Supporting Information. Determination of Equilibrium Constant and Relative Brightness in. FRET-FCS by Including the Third-Order Correlations

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1 Supportng Inforaton Deternaton of Equlbru Constant and Relatve Brghtness n FRET-FCS by Includng the Thrd-Order Correlatons Lngy Meng,, Shanshan He,3 and Xn Sheng Zhao,3 * Bodynac Optcal Iagng Center (BIOPIC; School of Lfe Scences; and 3 Bejng Natonal Laboratory for Molecular Scences, State Key Laboratory for Structural Chestry of Unstable and Stable Speces, and Departent of Checal Bology, College of Chestry and Molecular Engneerng; Pekng Unversty, Bejng 0087, Chna. These authors contrbuted equally. * Correspondng Author, E-al: zhaoxs@pku.edu.cn S. Dervaton of the thrd-order correlatons n FRET-FCS wth dffuson and scannng Followng the conventon n ref. and adoptng the odel and sybols of Fg. n the an text, we consder that n a soluton, reactant olecules at concentraton C fluorescently labeled by A and D dffuse wth the dffuson constant D, and product olecules at concentraton C fluorescently labeled by A and D dffuse wth the sae dffuson constant D. Meanwhle, all olecules have a lnear oton perpendcular to the laser bea at a constant speed v. When the reactant and product are converted back and forth sultaneously by reversble reactons, the te evoluton of the concentraton of the speces s descrbed by δ C (, r t t + δ C (, r t t + ( r, v ( r, ( r, ( r, D δc t δc t k+ δc t k δc t ( r, v ( r, ( r, ( r, D δc t δc t k+ δc t k δc t S. where δ C (, t C (, t C, j, r r S. j j j Wthn a certan te resoluton (bn te Dt, the detected photon nubers (photon S

2 countng of A, D, A and D under a gven experental condton are n Dt d ri(, r tq ( tc(, r t A D A D A n Dt d ri(, r tq ( tc(, r t D n Dt d ri(, r tq ( tc(, r t A n Dt d ri(, r tq ( tc(, r t D S.3 In Eq. (S.3 I(r,t s the laser ntensty descrbed by x + y z I( r, t I0 ( texp S.4 ωxy ω z where ω xy and ω z are the wdths of the laser focus n the xy plane and along the z axs, respectvely. Then, we have dn Dt d ri(, r tq ( tdc(, r t A D A D A dn Dt d ri(, r tq ( tdc(, r t D dn Dt d ri(, r tq ( tdc(, r t A dn Dt d ri(, r tq ( tdc(, r t D S.5 In the followng our language s adapted to the frst strategy n Secton.3 of the an text. The odfcaton for the second strategy should be straghtforward. In an experent, the A detector channel records etted photons fro A and A and the D detector channel records etted photons fro D and D. Usng our prevous ethodology, we wsh to calculate the thrd-order correlaton functons defned by G (3 jk ( t, t δn (0 δn ( t δn ( t j k S.6 nn n j k where, j, k refers to the A or D channel. In ref., consstent wth the defnton of G (3 δn(0 δn(0 δn( t ( t 3 S.7 n the ntal condton s δc ( r,0 δc ( r',0 δc ( r",0 Cδ δ δ( r r' δ( r r " S.8 l n l l ln S

3 where the ndces l,, n are duy varables n a suaton n the dervaton. They have dfferent eanng fro, j, k appeared n Eq. (S.6. In addton,, j, k are not duy varables. To be consstent wth Eq. (S.6, n current case the respectve ntal condton has to be changed to δc ( r, 0 δc ( r', t δc ( r", 0 γ ' ( t Cδ δ( r r' δ( r r " S.9a l n l l ln where γ ' ( t specfes the concentraton correlaton of speces l and between 0 and t l wth the condton γ ' l ( t 0 δl. Under the condton Dt <</k, when t Dt the dfference aong all γ ' ( t should be very sall. Therefore, we ay ake an approxaton that l δc ( r, 0 δc ( r', t δc ( r", 0 γ ( t Cδ δ δ( r r' δ( r r " S.9b l n jk l l ln where γ jk (t s an effectvely averaged concentraton correlaton factor between 0 and t assocated wth Eq. (S.6 wth the property γ jk (0. Then, the defnton of (S.6 dffers fro the defnton of (S.7 only by a factor of γ jk (t. Now, we can plug the above specfc expressons nto the general equatons derved n ref. and have G ( t, t G ( t G ( t G ( t, t S.0 (3 (3 (3 (3 jk D S R, jk The factor due to the dffuson s G (3 D / 8 4 t 4 t ( t V 3t D 3ωtD S. where τ D ωxy / 4D, V π ωxyωz 3, and ω ω z / ωxy. The factor due to the lnear oton s (3 4 t 4 t GS ( t exp + 3 ts 3t D S. where t S ω xy / v. The factor due to the reacton s where ( + + ( α G ( t, t A ( t + ( t e k k t S.3 (3 R, jk jk jk S3

