Homogeneity tests for Michaelis-Menten curves with application to fluorescence resonance energy transfer data

1 Homogeneity tests for Michaelis-Menten curves with application to fluorescence resonance energy transfer data Amparo Baíllo 1* , Laura Martínez-Muñoz 2 , Mario Mellado 2 1 Departamento de Matemáticas, Universidad A u tónoma de Madrid, Ciudad Universitaria de Cantoblanco, 28049 Madrid, Spain 2 Department of Immunology and Oncology, Cent ro Nacional de Biotecnología/CSIC, Ciudad Universitaria de Cantoblanco, 28049 Madrid, Spain * Corresponding author E-mail addresses: AB: amparo.baillo@uam.es LMM: lmmunoz@cnb.csic.es MM: mmellado@cnb.csic.es 2 Abstract Background Resonance energy transfer (RET) methods are in wide use for eval uating protein-protein interactions and protein c onformational changes in livi ng cells. Sensitized emission fluorescence resonance energy transfer (FRET) m easures energy transfer as a function of the acceptor-to-donor ratio, generating FRET saturati on curves. Modeling the saturation curves by Michaelis-Menten kinetics allows ch aracterization by two parameters, FRET max and FRET 50 . These parameters allow evaluation of a pparent affinity between two proteins and comparison of this affinity in different expe rim ental conditions. To reduce the effect of sampling variability, several replications (stati stical sam ples) of the saturation curve are generated in the same biological conditions. He re we study procedures to determ ine whether statistical samples in a collection are homogeneous, in the sense that the y are extracted from the same underlying saturation curve (or regression model). Results We used three methods to determine which statistical sam ples in a group are homogeneous. From the hypothesis testing view point, we considered two procedures: one based on bootstrap resampling and the other, a version of a classical F test. The third method analyzed the problem from the model selecti on viewpoint, and used the Akaike inform ation criterion (AIC). Although we only considered the Michael is-Menten model, all thre e statistical procedures would also be applicable to any other nonlinear regression model. W e compared the performance of the three hom ogeneity testing methods in a Monte Carlo study and through analysis in livin g cells of FRET satu ration curves for dimeric complexes of CXCR4, a seven-transmembrane receptor of the G protein-coupled receptor family. 3 Conclusions The simulation study and analysis of real FRET data showed that the F test, the bootstrap procedure and the model selec tion m ethod lead in general to similar conclusions, although AIC gave the best results when samp le sizes were small, whereas the F test and the bootstrap method were more appropriate for large sam ples. In practice, all three methods are easy to use simultaneously and show consistency, faci litating conclusions on sample homogeneity. Background Oligomerization, the formation of a com plex by two or more proteins, is a subset of protein-protein interactions that generates considerab le functi onal diversity. It frequently operates in the transduction of signals that beg in at the cell surface and continue to the nucleus, in pathways that participate in antigen receptor signaling, cytokine responses, regulation of gene transcription, an d in so many other processes that it clearly constitu tes a major mechanism in the regulation of cell re sponses. The classical biochemical methods for m onitoring these inte ractions include coimmunoprecipitation and wester n blot studies using untagge d or tagged proteins, or crosslinking analysis, which uses solubilizing detergents and disrupted cells; both can introduce artifacts. Difficulty increases when we evaluate proteins with a com plex structure at the cell membrane, such as the G protein-coup led receptors (GPCR). Th is protein family is characterized by seven α -helical dom ains that span the cell membrane, and is one of the m ost abundant in nature. It include s receptors for h ormones, neurotransm itters, chemokines and calcium ions, among others, and is thus a focal point of the pharm aceutical industry’s effort to develop antagonists for therapeutic use in man. New