Supplementary Figure 1. Numeric simulation results under a high level of background noise. Relative

Transcription

1 Suppleentary Inforation Suppleentary Figure. Nueric siulation results under a high level of bacground noise. Relative variance of lifetie estiators, var( ˆ ) /, noralised by the average nuber of photons, EN, is plotted as a function of T/τ for different lifetie fitting algoriths. The bacground noise included in the siulation was 0 ties higher than that used to obtain Figure a. Note that the LE-NF ethod failed to provide robust calculation in this case, which is explained in the rear about LE-NF in Suppleentary Note 5.

2 Suppleentary Figure. Spectral inforation about the LRET dyes. The eission spectru of Eu(TTFA) 3 and the excitation spectru of the hexafluorophosphate salt of cationic couarin show the required overlap for energy transfer.

3 Suppleentary Table. DNA sequences used in the deonstration of ultiplexed detection. Capture Probe Target Probe Nae DNA Sequence(5 to 3 )* odification(5 ) Control TTT GGG TTC CTC CAG ATT GAG GTC TTC HIV TTT GGG GTC ATG TTA TTC CAA ATA TCT EV TTT GGG ATA CTG TTC TCC AAC ATT TAC Aino odifier C HBV TTT GGG ATC ATC CAT ATA ACT GAA AGC HPV-6 TTT GGG AAT GCT AGT GCT TAT GCA GCA HIV AGA AGA TAT TTG GAA TAA CAT GAC EV GGA GTA AAT GTT GGA GAA CAG TAT HBV TTG GCT TTC AGT TAT ATG GAT GAT Biotin HPV-6 ATT TGC TGC ATA AGC ACT AGC ATT * Nucleobases ared in italic are copleentary sequences.

4 Definition of sybols used in the following Suppleentary Notes. β Luinescence decay rate; also as β i t Tie; t 0 N(t) Rando variable as the nuber of events occurring by tie t; also as N (t), N (t) and N (t) t N n τ T P λ λ(t) λ T n T * End tie for each detection channels; 0 < t < t Rando variable as the nuber of events occurred in the th detection channel Observed outcoes for the nuber of events occurred in the th detection channel Total nuber of detection channels Luinescence lifetie; also as τ i Width of every detection channel Probability, depending on the context Intensity (rate) of a hoogeneous Poisson process Intensity (rate) function of a nonhoogeneous Poisson process; also as λ (t) Paraeter for the Poisson distribution of N Rando variable as the elapsed tie between the (n )th and the nth event Rando variable as the waiting tie for a next photon to be eitted N 0 Nuber of eitters on the excited state at t = 0 A Intensity of the luinescence decay; A = N 0 β; also as A i B Intensity of the bacground noise as a hoogeneous Poisson process CRLB Craér Rao Lower Bound L Lielihood function I Fisher inforation atrix J Jacobian atrix for variable substitution x, S Substitutive variables for β and A; x = βt, S = A/β; also as x i and S i N C f(,t) N R Observed outcoe for the total nuber of photons in all channels Coponent in the CRLB related to the detection configuration of and T Rando variable of the total nuber of photons in all channels; N = N(T) Substitutive variable for B; R = BT I * Kullbac-Leibler iniu discriination inforation; also as I j * p,j * τ j γ j Q D, D, D 3 U, V b b i Y X z, z ^ Probability for the jth decay pattern that a photon is collected in the th channel; also for p,j Candidate lifetie value in the jth decay pattern used in the LE-PR algorith Candidate bacground level in the jth decay pattern used in the LE-PR algorith The greatest positive integer satisfying 3Q Partial suation used in the RLD algorith Successive integration (suation) used in the SI algorith Paraeter vector to be solved in the overdeterined equation used in the SI algorith Eleents in b Coefficient vector in the overdeterined equation used in the SI algorith Coefficient atrix in the overdeterined equation used in the SI algorith Auxiliary variables used in the SI algorith for dual-exponential decays Estiator of the paraeter under the caret

