cases. Our results for thefree energy profile as a function of Q can be compared tothe result obtained in [11] by weighted–histogram anal-ysis of molecular dynamics (MD) data. The qualitativepicture is very similar, although some differences can beobserved. The MD result show that some fluctuations innative contacts are allowed in both the native and un-folded states, a feature which is missing in our result dueto the extreme cooperativity of the model. Moreover,the unfolding barrier is predicted by MD to be around 15 k B T : this is consistent with the observation that ourmodel predicts systematically higher energy barriers andunfolding forces (see discussion below). ∆ G ( un i t s o f k B T ) M Figure 1. Free energy profile ∆ G , as a function of the fractionof native residues M , at unfolding temperature T = 356 Kand zero force. -5 0 5 10 15 20 25 30 35 0 0.2 0.4 0.6 0.8 1 ∆ G ( un i t s o f k B T ) Q Figure 2. Free energy profile ∆ G , as a function of the fractionof native contacts Q , at unfolding temperature T = 356 K andzero force. From now on we set the temperature at T = 293 K.The native structure of GFP is basically a β –barrel madeof 11 β -strands, with a N-terminal α –helix.In Fig. 3 the free energy profile ∆ G ( L ) is reportedfor a typical case, where the equilibrium unfolding force f = 35 . ≤ h m k ( L ) i ≤ k − thresidue is native–like when the molecule total elongationis L (data not shown), we find out that such bends corre-spond to the following structures: β and β (for L ≃ β β (18 nm) and β β β (25 nm). Here andin the following, β k · · · β n denotes an unfolded structureof the GFP, where β –strands from k to n are not in anative–like conformation, i.e. they are unfolded (in allthese structures the N–terminal α –helix is also not in anative–like conformation). −10 0 10 20 30 0 10 20 30 40 50 60 70 80 ∆ G ( un i t s o f k B T ) L (nm) Figure 3. Free energy profile ∆ G as a function of the proteinelongation L , with T = 293 K, and force f = 35 . SIMULATIONS AND COMPARISON WITHEXPERIMENTS
We first consider simulations at constant velocity,mimicking the effect of an AFM cantilever, which is re-tracted at a velocity v . The force is applied to themolecule ends and we set k = 30 pN/nm and considervelocities v = 0 .
3, 1 µ m/s, 2 µ m/s and 3 . µ m/s. Inref. [8] it was found that the most likely unfolding path-way is α → β → β β (the α –helix unfolds first, then β –strand 1, then strands 2 and 3 simultaneously, thenthe rest of the protein), observed in 72% of the trajecto-ries at v = 2 . µ m/s. A different pathway, α → β , wasobserved in the remaining 28% of the trajectories.The N–terminal α –helix is the first secondary struc-ture element to unravel in both pathways. This event istypically associated to very small signals, almost maskedby fluctuations, at odds with the clear jumps we observein the end–to–end length for the detaching of β –strands(see fig. 4). This is analogous to what occurs in the ex-periments where the unfolding of the helix is associatedto a very smooth “hump–like” transition with a shortcontour length increase of 3 . β is the first strand tounravel, while the remaining trajectories follow the ma-jor unfolding pathway found in experiments. In Fig. 4, we plot a typical unfolding trajectory of our GFP modelwhen the force is applied to the molecule ends. In partic-ular, we plot, as functions of time, the end–to–end length L , the force and several weighted fractions of contactsbetween adjacent β -strands φ β i − j , giving the fraction ofnative contacts between the strands β i and β j , see [16]. Major unfolding pathway.
Inspection of the top panelof Fig. 4, corresponding to the major unfolding pathway,provides clear evidence that there are three main unfold-ing events. (i) A drop in the number of contacts betweenstrands 1 and 2, signalling the unfolding of β (actuallythe α –helix has already unfolded, as discussed above, butthe corresponding fraction of native contacts φ α is not re-ported in the figure for the sake of clarity). The lengthof the corresponding intermediate state is in the range10 ÷ . G ( L ) showsa bend, see fig. 3. (ii) A drop in the number of con-tacts involving strands 2 and 3, signalling the unfoldingof these strands. The corresponding intermediate lengthis around 20 nm, where the free energy profile ∆ G ( L ) hasa local minimum. (iii) A drop in the number of contactsinvolving strands 10 and 11, signalling the unfolding ofthese strands. The corresponding intermediate length isin the range 30 ÷
37 nm: inspection of fig. 3 suggests thatfor such an elongation the molecules lies in the basin ofthe unfolded minimum.
Minor unfolding pathway.