methods based on resonance energy transf er (RET) are becoming widespread for the evaluation of protein-protein inte ractions in living cells. Thes e techniques can also be used to define protein rearrangement, conform ation dynamics or the role of ligand and receptor 4 levels, to screen for antagonists, and to study di merization sites within the cell [1]. There are two main types of RET, fluores cence resonance energy transf er (FRET) and bioluminescence resonance energy transfer (BRET); in the form er, the donor is a fluorescent protein that transfers energy to an acceptor fluorescent m olecu le and in the latter, the donor molecule is luminescent [2, 3]. FRET is based on non-radiative energy transm ission from a donor fl uorophore to a nearby acceptor, without photon emission. Energy tran sfer depends on the overlap between the donor emission spectrum and acceptor abso rp tion, on the distance between donor and acceptor (which m ust be in the 2-10 nm range ), and requires correct orientation between donor and acceptor fluorophores [4, 5]. FRET determ inations use the protein of in terest fused to distinct spectral variants of the green fl uorescent protein (GFP); the most commonly used variants are cyan (CFP) as donor an d yellow (YFP) as acceptor [6, 7]. Sensitized emission FRET allows measurem ent of energy transfer in reference to the acceptor-to-donor ratio to generate FRET saturation curves. These curves describ e FRET efficiency as a function of th e acceptor-to-donor ratio and are characterized by two important parameters. The first, B max (also frequently denoted FRET max ), is the (asym ptotic) maxim um of the curve. If energy transf er reaches saturation and the cu rve is hyperbolic, we define a second parameter, usually denoted K d or FRET 50 , that corresponds to the acceptor-to-donor ratio that yields half FRET max efficiency. The parameter K d allows estimation of the apparent affinity between the partners involved [8, 9]. Both parameters depend on the distance between donor and acceptor, and on their orientation in the complex; B max and K d are thus directly related to the energy transferred, a nd therefore to the numbe r of protein com plexes formed and/or to changes in complex conforma tions. Although here we have only considered FRET curves, the same analysis and conc lusions are applicab le to BRET titration. 5 B max and K d allow evaluation of oligom erization of tw o proteins (dimerization) in different experimental conditions. For example, they can be used to assess how the presence of a third coexpressed protein affects complex conformation. In practice, B max and K d are estimated using a statistical sample of points ) , ( i i y x , n i , , 1 K = , from the saturation curve. To avoid misunderstanding, we note that when we refer to “sample” in the statistical sense, w e refer to that which in biology is termed a “saturati on curve”. To reduce the effect of sampling variability on the com parison of two distinct ex perimental conditions, several replications (or statistical samples) of the saturation curve are usually generated in each condition. For this study, we used seven statistical samples ) , ( ij ij y x , 7 , , 1 K = i , i n j , , 1 K = of a FRET saturation curve for CXCR4 dimers in living ce lls (Figure 1; see Results an d Discussion for more inform ation on the data). CXCR4 is a chemokine receptor of the GPCR family, with key roles in homeostasis and pa thology. Mice lacking CXCR4 die perinatally and have defects in vascular development, he matopoiesis and cardiogenesis [10]. CXCR4 is also implicated in cancer [ 11], rheumatoid arthritis [12] and pulmonary fibrosis [13]; finally, together with CCR5, CXCR4 is one the main coreceptors for HIV-1 infection [14]. Statistical analysis is necessary to determ ine which samples are hom ogeneous, in the sense that they are extracted from the same underl ying saturation curve (regression model). The homogeneous samples from each saturation curv e will be those finally considered for comparison of distinct experim ental conditions, e. g., alone or in the pres ence of an additional protein. Our goal was to define a method for reliable co m parison of protein dimerization before