5 Suppleentary Note The proble to be solved for lifetie coputation. We consider a syste of a nuber of independent eitters that are all excited at t 0 = 0, and assue that each of the has an identical rate β [s - ] to produce a single photon. Let N(t) represent the total nuber of events i.e. eitted photons that occur by tie t. {N(t), t 0} is a stochastic process influenced by β, and possibly by soe bacground noise as well. For any increental tie series t > 0 ( =,, ), N(t ) are rando variables; so are the nubers of photons in adjacent tie intervals, N = N(t ) N(t - ). In reality, we can only easure the nubers of photons, n ( =,, ), in tie intervals of (0, t ], (t, t ],, (t -, t ], respectively; these intervals are referred to as detection channels used to onitor our syste. We are assuing these channels have equal width T, hence t = T ( = 0,,, ); while T is the entire length of the detection window. We wish to use these observed values to estiate the rate paraeter β, (the reciprocal of the lifetie τ, as shown below). The objectives of this wor are to deterine: ) How the rate paraeter β affects the rando variables N ; ) The best ethod of utilising n to estiate β and thus τ, in ters of accuracy, speed and robustness; 3) The effect of detection configuration, i.e. the selection of T and, upon the above two aspects. Objective () is addressed in Suppleentary Note 4 when discussing the lielihood function. Objective () and (3) are addressed in the ain text.

6 Suppleentary Note Bacground nowledge about the Poisson process. A Poisson process is a counting process {N(t), t 0} of independent events where two events cannot occur siultaneously. In an infinitesial tie interval of (t, t + dt], the probability of a single event is proportional to an intensity λ(t) ties the interval, dp P{ N( t dt) N( t) } ( t) dt (Suppleentary Equation ) If λ(t) λ is a constant, the Poisson process is referred to as hoogeneous. Otherwise, in ore general cases where the intensity λ(t) is a function of tie, the Poisson process is referred to as nonhoogeneous. The following leas about Poisson Process can be found in coon textboos on stochastic processes (e.g. Stochastic Processes, and Introduction to Probability odels, both by S.. Ross). Lea (the Poisson property): For Poisson processes, no atter whether hoogeneous or nonhoogeneous, one of their unique properties is that the rando variables N are independently Poisson distributed with paraeters λ, respectively, denoted as N ~ Po(λ ), so that n e P( N n) n! T ( ) T () t dt (Suppleentary Equation ) In addition, the total nuber of events in all channels, N( T ) N T variable, N(T) ~ Po ( t) dt 0 Poisson-distributed rando variable is equal to its paraeter, therefore, is also Poisson distributed as a rando, and independent with all N. Note that the expected value of a E N,,..., E N( T ) ( t) dt 0 T (Suppleentary Equation 3) Lea (the eoryless property): Let us denote the tie of the first event by T, and the elapsed tie between the (n )th and the nth event by T n for n >. Another unique property, associated with the independent increents of Poisson processes, is that the ties between each pair of consecutive events, T n, are independent rando variables. In particular, for hoogeneous Poisson process with a constant intensity λ, T n have identical exponential distribution with paraeter λ, denoted as T n ~ Exp(λ), i.i.d. for any positive integer n.

7 Their probability density function is d t p( t) P( Tn t) e, t 0 (Suppleentary Equation 4) dt Note that the expected value of an exponential-distributed rando variable is equal to the reciprocal of its paraeter, therefore ET n (Suppleentary Equation 5) Lea 3 (the additivity property): If {N (t), t 0} and {N (t), t 0} are independent Poisson processes, with respective intensity functions λ (t) and λ (t), {N (t) = N (t) + N (t), t 0} is also a Poisson process with intensity function λ (t) + λ (t). It is obvious that the su Poisson process is hoogeneous if only both Poisson processes are hoogeneous; otherwise it is nonhoogeneous.