The bottom panel of Fig. 4corresponds to the minor unfolding pathway and we cansee that the first strand to unravel is β followed by β .In [8] the unfolding pathway was only traced up to the∆ β intermediate because the subsequent event is theflattening of the barrel but, after the barrel flattens, thereis at least another rupture event as the last force jump inFig. 1b of [8] shows. It is reasonable to assume that thisevent is related to the breaking of native-like contactsbetween the beta strands, which were not ruptured dur-ing the flattening of the barrel. Our model, which lacksa fully three-dimensional representation, cannot describethe flattening of the barrel, while it can describe witha high time resolution the breaking of the beta strandcontacts, which here yield in a few distinct steps (Fig. 4,bottom panel).We can now put the local minima and bends of thefree energy landscape of Fig. 3 (which is a thermody-namic equilibrium property of the system) in correspon-dence with intermediates found in our simulations andin experiments (which are performed in non-equilibriumconditions). Some of these features of the free energy pro-file are indeed barely visible, but the equilibrium proba-bilities h m k ( L ) i , we introduced in the previous section togive a structural interpretation of the various minima andbends, are perfectly consistent with the nonequilibrium m k values obtained from the simulations, which allow usto identify the structures of the nonequilibrium interme-diates.In ref. [3] the authors observed two intermediates with φ L ( Å ) , f ( p N ) Time (Mc steps) β β β β β β f L φ L ( Å ) , f ( p N ) Time (Mc steps) β β β β β β β fL Figure 4. (color online) Typical unfolding trajectories of aGFP module under constant velocity pulling ( v = 0 . µ m/s).Length L , force f and weighted fractions φ β i − j of strand–strand contacts as functions of time for two typical cases:major (top panel) and minor (bottom panel) unfolding path-way, see text. separation values from native configuration of 3 . α –helix detached that profile of Fig. 3 doesnot show, while the second is an intermediate with theN–terminal α –helix detached and a β –strand detachedwhich corresponds to the bend at 11 nm (9 . α –helix and first, second and third β –strands detached)with a distance of 26 . . Different directions.
We now consider simulationswhere the points of force application are not the moleculeends, so that the direction of the force with respect tothe molecule is varied. Table I reports, for different di-rections (specified through the application point residuenumbers), the mean unfolding forces, where unfolding isdefined as unravelling of the first β strand. Since mostof these directions were considered in experiments [4], at least at v = 3 . µ m/s, it is interesting to compare ourresults to the experimental ones. Our unfolding forcesare systematically larger than the experimental values,with the largest discrepancies (a factor 2 to 3) occur-ring for directions 3–212 and 132–212. However, it isinteresting that in spite of the simplicity of the model,which lacks a fully three-dimensional representation, theorders of magnitude for the rupture forces are correctand many qualitative aspects are reproduced. In partic-ular, by analyzing the experimental data one finds thatthe unfolding force increases with the following order: (i)pulling along the end–to–end direction (it must be notedthat the rupture force along this direction was measuredfor v = 0 . µ m/s instead of 3 . µ m/s as most other di-rections); (ii) pulling along the 3–212 and 132–212 direc-tions, the corresponding rupture forces are equal withinthe experimental error; (iii) pulling along the 182–212and 3–132 directions, the corresponding rupture forcesare equal within the experimental error (though the lat-ter was measured for v = 2 µ m/s); (iv) pulling along the117–182 direction. This hierarchy is respected by our re-sults: we find that the rupture force increases when weconsider the pulling directions as ordered above, the onlyexception being for 3–212 and 132–212, whose unfoldingforces are not equal (we obtain a smaller force for thelatter), and the same holds for 182–212 and 3–132 (weobtain a larger force for the latter). Table I. Unfolding forces at different velocities for differentdirections. Experimental values ( ∗ from ref. [3] and † fromref. [4]) in parentheses. Unfolding force (pN)Direction v = 0 . µ m/s v = 2 µ m/s v = 3 . µ m/send–end 140 ± ± ∗ ± ± ± ± ± ±
12 298 ±
12 317 ± ± † ± ± ± ± † ±
12 360 ±
20 381 ± ± ± ± ± † ±
16 471 ± ± † ± ± ±
11 512 ± ± † Finally, in Table II we report the potential width values x u corresponding to the rupture of the first β –strand fordifferent directions. These were obtained through a fitof the most probable unfolding force f M as a function ofvelocity to the Evans–Ritchie theory [21, 22], which gives f M = k B Tx u ln (cid:18) τ x u k B T r (cid:19) (2)where τ is the unfolding time at zero force. It must bekept in mind that in the Evans–Ritchie theory the forcegrows with a constant rate r = k · v and hence its ap-plicability to the present case (harmonic potential whosecenter moves at constant velocity v ) is only approximate. Table II. Potential width x u obtained from a fit to Eq. 2.Experimental values between parentheses.direction x u (nm) direction x u (nm)end–end 0 . ± . . ± .
03) 132–end 0 . ± . . ± .
03 182–212 0 . ± . . ± . . ± . . ± .