and after a specific change in expe rimental conditions (statistica lly also term ed “treatments”). If there is only one version or statistical sam ple of each distinct experim ent, it is reasonable to use any of the procedures (a t test, for instance) described by Motulsky & Christopoulos [15]. In other cases, several statistical samples are obt ained of the experiment before and after the 6 change, which gives rise to several estimat ed FRET curves in the two experimental conditions. Curves in these two groups are sometimes naturally paired, for example, when dimerization is evaluated in the same group of cells before and after the change in experimental conditions. We then have I pairs of saturation curves, and a t test can be used to compare the components of each pair (for exampl e, see [15-17]). If the m ajority of the I p- values from the t tests is <0.05, the conc lusion is th at the effect of the treatment is statistically significant. When the curves before and after the treatment are not paired, it is reasonable to focus first on each of the two samples of curv es separately. A sam ple of I saturation curves represents I realizations in th e same experim ental conditions, wh ich (intuitively) should correspond to observations of the same probability model. One possibl e procedure is to fit two random- effects models, one to the experiment before the treatm ent and another to that after th e treatment [15]. The realizati ons of the random effects in a specific model would account for the differences between versions of the same experiment (s ee [18] for the rando m-effects version of the Michaelis-Menten model (1), us ed to describe saturatio n curves). Although this idea is appealing, the appropriateness of fitting a random-effects m odel when the number of samples I is low (e.g., I = 3) is questionable. Here we propose to verify, via a homoge neity test of hypotheses, which of the I realizations of an experiment can be accepted to com e from the same Michaelis-Menten model. Statistical samples co rresponding to homogeneous outcom es can then be pooled into a unique sample from the common underlying model. Once we have determ ined a homogeneous sample for each experiment (bef ore and after the change), we can apply a t or an F test to determine whether th ere are differ ences due to the change in exp erimental conditions. For the homogeneity test, we cons ider two different testing procedures (an F test 7 and a resampling-based scheme) and compare th eir behavior via a simulation study and the analysis of real FRET data; in all cases, we also com puted the Akaike information criterion (AIC) for the models implied by the null and a lternative hypotheses. Although the theories underlying the information-theoretic approach and null-hypothesis tes ting differ [19], the conclusions derived from all the approaches are in general the same. Methods Michaelis-Menten model The saturation curve, that is, FRET efficiency ( Y ) as a function of the acceptor-to-donor ratio ( X ), is usually described via the nonlinear re gression model of Michaelis-Menten [5, 15, 20, 21] Y = B max X K d + X + σε , (1) where ε follows a standard normal distribution. Throughout this study, we estimate the unknown parameters ) , ( max d K B = θ and σ by maximum likelihood. For a sample ) , ( i i y x , n i , , 1 K = , of independent observations of model (1) , the max imum l ikeli hood e stimat ors (m.l.e.) are given by ) ( min arg ) ˆ , ˆ ( ˆ max θ θ θ S K B d = = and n S / ) ( ˆ 2 θ σ = , where 2 max 1 )) /( ( ) ( i d i n i i x K x B y S + − = ∑ = θ is the residual sum of squares. The homogeneity test for Michaelis-Menten curves We have data ) , ( ij ij y x , I i , , 1 K = , i n j , , 1 K = , from I realizations of an experim ent. For each i , the sample ) , ( ij ij y x , i n j , , 1 K = , is assumed to be observed from the model Y = B max; i X K d ; i + X + σε , (2) where ε follows a standard normal distribution. 