8 Suppleentary Note 3 The Poisson process behind fluorescence/luinescence. Here we suarise the atheatical odel of fluorescence/luinescence intensity in our syste under exaination. Step : In quantu echanics it states that β is related to the probability per unit tie that a photon will be eitted, so that dp dt (Suppleentary Equation 6) wherein dp is the differential probability that a photon will be eitted in the infinitesial tie interval dt. (This approach ensures that the nuber of photons actually observed ust be a nonnegative integer). By coparing Suppleentary Equation 6 with Suppleentary Equation, it is apparent that the counting process of eitted photons in this case is a hoogeneous Poisson process, with a constant intensity equals to the decay rate β, as it is well nown. According to Lea, the waiting tie T * for the next photon to be eitted is an exponentially-distributed rando variable satisfying Suppleentary Equation 4, so that t dp( T t) e dt (Suppleentary Equation 7) wherein dp(t) is the differential probability for a next photon to be eitted in the infinitesial interval dt. Note that by using Suppleentary Equation 5, the expected value of T * is given by ET (Suppleentary Equation 8) This indicates the reciprocal of the decay rate reflects the average tie for a photon to be eitted. Thus, this average tie is called the lifetie of the excited state, as denoted by τ. Step : Now let us consider the case that ultiple (N 0 ) eitters exist at excited state. The ties for each of the to eit a photon are rando variables, and these are independently distributed with the identical exponential distribution. Therefore, the probability per unit tie that a photon will be eitted is not only proportional to the decay rate β, but also proportional to the nuber of eitters reaining in the excited state at that tie, which is N 0 e -βt. Thus, if we let A = N 0 β,

9 t t dp N0e dt Ae dt (Suppleentary Equation 9) By coparing Suppleentary Equation 9 with Suppleentary Equation, it is clear that the counting process of luinescence eission with ultiple eitters at the beginning can be regarded as a nonhoogeneous Poisson process, with its intensity function given by () t Ae t (Suppleentary Equation 0) Step 3: In general, this atheatical odel is appropriate not only for onoexponential decays, but also for ultiexponential decays, as well as either with or without bacground noise (which can be treated as an independent hoogeneous Poisson process with a constant intensity B), according to Lea 3. In such case, the intensity function of the nonhoogeneous Poisson process changes to i () t Ae t i B (Suppleentary Equation ) i

10 Suppleentary Note 4 The Craér Rao Lower Bound for lifetie estiators. Principle: The Craér Rao inequality of the estiation theory states that, the lowest possible variance of any unbiased estiators (i.e. rules to calculate estiates for unnown paraeters), called Craér Rao Lower Bound or CRLB in short, is given by the inverse of the Fisher inforation atrix (ref. Fundaentals of Statistical Signal Processing: Estiation Theory, by S.. Kay). We will calculate this CRLB in our odel of onoexponential decay without bacground, as described by Suppleentary Equation 0, where the unnown paraeters are τ ( /β) and A. The lielihood function of τ and A, given the observed nubers of photons in the detection channels n ( =,, ), is equal to the probability of those observed outcoes given those paraeter values, that is L(, A n, n,, n ) P( N n, N n,, N n, A) (Suppleentary Equation ) The Fisher inforation atrix is given by ln L(, A) L(, A) A I (, A) (Suppleentary Equation 3) L(, A) L(, A) A A The Craér Rao inequality states that any unbiased estiators ˆ and  ust satisfy ˆ ˆ cov(, A) I (, A) (Suppleentary Equation 4) In the case if variable substitution is conducted, and if the Jacobian atrix for variable substitution is given by x S J, A( xs, ) (Suppleentary Equation 5) A A x S we get cov( ˆ, Aˆ ) J ( x, S) I ( x, S) J ( x, S) (Suppleentary Equation 6) T, A, A where the Fisher inforation atrix is a function of x and S,