01) 3–132 0 . ± . . ± . . ± . . ± . . ± . . ± . Our potential widths are consistent with experimentalones only in a few cases (end–end, 3–132): once again,this might be attributed to the fact that our model lacksa fully three–dimensional representation.It must also be observed that the Evans–Ritchie theoryis built on the assumption that x u is independent of theapplied force, and this can be another source of error inthe determination of x u . This assumption was relaxed inmore recent theories [24, 25] which yield generalizationsof Eq. 2, which predict that the f M vs ln v plot is non-linear, with the slope being an increasing function of v ,as observed in many experiments. Indeed our data showsome nonlinearity, but this is too small to apply thesetheories, probably because our velocities span only 1 or-der of magnitude. Previous applications of these theories[9, 24, 25] were done on data sets with velocities spanning4–5 orders of magnitude, such that the nonlinear effectswere much more important, but such a wide range ofvelocities is beyond the scope of the present paper. GFP POLYPROTEIN
As we discussed above, the equilibrium properties ofthe GFP at constant force, can be obtained exactly, forany pulling direction. Exploiting this result, we study apolyprotein where each module is connected to the neigh-bouring ones through different points of force application,as illustrated in Fig. 5.For example Dietz et al. [4] already proposed a copoly-mer with mixed linkage geometries GFP(3,212)(132,212),
PSfrag replacements f Figure 5. (color online) Sketch of a polyprotein made ofvarious modules connected between them through differentresidues. made up of several GFP modules, where a module linkedby its (3,212) residues to the main structure was alter-nated with a module linked by its (132,212) residues.Such a molecule can be easily obtained by using the cys-teine engineering method discussed in ref. [26], which al-lows one to construct polyproteins with precisely con-trolled linkage topologies: the points of force applicationto each module correspond to the position of the linkingcysteines in the folded tertiary structure. In order to un-derstand the general behavior of our model polyproteinunder a constant force we first investigate the responseto a constant force of a single GFP module. The corre-sponding unfolding forces are reported in Table III.
Table III. Equilibrium unfolding force for different directions.direction unfolding force (pN) direction unfolding force (pN)end–end 35 . . . . . . . . . . We then proceed by studying the response to a con-stant force of a polyprotein made up of 10 modules, eachwith different linkage topologies. It is worth to note thatat equilibrium a force applied to the free ends of thepolyproteins will have the same value throughout thewhole chain. Thus, the different modules will unfold atdifferent values of the force, according to the hierarchyshown in Tab. III, and thus the luminescence will be dif-ferent for different values of the force. If we assign avalue 1 (in an arbitrary scale) to the maximum possibleluminescence, where each module is emitting green light,a luminescence of 0.5 will correspond to a configuration,and thus to a force, where half of the modules are un-folded (non intact structure). Given that each modulewith a different linkage has a different unfolding force,we obtain a curve like the one shown in fig. 6, relatingthe luminescence of the polyprotein to the force appliedto its free ends, where the force ranges from 35.9 to 96.7pN, see Table III. It is worth to note that interface in-teractions and aggregation effects between neighbouringunits in polyproteins similar to the one we propose, havebeen ruled out by experimental investigations, see [4]. fr ac ti on o f n a ti v e m odu l e s f (pN) end-end3-2123-182117-end3-132182-end117-190102-190117-182 132-182 Figure 6. Fraction of native–like modules as a function offorce at T =293 K. Each “step” corresponds to the unfoldingof a different module in the polyprotein and thus to a decreasein the luminescence by a “unit”. Figure 6 represents an important result of this paper.It is worth to note that more modules, with different link-ages, can be added, and this would give a more precisedetermination of the force. Once the polyprotein we pro-pose has been engineered, a curve like the one shown infig. 6 can be very easily obtained in an optical tweezersexperiment at constant force as those discussed, e.g., inref. [1]. This approach would also allow one to calibratethe device. Finally, we want to emphasize that, althoughunfolding studies of GFP along different directions wherealready performed in, e.g., [4, 8], those previous studiesconsidered the dynamic-loading set up, with a constantretraction speed of the AFM cantilever. On the contrarywe investigate here for the first time the unfolding at constant force of GFP. The unfolding force of a moleculeunder dynamic loading depends not only on the molecu-lar features, but also on the force rate, and thus a forceprobe based on those data must be able to measure at thesame time the loading rate and the rupture force. Ourconstant force probe does not exhibit this drawback.