8 We wish to test whether the data fro m the I samples are observations from the same model of type (1), that is, we are interested in the hypothesis test : 0 H ) ( 0 2 1 θ θ θ θ = = = = I K (3) : 1 H j i θ θ ≠ for some k j ≠ where ) , ( ; max; i d i i K B = θ . Accepting 0 H means that all the observations ) , ( ij ij y x , I i , , 1 K = , i n j , , 1 K = are outcomes of the same experiment and can be pooled into a sam ple of size I n n n + + = K 1 to give a single estimation of 0 θ . This would be the desirable conclusion when the I samples are observed in the same experim e ntal conditions (as in Figure 1). Since the data are observations of pr otein-protein interactions in live cells, however, it is frequent that, due to uncontrollable f actors, at least one of the I statistical sam ples appears to be different from the m ajority (see, for exam ple, sample 7 in Figure 1). As the final a im is to compare results in distinct experimental condi tions, it is important fi rst to decide which estimated saturation curves are hom ogeneous in the same experimental conditions. Rejection regions Let us first fix the following notation. Under 1 H the m.l.e. of i θ is given by ) ( min arg ) ˆ , ˆ ( ˆ ; max; θ θ θ i i d i i S K B = = , where 2 1 max ) ( ∑ = ⎟ ⎟ ⎠ ⎞ ⎜ ⎜ ⎝ ⎛ + − = i n j ij d ij ij i x K x B y S θ and the m.l.e. of 0 θ under 0 H is ∑ = = = I i i d S K B 1 0 ; 0 max; 0 ) ( min arg ) ˆ , ˆ ( ˆ θ θ θ . We consider two possible ways of constructing a rejection region for th e test (3); one is based on a bootstrap resampling scheme [2 2] and the other is derived from an F test. Bootstrap rejection region Let us consider the test statistic 9 ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ − ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = ∑ ∑ = = I i i i I i i S S T 1 1 0 ) ˆ ( log ) ˆ ( log θ θ (4) proportional to the log-like lihood ratio. We reject 0 H when T > T 0; α , the ) 1 ( α − quantile of T under 0 H . The value of T 0; α has been approximated via boot strap, with the following algorithm: 1. For the original sample ) , ( ij ij y x , I i , , 1 K = , i n j , , 1 K = , compute the m.l.e. 0 ˆ θ and 2 ˆ σ under 0 H . 2. Fix a (typically large) number B of bootstrap samples. In this study, we chose B = 1000. 3. For every B b , , 1 K = draw a sample y ij ( b ) = ˆ B max;0 x ij /( ˆ K d ;0 + x ij ) + ˆ σ ε ij ( b ) , I i , , 1 K = , i n j , , 1 K = , where ) ( b ij ε follows a standard normal distribution. 4. Compute ) ( b T , the value of the test s tatistic (4), for the b -th bootstrap sample ) , ( ) ( b ij ij y x , I i , , 1 K = , i n j , , 1 K = . 5. As an approximation to α ; 0 T , take α ; 0 ˆ T , the [(1 – α ) B ]-th order statistic of the sample ) ( ) 1 ( , , B T T K , where [ a ] denotes the least in teger greater than o r equal to a . F test There are several proposals of F test statistics in the nonlinear regression literatu re (see [21] for a review). To appl y these ideas to our problem , we reparameterize the mo del given in (2) as follows. W e consider the global vector of parameters )' , , ( ) 1 ( 2 1 + = I γ γ γ K , where 0 max; 1 B = γ , 0 ; 2 d K = γ , 0 max; max; 1 2 B B i i − = + γ , 0 ; ; 2 2 d i d i K K − = + γ , for I i , , 1 K = and ∑ = + = I i j i 1 2 0 γ for j = 1, 2. Then the test given in (3) is equivalent to : 0 H 0 ) 1 ( 2 3 = = = + I γ γ K (5) 10 : 1 H 0 ≠ k γ for some 3 ≥ k . We consider the test statistic F = S i ( ˆ θ 0 ) − S i ( ˆ θ i ) () i = 1 I ∑ S i ( ˆ θ i ) i = 1 I ∑ n − 2 I 2( I − 1) , (6) which, under 0 H in tests (3) or (5), follows appro ximately an I n I F 2 ); 1 ( 2 − − distribution [21]. Consequently, we reject the null hypothesis of homogeneity when α ; 2 ); 1 ( 2 I n I F F − − > . Model selection Between models (1) and (2), deciding which is the m ost appropriate to fit and analyz e the information contained in the sample can also be viewed as a model selection problem [23]. In this case, we can use AIC to select the m o del best approximating the data [24]. The information criterion corresponding to the general model (2) is given by ) 2 1 ( 2 / ) ˆ ( log 1 I n S n AIC I i i i + + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = ∑ = θ . The more restrictive model (1) is nested in this class, satisfies 0 1 θ θ θ = = = I K and its information criterion is 6 / ) ˆ ( log 1 0 0 + ⎟ ⎠ ⎞ ⎜ ⎝ ⎛ = ∑ = I i i n S n AIC θ . To select the best model, we compute the AIC difference AIC AIC − = Δ 0 . A small value of Δ (say 2 0 ≤ Δ ≤ ) indicates that m odel (1) is be tter than (2). If Δ is large, then m odel (2) is to be preferred. We refer the reader to [19, 23 ] for m ore details on inform ation criteria-based decisions. We are aware that this info rmation-theoretic paradigm a nd hypothesis testing are very different approaches to the pr oblem at hand, and should not be mixed. We have nonetheless found that the AIC can serve to corroborate the decision of accepting or rejecting the homogeneity of Michaelis-Menten curves, especially in the analysis of real data. W e consider 11 it interesting to focus on the probl em from this point of view a nd to notice that, in this study, the information criterion and hypothesis te sting lead to similar conclusions. Results and Discussion We compared the p erformance of the hom ogene ity testing procedures described above via a Monte Carlo study and analysis of FR ET data obtained in the laboratory. Simulations We consider the homogeneity test ( 3). In th is subsection, we describe and interpret the results of a simulation study carried out to compare the power of the testing procedures introduced above. In all cases , the significance level is α = 0.05 and the number of Monte Carlo runs is 1000. Let us first describe how the observations were generated. We fix the number I of samples whose homogeneity we want to test, the values of i n , i B max; and i d K ; , for I i , , 1 K = , and σ . Then, for each I i , , 1 K = , we generate i n independent observations X from a uniform distribution on the interval 0 , 4 K d ; i [] and i n independent ε from a N(0,1) distribution. The random variables X and ε are independent. The correspon ding values of the response Y are computed according to model (2). For simplicity, in each sim ulation we fix equal sam ple sizes I n n n = = = K 2 1 . We consider three values for i n : 20, 50 and 100. To illu strate the case in which 0 H is true, w e chose I = 2 and I = 5, θ = ( B max , K d ) = (0.75 , 0.5) , θ = (1 , 1 . 5 ) and σ = 0.01 or 0.001. The results of the corresponding simulations are s hown in Table 1. In the second and third columns from the right, we reco rd the proportion of tim es that 0 H is rejected using the bootstrap rejection region and the F test, respectively. The la st column shows the proportion of times that the AIC difference, Δ , is greater than 2. In the table, the proportion of 0 H 12 rejections is alw ays near the nominal value of α = 0.05 or well below it. For I > 2, this proportion is much lower for th e AIC-based method than f or th e bootstrap procedure or the F test. When 1 max = B , the number of times that 0 H is rejected is lower f or the F test than for the bootstrap method. This ag rees with th e results for 1 max; ≈ i B shown in Table 2. Simulation results on the proportion of 0 H rejections when the null hypothesis is false appear in Tables 2 and 3. W e compared I = 2 and I = 3 curves (see Tables 2 and 3, respectively), since the case I ≥ 4 involves choosing many differe nt parameters, but does not provide any more useful information than I = 3. The sample sizes i n coincide with thos e of Table 1. Here we only used σ = 0.01, as σ = 0.001 yielded a considerable number of (less interesting) cases in which the power was nearly 1. The value of the parameters i θ chosen for these simulations is in the ne ighborhood of those in Table 1. The simulation results in Tables 2 and 3 show that, for i B max; near 0.75, the power of the AIC-based procedure is greater than that of the bootstrap method and the F test when 20 = i n , for I i , , 1 K = ; that is, for a sm all sample size. This is due to the facts th at bootstrap is a resampling-based technique [22] and the F test statistic (6) only follow s Fisher’s distribution asymptotically (for larg e sample sizes i n ) under the null hypothesis [21] . For larger sample sizes ( 50 = i n or 100) and I = 3, the bootstrap procedure and the F test yield very sim ilar powers, in general slig htly better than those a ttained by AIC . When i B max; is near 1, the boundary of the parameter space for max B , then the bootstrap procedur e is superior to both other methods in all cases, as also happened when 0 H was true. In the case of the F test, this is because the