11 ln L ln L x xs I ( xs, ) (Suppleentary Equation 7) ln L ln L xs S Calculation: Assuing our onoexponential decays and the detection configuration described in Suppleentary Note, it is apparent based on Lea that N ~ Po(λ ), with λ derived by using Suppleentary Equation and 0, n e P( N n) n! A e T ( e T ) (Suppleentary Equation 8) Lea also states the independency of N, so that the joint probability function is given by P( N n, N n,, N n) e (Suppleentary Equation 9) n! n By using Suppleentary Equation, the log lielihood function is given by ln L ln P( n, n,, n ) C n ln ln( n!) x x N ln[ S( e )] x n S( e ) ln( n!) (Suppleentary Equation 0) where x = βt, S = A/β; and N C n is the total nuber of collected photons, as an observed outcoe for the rando variable N(T). According to Lea, Suppleentary Equation 3 and 0, N(T) ~ Po S( e x ), thus E N( T ) S( e x ) (Suppleentary Equation ) Taing partial derivatives for Suppleentary Equation 0, we have

12 x ln L e NC n x Se x e ln L N S S ln L N x C x e C x ln L e xs ln L N S S C x e ( e ) x x S e x (Suppleentary Equation ) Substituting Suppleentary Equation into Suppleentary Equation 7, the Fisher inforation atrix (as a function in x and S) can be derived as x x e ( e ) x x S S e e x ( e ) I ( xs, ) (Suppleentary Equation 3) x x e e S The Jacobian atrix for the variable substitution is T 0 (, ) x J, A xs (Suppleentary Equation 4) S x T T Substituting Suppleentary Equation 3 and 4 into Suppleentary Equation 6, we obtain x T e x T S 0 e ˆ ( ˆ x S x T cov, A) x x x x e ( e ) x S x x e ( e ) x x e x e S S e 0 x ( e ) T T ( e ) T (Suppleentary Equation 5) Thus, the variance for any lifetie estiators ˆ (which is equal to the first eleent in the covariance atrix), is liited by ˆ T var( ˆ ) cov ( ˆ, A), (Suppleentary Equation 6) 4 x x Sx e ( e ) x x ( e ) e In other words, the Craér Rao Lower Bound (CRLB) for lifetie (τ) is T CRLB( ) Sx 4 x x e ( e ) x x ( e ) e (Suppleentary Equation 7)

13 Suppleentary Equation 7 can be rewritten with the help of Suppleentary Equation to better reflect the contributing factors to the CRLB, CRLB( ) (, ) E N( T ) f T (Suppleentary Equation 8) wherein CRLB( ) f (, T ) E N( T ) T / T / T e e T / T / ( e ) ( e ) (Suppleentary Equation 9) Note: For siplicity purpose, the rando variable of the total nuber of collected photons is expressed by N, instead of N(T), in the ain text, as well as in the following sections. Rear: The Objective () proposed in Suppleentary Note, which is to deterine how the decay rate β affects the observed outcoes of the rando variables of N, is achieved by the derivation of Suppleentary Equation 9 and 0. The CRLB obtained in Suppleentary Equation 7 9 is applicable to the case without bacground noise. It provides a valuable reference when coparing the accuracy of different lifetie fitting algoriths, since no algorith can achieve a better variance than the CRLB. If a bacground noise baseline is considered in the odel, no close for can be derived for the CRLB. However, fitting algoriths can still be copared to deterine which one is the ost accurate.