CONCLUSIONS
The Ising–like model of protein mechanical unfoldingdescribes correctly the most important qualitative as-pects of the direction–dependent mechanical unfoldingof the Green Fluorescent Protein, namely the unfold-ing pathways and intermediates observed when pullingat constant velocity from the molecule ends, and the or-ders of magnitude and ranking of the unfolding forcescorresponding to different directions. Some features,like the flattening of the barrel or the potential widthscorresponding to many directions, cannot however bedescribed by our model, which lacks a fully three– dimensional representation. Moreover, from a morequantitative point of view, our energy barriers and un-folding forces are systematically larger than those ob-served in experiments.We have exploit the dependence of the unfolding forceon the pulling direction to investigate a force sensor basedon a GFP polyprotein where each module is linked witha different geometry to the nearest neighbouring mod-ules, so as to experience the force along different direc-tion, yielding a device whose luminescence depends (ina discrete way) on the force. It is worth noting thatsuch a device may be used in in-vivo experiments, tomeasure forces at molecular level, e.g. inside cells, in anon–invasive way.MC gratefully acknowledges the financial support ofAarhus Universitets Forskningsfond (AUFF) during hisstay at Aarhus University. AI gratefully acknowledges fi-nancial support from Lundbeck Fonden. AP thanks An-tonio Trovato for a useful discussion. ∗ [email protected] † [email protected] ‡ [email protected][1] J. Liphardt et al , Science , 733 (2001).[2] B. Onoa et al , Science , 1892 (2003).[3] H. Dietz and M. Rief, Proc. Natl. Acad. Sci. USA ,16192 (2004).[4] H. Dietz, F. Berkemeier, M. Bertz, and M. Rief, Proc.Natl. Acad. Sci. USA , 12724 (2006).[5] A. Imparato, F. Sbrana, M. Vassalli, Europhys. Lett. ,58006 (2008).[6] S. Kumar and M. S. Li, Phys. Rep. , 1 (2010).[7] A. Irb¨ach, S. Mitternacht, and S. Mohanty, Proc. Natl.Acad. Sci. U.S.A. , 13427 (2005).[8] M. Mickler, R. Dima, H. Dietz, C. Hyeon, D. Thiru-malai, and M. Rief, Proc. Natl. Acad. Sci. USA ,20268 (2007).[9] A. Imparato, A. Pelizzola and M. Zamparo, Phys. Rev.Lett. , 148102 (2007); A. Imparato, A. Pelizzola andM. Zamparo, J. Chem. Phys. , 145105 (2007).[10] A. Imparato, S. Luccioli, A. Torcini, Phys. Rev. Lett. ,168101 (2007).[11] B.T. Andrews, S. Gosavi, J.M. Finke, P.A. Jennings,Proc. Natl. Acad. Sci. USA , 12283 (2008)[12] C. Hyeon, G. Morrison, and D. Thirumalai, Proc. Natl.Acad. Sci. U.S.A. , 9604 (2008).[13] A. Imparato and A. Pelizzola, Phys. Rev. Lett. ,158104 (2008).[14] S. Mitternacht, S. Luccioli, A. Torcini, A. Imparato, A.Irb¨ack, Biophys. J. , 429 (2009).[15] A. Imparato, A. Pelizzola, and M. Zamparo,Phys. Rev. Lett. , 188102 (2009).[16] M. Caraglio, A. Imparato, and A. Pelizzola,J. Chem. Phys. , 065101 (2010).[17] S. Luccioli, A. Imparato, S. Mitternacht, A. Irb¨ack andA. Torcini, Phys. Rev. E , 010902 (2010).[18] M. Zimmer, Chem. Rev. , 759 (2002).[19] J. Dopf and T. M. Horiagon, Gene , 39 (1996). [20] X. Li, G. Zhang, N. Ngo, X. Zhao, S. Kain, and C. C.Huang, Biol. Chem. , 28545 (1997).[21] E. Evans and K. Ritchie, Biophys. J. , 1541 (1997).[22] E. Evans and K. Ritchie, Biophys. J. , 2439 (1999).[23] H. Wako and N. Saitˆo, J. Phys. Soc. Jpn , 1931 (1978);H. Wako and N. Saitˆo, ibid. , 1939 (1978); V. Mu˜noz et al. , Nature , 196 (1997); V. Mu˜noz et al. Proc.Natl. Acad. Sci. USA , 5872 (1998); V. Mu˜noz andW.A. Eaton, ibid. , 11311 (1999); P. Bruscolini and A.Pelizzola, Phys. Rev. Lett. , 258101 (2002). [24] O. K. Dudko, A. E. Filippov, J. Klafter, and M. Urbakh,Proc. Natl. Acad. Sci. U.S.A. , 11378 (2003); O. K.Dudko, G. Hummer, and A. Szabo, Phys. Rev. Lett. ,108101 (2006).[25] M. Schlierf, Z.T. Yew, M. Rief and E. Paci, Biophys. J. , 1620 (2010).[26] H. Dietz, M. Bertz, M. Schlierf, F. Berkemeier, T. Born-schl¨ogl, J. P. Junker, and M. Rief, Nat. Protocols1