asymptotic distribution of the st atistic s involved (such as the m .l.e. of the parameters) is obtained under the restriction that the unknown param eters lie in the interior of the parameter space [21]. In FRET data analys is, th is restriction is not important f rom a 13 practical point of view, since values of 1 max ≥ B do not occur. Values of 1 max = B would indicate that all energy em itted by the donor protein is absorbed by the acceptor protein. This is only the case when CFP-YFP are f used in tandem, as for a positive control. W hen CFP and YFP are fused to the C-terminal end of the GPCR, the theor etical di stance between CFP-YFP (5.28 nm) and the orientation allowing 100% en ergy transfer between donor and acceptor are modified; real max B is therefore always < 1 [25]. As the case 1 max; ≈ i B is thus only interesting from a statistical po int of vi ew, Table 3 shows fewer simulations with 1 max; ≈ i B than with 75 . 0 max; ≈ i B . FRET data In this subsection, we study the homogeneity of the I = 7 FRET statistical samples in Figure 1. Laboratory material and methodology HEK293T (human embryonic kidney) cells from the American Type Culture Collection (CRL-11268) were plated in 6-well plates (Nunc) 24 h be fore transfection (5 x 10 5 cells/well). Cells were transient ly transfected with c DNA encoding the fusi on proteins (CXCR4-CFP or CXCR4-YFP) by the polyethylenimine method (PEI ; Sigm a-Aldrich). Cells were incubated with DNA and PEI (5.47 mM in nitrogen residu es) and 150 mM NaCl in serum -free medium, which was replaced after 4 h with complete medi um . At 48 h post-transfection, cells were washed twice in Hank’s balance salt solu tion supplemented with 0.1% glucose, and resuspended in the same solution. Total protei n concentration was determined f or whole cells using a Bradford assay kit (B ioRad). Cell suspensions (20 μ g protein in 100 μ l) were pipetted into black 96-well microplates and read in a Wallac Envision 2104 Multi label reader (Perkin Elmer) equipped with a high-energy xenon flash lamp, using an 8 nm bandwidth excitation filter at 405 nm (393-403 nm ) and 10 nm bandwid th emission filters at 486 nm (CFP channel) and 530 nm (YFP channel). Gain settings were optimized for each experiment to m aintain a 14 constant relative contribution of fluorophores to the detection channels for spectral imaging and linear unmixing. To determine the spectra l signature, HEK293T ce lls were transiently transfected wit h CXCR4-CFP or CXCR4-YFP. The contributions of CFP and YFP alone were measured in each detection channel (sp ect ral signature), and norm alized to the sum of the signal obtained for both channels [17, 26, 27]. The spectral signatures of CXCR4-CFP and CXCR4-YFP did not vary significantly (p >0 .05) from the signatures determ ined for each fluorescent protein alone. For FRET quantita tion, the spectral signature was tak en into consideration for linear unmixing to separate the two em ission spectra. In each experimental co ndition, we thus meas ured FRET efficien cy for multiple acceptor- to-donor ratios that are obtai ned by maintaining a consta nt donor amount and increasing amounts of acceptor. To quantify acceptor-to-donor ratio, we first determined the tota l amount of donor protein by excitation with its specific wavelength (405 nm) and measurement at 486 nm , as well as the total amount of acceptor pro tein by excitation at 515 nm and measurem ent at 530 nm. FRET efficien cy was th en calculated for each acceptor-to - donor ratio using the formulas CFP = S/(1+1 /R ) and YFP = S/1+R, where S = ChCFP + ChYFP, R = (YFP 530 Q - YFP 486 )/(CFP 486 - CFP 530 Q) and Q = ChCFP/ChYFP. ChCFP and ChYFP represent the signal in the 486 nm and 530 nm detect ion channels (Ch). CFP 486 , CFP 530 , YFP 530 , and YFP 486 represent the norm alized cont ributions of CFP and YFP to channels 486 and 530, as determined from sp ectral signatures of th e fluorescent proteins. Statistical analysis In each realization i in the same experim ental conditi ons, we obtain a statistical sample ) , ( ij ij y x , i n j , , 1 K = , where the response variable Y represents FRET efficiency and the explanatory variable x is the acceptor-to-donor ratio. Figure 