4 A ( t γ j j jk ( t n(0 n ( t ( + K + K j n (0 n ( t jk j C n n ( + KQ ( + KQ n (0 n ( t j S.4 and j n(0 n( t ( Qk K j (0 ( α ( n n t jk t j n(0 n( t + K ( + KQk j (0 ( n n t S.5 S S where n ( τ τi 0 (0, τ Q ( τ (S, j; λ, ; t 0, t. λ λ In Eq. (S.5, f when j one takes t 0 as n ref., n λ (0 s dependent on the photon fluctuaton statstcs. However, f j but t 0 n λ (0 and nλ ( t are no longer correlated. j Slarly, when j n λ (0 and nλ ( t are also not correlated regardless t 0 or not. Therefore, we sply have n (0 n ( t n (0 n ( t n n, when t 0 λ λ λ λ λ λ n (0 n ( t n (0 n ( t n n, when j j j j λ λ λ λ λ λ S.6 Then, Eq. (S.5 becoes ( ( KQ Q QQ j ( Qk K αjk + ( + KQ j k S.7 Specfcally, they are: α α AAA AAD ( QA( QA K ( QD( QD K α DDD + KQA + KQA + KQD + KQD, ( ( ( ( ( QA( QD K ( QD( QA K α DDA + KQA + KQD + KQD + KQA, ( ( ( ( ( QQ A D( QD K ( QQ A D( QA K αadd αdad, αdaa αada ( + KQ Q ( + KQ ( + KQ Q ( + KQ A D D A D A S.8 In the sae anner, under the condton that f j and t 0 or f j Eq. (S.4 becoes ( γ ( t ( + K + KQ Q Ajk ( t C ( + KQ ( + KQ jk j j S.9 S4

5 S. Dervaton of the correlaton functons for a fxed olecule In the followng our language s adapted to the frst strategy n Secton.3 of the an text. The odfcaton for the second strategy should be straghtforward. Consder a fxed speces (A or D converts n a te course between state and state wth the fluorescence brghtness of Q and Q, respectvely (Fg. S. The laser ntensty s I 0 (t, and the bn te s Dt. The forward and backward rate constants are k + and k -. Let P (t be the probablty that the speces are at state. P (t satsfes the followng relaxaton rate equaton dp k+ P+ k ( P ( k+ + k P+ k S. dt If at t0 the speces are at state, the soluton of S. s k+ kt k Pt ( e + S. k k where kk + +k -. Slarly, f when t0, the speces are at state, the te course of P (t s k kt k+ P ( t e + S.3 k k When obtanng Eqs. (S. and (S.3 we used the condton that as t, under the equlbru, the probabltes of the speces at state and, respectvely, are P k k, P S.4 k k + There are 4 possble transtons between the fluorescence brghtness for speces,.e., Q Q, Q Q, Q Q, and Q Q. Now, we calculate the second-order correlaton usng the total photon countng n (0 ( ( nj t G' j ( t S.5 nn j where, ja or D channel. n ( (0 + (0( ( + ( j j (0 nj( t n n n t n t ( ΔtI ( ΔtI 0 0 S5