14 Suppleentary Note 5 Brief descriptions of the four lifetie fitting algoriths. The detection configuration reains the sae as described in Suppleentary Note. The coputation forulae for each fitting algorith are given for the case of onoexponential decay with bacground noise (for SI, the coputation forula for the case of dual-exponential decay with bacground noise is also derived in Suppleentary Note 6). The rate function of the Poisson process for the onoexponential decay with bacground noise is: t () t Ae B (Suppleentary Equation 30) Thus, for =,,, N ~ Po(λ ), and x x Se ( e ) R EN (Suppleentary Equation 3) where x = βt, S = A/β, R = BT Algorith A. Nonlinear Fitting based on axiu Lielihood Estiation (LE-NF) Due to the introduction of bacground noise, by using Suppleentary Equation 3, the log lielihood function in Suppleentary Equation 0 becoes: ln L( x, S, R) n ln ln( n!) x x x n ln[ Se ( e ) R] S( e ) R ln( n!) (Suppleentary Equation 3) The axiu lielihood estiation requires us to find estiators ˆx, Ŝ and ˆR, so that the log lielihood (lnl) is axiised; in this case the lifetie estiator is thus ˆ T / xˆ. In general, no close-for forula exists for ˆx, therefore it requires solving a nonlinear optiisation proble. This can be realised by a range of nuerical algoriths, such as the Nelder-ead siplex algorith used in our siulation (by finsearch function in ATLAB), or other iterative ethods. Note that the last ter in Suppleentary Equation 3 is constant for a recorded decay curve, and thus can be oitted during coputation. Rear: LE-NF provides supree accuracy even for noisy decay curves if the fluctuations are caused by the nature of Poisson process with wea signals (i.e. shot noise). However, when the noise is fro external

15 bacground (e.g. dar counts), once the signal-to-bacground ratio is insufficient, the accuracy of LE-NF ay be ipaired due to iproper selection of initial values as well as constraint conditions for nonlinear fitting algoriths, which ay find local axia instead of global axia, or, soeties, ay siply fail to converge (refer to Suppleentary Figure ). Algorith B. Pattern Recognition based on axiu Lielihood Estiation (LE-PR) LE-PR is an approach to iprove the robustness of LE, by proposing a discrete set of possible patterns and seeing the pattern with axiu lielihood (the one that loos ost siilar) to the observed data. To achieve this, it requires easuring the extent of siilarity between a proposed pattern and the observed data curve. Siple easureents such as cross correlation could be used; however, the optiised easureent fro a statistic point of view is the Kullbac-Leibler iniu discriination inforation: I j n (Suppleentary Equation 33) nln Np, j in which p,j ( =,, ) is the series of probability for the jth decay pattern that a photon is collected in the th channel, provided it is observed (i.e. p, j ). The observed data has the axiu lielihood of being the jth decay pattern if its iniu discriination inforation I j gives the sallest value of all I *. The values of p,j can be calculated fro a pair of decay paraeters (τ j *, γ j ) for a given detection configuration, in which γ j reflects the aount of bacground noise. If the bacground is excluded, it can be derived that p e ( ) T / j T / j, j T / j e e (Suppleentary Equation 34) When the bacground is considered, we obtain: p ( ) p / (Suppleentary Equation 35), j j, j j Note that the sus (upon ) of p,j as well as p, j are noralised to unity. The candidate patterns are typically selected with a sequence of τ j * (with several levels of γ j ), so that they cover a range where we expect our lifetie to be. In our siulation, τ j * were set as,,, 400 μs; γ j were set

16 as 0, 0.0,, 0.0 for Figure a and up to 0.30 for Suppleentary Figure. Rear: Copared with LE-NF, LE-PR guarantees a robust coputation at slightly reduced precision, depending on the nubers of decay patterns for calculating the iniu discriination inforation. However, the coplexity of LE-PR is quite substantial if high precision is required, especially if it is applied to fit ultiexponential decays, since every extra paraeter introduces one additional diension for the atrix of decay patterns. Algorith C. Rapid Lifetie Deterination (RLD) This ethod inherently taes advantage of a high level of data binning (regardless of the binning pretreatent described in the ain text), so that the lifetie coputation is extreely siple and fast. It partitions the photon counts in different detection channels into three parts: Q Q Q 3Q Q Qx EN S Se RQ E E Qx Qx N Se Se RQ Qx 3Qx N Se Se RQ (Suppleentary Equation 36) in which Q is the greatest positive integer satisfying 3Q. When using the observed n ( =,, 3Q) to substitute for the expected values of the rando variables, we have a lifetie estiator QT ˆ D D ln D D 3 (Suppleentary Equation 37) in which Q Q 3Q. D n, D n, D n 3 Q Q Rear: RLD is probably the fastest and siplest lifetie fitting algorith. However, it is evident that RLD ay provide inaccurate results, depending on the length of the detection window. On one hand, if the window is too short, D D will be siilar to D 3 D, so that the ajor denoinator is approaching zero, causing significant errors in the lifetie estiates. On the other hand, if the window is too long, another denoinator D 3 D approaches zero, also leading to incorrect results. In extree cases when the rando fluctuations of the decay curve are large, physically ipossible results of negative lifeties (exponential growth, if D 3 D greater