1 shows I = 7 samples of the saturation curve corresponding to the experiment described above. In Table 4, we summarize statistical inform ation on the data. 15 We study the homogeneity of these sam ples via the three test procedures described (see Methods). Since estimation of ) , ( max d K B = θ is highly sensitive to the presence of outliers, we used the ROUT outlier detection procedure [ 28] to identify and rem ove this type of data before further analysis. In Figure 2, we show the estim ated Michaelis-Menten curves for the samples in Figure 1 after rem oval of the outly ing observations. W e marked the values of i B max; ˆ and i d K ; ˆ , for 7 , , 1 K = i , on the vertical and horizontal ax es of Figure 2, respectively. The seventh curve clearly differs from the first si x, wh ich are almost iden tical to one another. In effect, the bootstrap and the F test procedures reject the nu ll hypothesis (3) of homogeneity for significance level α = 0.01. In the bootstrap procedure, the test statistic (4) takes the value T = 1.08 and the critical value is 23 . 0 ˆ 01 . 0 ; 0 = T . In the F test, the statistic (6) is F = 16.5 and the critical value is 4 . 2 01 . 0 ; 102 ; 12 = F . The AIC difference is Δ = 100.94, so according to AIC, we would choose the general model (2). The three procedures thus lead to the sam e conclusion: the seven curves ar e not homogeneous. Let us now remove the seventh curve and carry out the hom ogeneity test (3) with the first I = 6 curves. The bootst rap test statistic is T = 0.16, less than the critical v alue 24 . 0 ˆ 01 . 0 ; 0 = T . The F test statistic is F = 1.5, also less than the critical value 5 . 2 01 . 0 ; 87 ; 10 = F . The AIC difference is Δ = –3.84. Consequently, the three procedures agree that the six curves are reali zatio ns from the sam e Michaelis-Menten model. Conclusions FRET and/or BRET are techniques widely used to study protei n-protein inte ractio ns in living cells. To evaluate these intera ctions, we used sample inf ormation to estim ate the Michaelis-Menten parameters, B max and K d . Here we considered a nd compared three ways of contrasting the homogeneity of several statis tical samp les obtained from FRET efficiency curves. Our aim was to determine whether I > 1 realizations of FRET experiments are 16 homogeneous, in the sense that they are sam ples from a common underlying regression model. We focused on the Michaelis-Menten non linear regre ssion model, since it is the m ost commonly used to fit this type of data, but th e ideas can be extended to any other regression model. From the hypothesis testing point of view , we considered two test procedures for the null hypothesis of homogeneity, one based on bootst rap resampling and the other, a version of the classical F test. We also used the AIC to decide which of these m odels (under the null or the alternative hypothesis) best fitted the data. Observations from homogenous statistical sa mples obtained in the sam e experimental conditions can be pooled into a single sample. Once there is just one sam ple from each of two experiments performed in diffe rent experimental conditions, we m ight also use any of the statistical procedures considered here to dete rm ine whether there are differences due to the change in conditions. A simulation study and analysis of real FRET data showed that the three methods used to study the homogeneity of FRET curves usually l ead to the same conclusions. This, and the short time required to execute the program, s uggests that for the an alysis of real FRET saturation curves, it is feasible to use all thr ee testing methods and veri fy that they lead to similar conclusions on samp le homogeneity. It should nonetheless be taken into account that selection with AIC gave the best results for small sam ple sizes, whereas the F-test and bootstrap method should be selected for comparis on of large sam ples. The Matlab code to implement the procedures described has b een developed and tested by the authors. Authors' contributions MM suggested the need for a homogeneity te st. AB proposed the testing procedures, implem ented them in Matlab and perform ed th e computational experiments. LMM carried out the laboratory