6 j j ( (0 (0 ( ( (0 (0 ( ( ( j j ( + P Q P Q tpt + Q P Q t Pt + P Q (0 P (0 Q ( t( P ( t Q (0 P (0 Q ( t P ( t S.6 where I0 I0(0 I0( t. Pluggng Eqs. (S. (S.4 nto Eq. (S.6 and notcng that when t 0, Q (0 Q j ( t QQ j, we have l l n (0 n ( t k k k k k k k k k k j j + kt j + + kt + j kt j kt + QQ e + + QQ e + QQ e + QQ e + ( ΔtI0 k k k k k k k k k k ( ΔtI 0 Therefore, k ( kq + kq ( kq + kq kk + ( Q Q( Q Q j j j j kt j j j j ( kq ( kq kk ( Q Q( Q Q j j k ( kq + ( kq + j j ( kq + ( kq + ( Q( Qj K k ( + KQ ( e kt + KQ kt + e + j e S.7 j j nn j j j ( kq + ( kq + ( P Q + P Q( P Q + P Q S.8 k ( j ( Q ( Q K ( + KQ( + KQj j kt kt j G' ( t + e + α e S.9 where Kk + /k - and S S Ql Q / Q ; S, j. The α j n Eq. (S.9 s dentcal to that n Eq...6 n the an text derved fro the dfference of the photon countng when dffuson and/or scannng are consdered. Next, let us calculate the thrd-order correlatons usng the total photon countng G' ( t (3 jk n (0 n ( t n ( t j k S.0 nn n j k where, j, k A or D channel. ( (0 + (0( ( + ( ( ( + ( j j k k (0 j( k( n n n t n t n t n t n n t n t ( ΔtI ( ΔtI S6

7 ( ( j j ( ( + P Q(0 P(0 Q( t Pt ( + Q( t( Pt ( Q( tpt ( j j k + P Q(0 P(0 Q( t Pt ( + Q( t( Pt ( Q( t( Pt ( j j k + P Q (0 P (0 Q ( t P ( t + Q ( t ( P ( t Q ( t( P ( t k + P Q (0 P (0 Q ( t P ( t Q ( t ( P ( t Q ( t P ( t j j k S. where I I (0 I ( t I ( t j. In Eq. (S., the ters contanng Q ( t ( P ( t, κ λ ( κλ,, ; κ λ, can be gnored because when t Dt they are extreely sall. By notcng the slar propertes as that when Eq. (S.7 s derved and approxatng P(0 Pt ( and P(0 Pt ( by kt γ e + jk ( t ( P(0 Pt ( + P(0 Pt ( S. we then have (0 j( k( γ jk ( 3 ( j j ( k k ( j j ( k k e kt k+ QQ + k QQ k+ Q + k Q + k+ k QQ QQ Q Q n n t n t t ( ΔtI 0 k j j k k j j k k γ jk ( t( kqq + Q ( kq + k+ k ( QQ QQ ( Q Q + e ( j j ( k k k kqq + Q kq + γ ( t( kqq Q( kq + ( QQ( Q K j j k k jk + + j k kt e k ( + KQQ j( + KQk kt S.3 nn n ( ΔtI 0 3 ( P Q P Q( P Q P Q ( P Q P Q j k j j k k j j k k ( kq + ( kq + ( kq + 3 k S.4 Therefore, G γ ( t ( k Q Q + k QQ k ( QQ ( Q K + + j j (3 jk + j k kt ' jk ( t, t e j j + ( kq kq + ( kq kq + ( + KQQ j( + KQk γ ( t ( + KQQ ( + K ( QQ ( Q K + e ( + KQ( + KQj ( + KQQ j( + KQk jk j j k kt ( jk A ( t + α e kt jk S.5 Especally, the α jk n Eq. (S.5 s dentcal to that n Eq. (.3.8 n the an text derved fro the dfference of the photon countng when dffuson and/or scannng s consdered. S7