17 than D D ) ay be obtained, or the estiator ay have no solution (if either D 3 D or D D becoes negative). Although other odified forulae ay be utilised (ref. S.P. Chan, et al., Analytical Cheistry 73 (8): ), in general RLD is not sufficiently accurate and robust. oreover, it is not flexible enough for decoding lifeties in a large dynaic range, since the length of the detection window has to be altered accordingly in order to optiise the precision for calculating different lifeties. Algorith D. ethod of Successive Integration (SI) For the SI ethod, first consider the total nuber of photons in the first channels. Based on Suppleentary Equation 3, we have: x EN S Se R (Suppleentary Equation 38) On the other hand, Suppleentary Equation 3 also gives Se x EN R x e Substituting Suppleentary Equation 39 into Suppleentary Equation 38 gives (Suppleentary Equation 39) EN R EN S R (Suppleentary Equation 40) x e or alternatively, x x x E N ( e ) E N R( e ) S( e ) R (Suppleentary Equation 4) The observed n ( =,, ) are now used to substitute for the expected values of the rando variables. The ensuing set ( equations) of Suppleentary Equation 4 gives: n U b n U b b3 n U (Suppleentary Equation 4) where U x x x n, b e, b R( e ), b3 S( e ) R. Suppleentary Equation 4 is an overdeterined equation which has the atrix for of Y = Xb. It has an approxiate solution (in the sense of least squares) given by:

18 ˆ ( ) T - T b X X X Y (Suppleentary Equation 43) Hence, the lifetie estiator via the SI is given by ˆ T / ln( b ˆ ). Rear: The nae of Successive Integration coes fro its initial description in 993 by K. Tittelbach-Helrich, in which integration was used for analog wavefors of exponential decays. However, although atheatically rigorous, integration is generally replaced by suation in ost cases, since discrete (sapled) wavefors are typically obtained in reality.

19 Suppleentary Note 6. SI algorith for ultiexponential decays The SI forulae can be generalised to fit ultiexponential decays. As an exaple we now consider a dual-exponential decay with bacground noise. Its rate function of Poisson process is () t Ae A e B (Suppleentary Equation 44) t t Suppleentary Equation 3 now becoes E N S e ( e ) S e ( e ) R (Suppleentary Equation 45) x x x x in which x T, x T, S A /, S A /, R BT. Therefore, l l EN S S e S S e R x x x x e e ( ) x x EN S S S S R e e (Suppleentary Equation 46) Siilarly, we can use the observed n ( =,, ) to substitute for the expected values of the rando variables. Let U n, V x x I, z e and z e. This gives: l l ( ) n zzv ( z z) U zzr ( S S) zz R( z z) zs zs R (Suppleentary Equation 47) or, in the atrix for, n V U b b n V U 3 b3 b 4 n V U ( ) / b5 (Suppleentary Equation 48) Liewise, this overdeterined equation Suppleentary Equation 48 has an approxiate solution ˆb in the sense of least squares. Here, ẑ and ẑ as the two roots of the equation ˆ ˆ 0. Hence, the lifetie z bz b estiates fro the SI is given by ˆ T / ln( zˆ ) and ˆ T / ln( zˆ ). By applying the above integration (suation) in a recursive way thus called Successive Integration one can fit a ultiexponential decay with the nuber of paraeters (including the lifeties, the aplitudes, and the bacground) up to the nuber of detection channels, which is the degree of freedo of the overdeterined syste.