8 S3. The effect of background and cross talk Ref. has dscussed the effect of background photons. Denotng the average sgnal photon countng of (A, D channel be n and the respectve average background photon countng be n b,, t s shown that when the dfference of photon countng s used n the calculaton of the correlaton functons, there are relatons nn ( ( j Gj ( WB Gj ( NB n n n n ( + b, ( j + b, j nn n (3 (3 j k Gjk ( WB Gjk ( NB n n n n n n ( + b, ( j + b, j( k + bk, S3. where WB and NB stand for wth and wthout background respectvely. S3. shows that the background only generates a proportonalty factor. Especally, t does not affect the pre-exponental factors. We have deonstrated that whle workng wth fxed olecules, the correlaton usng total photon countng s convenent. In ths case, one ay fnd out the average background photons fro control experent, subtract t fro the saple fluorescence trace, calculate the background-free FCS functons, and ft the to get K and Q. On the other hand, as deonstrated n Fg. SA, the background can be consdered as part of the sgnal, and one can get the pre-exponental factors drectly fro the correlaton functons that nclude the background. Ths procedure leads to dfferent value on Q but has no effect on K. Ths wll not bother us at all because all the Qs dscussed n ths paper are dependent on the experental condtons and are not calbrated to unque olecular quanttes. In the slar fashon, f there s cross talk between channels A and D and f there s only one relaxaton process, the photons fro speces A or D to channel jd or A can be consdered as part of the sgnals of channel j, and vce versa (Fg. SB. It wll change the effectve relatve brghtness of the speces, but wll not affect the value of K. S8

9 S4. The nfluence of olecular densty By nspectng the dervaton n Secton S, we know that when olecules are n soluton or randoly dstrbuted on surface, the effect of olecular densty appears to be a proportonalty factor n the correlaton functons when dffuson and/or scannng occur. Especally, t does not affect the value of the pre-exponental factors. However, f the sae knd of speces at the sae pont oves together or do not ove, the pre-exponental factors wll be dfferent, whch s very portant for the experents that are conducted on the fxed olecules. If the nfluence of the olecular densty s not carefully and accurately calbrated, the results on the equlbru constant and relatve brghtness wll not be relable, or even no answer can be offered fro the experental data. Let us use the second-order auto-correlaton functon to llustrate the effect of the olecular densty when the experent s carred out on the olecules fxed on a surface and randoly dstrbuted. Denote the olecular relatve fluorescence brghtness be Q(t and the bn te be Dt. Agan, the laser feld s descrbed by Eq. (S.4, but now the z-coponent s not relevant. If, on average, there are N olecules n the laser focus area S 0 πω, the olecular densty s xy ρ N S 0 S4. Consder a suffcently large area S, n whch there s, at least, one olecule and outsde whch the fluorescence sgnals are neglgble. Suppose there are olecules n ths area, we calculate the correlaton G ( aa ( ( 0 nt, j n n n ( r,0 n ( r, t j j n ( r,0 S4. where aa stands for auto-correlaton,, j are ndces for the th and jth olecules, n ( r, τ τi( r, τ Q( τ (l, j s the photon countng of lth olecule located at r l at t (0, t, l l l S9

10 stands for the average over te as a te correlaton does. The nuerator n Eq. (S4. s ( ( n 0 n t n ( r,0 n ( r, t + n ( r,0 n ( r, t S4.3 where n (,0 n (, t j j j j j r r and n ( r,0 n ( r, t are the correlatons between dfferent olecules and wthn the sae olecule, respectvely. Because the relaxaton and photon fluctuaton of dfferent olecules are not correlated, we have r r ( S4.4 n(,0 n(, t Δ t Q II, j j j j where I I( r,0 I( r, t, l, j. The correlaton wthn the sae olecule s exactly that l dscussed n S,.e., Therefore, l l ( ( n ( r,0 n ( r, t Δt Q I +α e kt S4.5 αα n n t t Q II I j ( ( ( 0 Δ kt j + ( + αααe ( Δt Q I + I ( kt α e αα I kt α αα Δt Q I + e I S4.6 The denonator n Eq. (S4. s So, we have n n(,0 Δt Q I r ( S4.7 G I ( t + α e aa I ( aa kt S4.8 Averagng over the olecular spatal dstrbuton, we have S0

11 G I, p ( aa t kt ααα ( + aae + e ' I, p α kt S4.9 where stands for the enseble average and ' s the factor due to the olecular densty. Now, for a randoly dstrbuted saple, there wll be, on average, MρSNS/S 0 olecules n the area S. Let the probablty to have olecules n the area S be P(M,. If we repeatedly carry out the easureent on the saple at dfferent locatons, we wll have G ( aa PM (, kt ααα ( t + αaae + e ' N' kt S4.0 where we call N ' the effectve nuber of olecules n the laser focus area. Obvously, as ρ 0, N '. On the other hand, when ρ s large, the suaton n Eq. (S4.0 can be represented by the ter of M, and the respectve average n Eq. (S4.9 can be approxated by an ntegral, and we have N exp( 4 r / ωxy p d pω 0 xy r r N' N N exp( r / ωxy prdr pω 0 xy S4. Ths s why we call N ' the effectve nuber of olecules. It s nterestng to observe that Eq. (S4. reaches the slar concluson as that n ref. even though stuatons are dfferent. We ust ephasze that Eq. (S4.0 s derved for repeated saplngs of randoly dstrbuted olecules on a surface. In a FCS experent wth fxed olecules, we norally wll put one olecule at the center of the laser focus. The results wll be slghtly dfferent fro that predcted by Eq. (S4.0. Fg. S3 presents the coparson of sulated results aong rando dstrbuton vs. rando dstrbuton but wth one olecule at the center of the laser focus, and the /N approxaton by ntegraton. The sulaton was conducted wthout background. Hgher the background photon countng s the saller S wll be. Then as ρ 0, N ' approaches faster than the background free stuaton. S

12 The nfluence of olecular densty on cross-correlatons and hgher-order correlatons can be derved n the sae anner, and ther relatons are ore coplcated. Because t s very dffcult to obtan accurate olecular densty and because the coplex relatons between the olecular densty and the pre-exponental factors, we recoend that the FCS experent on fxed olecules to be carred out by trackng the fluorescence trajectory fro one olecule at a te under a sngle olecule condton. Even though when the experents are conducted wth dffuson and/or scannng the olecular densty does not, n prncple, affect the pre-exponental factors, we recoend to keep the olecular densty to be about olecule n the laser focus volue (area, because the fttng paraeters can hardly be accurate when N devates too uch fro. References. Wu, Z. Q.; B, H. M.; Pan, S. C.; Meng, L. Y.; Zhao, X. S. Deternaton of Equlbru Constant and Relatve Brghtness n Fluorescence Correlaton Spectroscopy by Consderng Thrd-Order Correlatons. J. Phys. Che. B 06, 0, Yn, Y. D.; Yuan, R. F.; Zhao, X. S. Apltude of Relaxatons n Fluorescence Correlaton Spectroscopy for Fluorophores That Dffuse Together. J. Phys. Che. Lett. 03, 4, S

13 Suppleentary Fgures Fgure S A fluorescence te trace of a fxed speces. Q and Q are the fluorescence brghtness of state and state, respectvely. The fluorescence brghtness, Qt (, vares wth the te. S3

14 Fgure S The effect of background and cross talk for a fxed olecule. A When there s background, the background can be consdered as part of sgnal so that the nonal sgnal s sgnal+background. B When there s cross talk due to the fluorescent spectral overlap, A channel can detect photons etted fro speces D. Let n DA be the photon countng due to the cross talk and n A be the true photon countng fro speces A. Then, the total photon countng n A channel wll be n A + n DA. Vce versa. Both these effects n Fgs. SA and SB change the relatve brghtness of the speces but wll not affect the value of K. S4

15 Fgure S3 Coparson on the / N ' factors. The coparson s ade aong the sulatons of the rando dstrbuton (blue squares and of the rando dstrbuton but wth one olecule at the center of the laser focus (red crcles and the approxaton by ntegraton (black lne. As N decreases, both sulatons approach. As N ncreases, both sulatons approach the /N approxaton. S5

16 Fgure S4 Sulated FCS curves on fxed and scanned olecules. A and B are for fxed olecule. C and D are for scanned olecules at N 0.56 olecules/µ, ω xy 0.3 µ, and v µ/s. The sulatons were carred out by takng the photon countng wthout Posson dstrbuton. The nput paraeters were k00 s -, K, Q A 0., Q D 4. S6

17 Fgure S5 FCS curves on the FRET labeled DNA syste wth 0 µs bn te. A The auto-correlaton curves. B The cross-correlaton curves. Rch knetc processes were contaned n the FCS curves wthn the broad te range. S7