expe riments and perform ed the homogeneity tes ts on the real data. M M, 17 LMM and AB prepared the manuscript. All au thors analy zed the results, read and approved the final document. Acknowledgements This work was partially supported by the Funda ción Genoma España (MEI CA), the European Union (FP7 Integrated Project Masterswitch no. 223404), the Spanish Ministry of Science and Innovation (SAF2008-02175 and MTM2010-17366), the DGUI de la Comunidad de Madrid/Universidad Autónom a de Madrid (CCG10-UAM/E SP-5494) and by the RETICS Program (RD08/0075 RIER; RD07/0020) of the Spanish Instituto de Salud Carlos III (ISCIII). References 1. Harrison C, van der Graaf PH: Current methods used to investigate G protein cou pled receptor oligomerisation . 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BMC Bioinformatics 2006, 7 :123. 21 Figures Figure 1. Samples of the saturation curve for CXCR4-CXCR4 dimer data Seven samples of the FRET saturation curve f o r CXCR 4 dimer com plexes in living cells. Each sample was obtained using HEK293T cells transiently cotransfected with a constant amount of CXCR4-CFP (2.0 μ g, 4000 fluorescent unit [FU]) and increasing amounts of CXCR4-YFP (500-10,000 FU). Each sample corre sponds to an individual experiment and is represented in a different color. The possi ble outliers have no t yet been removed. 22 Figure 2. Michaelis-Menten curves for CXCR4-CXCR4 dimer data The figure shows the estimated Michae lis-Menten cur ves for the data in Fi g. 1. The parameters i B max; and i d K ; were estimated by maxim um likelihood after removing the outliers. The estimates i B max; ˆ and i d K ; ˆ are marked on the vertic al and horizontal axis, respectively. 23 Tables Table 1. Simulation results when H 0 is true We display the proportion of 0 H rejections under the null hyp othesis using the bootstrap procedure, the F test an d the AIC (see Methods). max B d K σ I n i Bootstrap F test 2 ≥ Δ 20 0.048 0.045 0.066 50 0.050 0.051 0.055 2 100 0.049 0.050 0.052 20 0.051 0.052 0.033 50 0.049 0.050 0.024 0.01 5 100 0.054 0.047 0.027 20 0.040 0.039 0.063 50 0.048 0.047 0.058 2 100 0.058 0.057 0.060 20 0.063 0.057 0.040 50 0.060 0.059 0.034 0.75 0.5 0.001 5 100 0.053 0.051 0.020 20 0.031 0.021 0.029 50 0.033 0.018 0.022 2 100 0.036 0.023 0.023 20 0.043 0.015 0.010 50 0.051 0.023 0.010 0.01 5 100 0.038 0.008 0.003 20 0.023 0.012 0.015 50 0.035 0.023 0.026 2 100 0.027 0.016 0.016 20 0.032 0.010 0.006 50 0.020 0.007 0.004 1 1.5 0.001 5 100 0.035 0.006 0.001 24 Table 2. Simulation results when 0 H is false, for I = 2 curves We display the proportion of 0 H rejections under the alte rnative hypothesis using the bootstrap procedure, the F test and the AIC. 1 max; B 2 max; B 1 ; d K 2 ; d K σ i n Bootstrap F test 2 ≥ Δ 20 0.236 0. 235 0.285 50 0.560 0. 553 0.571 0.75 0.76 0. 50 0.50 0.01 100 0.990 0.990 0.992 20 0.109 0. 104 0.130 50 0.247 0. 244 0.253 0.75 0.75 0. 50 0.51 0.01 100 0.496 0.489 0.498 20 0.055 0. 033 0.048 50 0.090 0. 051 0.059 1 1 1.50 1.51 0.01 100 0.121 0.084 0.097 20 0.270 0. 203 0.242 50 0.596 0. 531 0.550 0.99 1 1.50 1.50 0.01 100 0.977 0.974 0.974 25 Table 3. Simulation results when 0 H is false, for 3 = I curves We display the proportion of 0 H rejections under the alte rnative hypothesis using the bootstrap procedure, the F test and the AIC. 1 max; B 2 max; B 3 max; B 1 ; d K 2 ; d K 3 ; d K σ i n Bootstrap F test 2 ≥ Δ 20 0.134 0. 136 0.148 50 0.243 0. 242 0.234 0.75 0.75 0. 75 0.50 0.50 0.51 0.01 100 0.473 0.475 0.450 20 0.391 0. 391 0.416 50 0.794 0. 792 0.784 0.75 0.75 0. 75 0.50 0.50 0.52 0.01 100 0.983 0.985 0.982 20 0.262 0. 261 0.276 50 0.645 0. 650 0.636 0.75 0.75 0. 75 0.50 0.51 0.52 0.01 100 0.945 0.945 0.937 20 0.359 0. 356 0.387 50 0.839 0. 832 0.825 0.75 0.75 0. 76 0.50 0.50 0.50 0.01 100 0.985 0.985 0.980 20 0.869 0. 870 0.879 50 0.999 0. 999 0.999 0.74 0.75 0. 76 0.50 0.50 0.50 0.01 100 1.000 1.000 1.000 20 0.094 0. 048 0.046 50 0.195 0. 126 0.121 1 1 1 1.50 1.50 1.52 0.01 100 0.445 0.324 0.312 20 0.403 0. 293 0.319 50 0.841 0. 770 0.761 0.99 1 1 1.50 1.50 1.50 0. 01 100 0.903 0.868 0.862 26 Table 4. Summary of the FRET data We show sample sizes and estimated parameters in the Michaelis-Menten m odel for the FRET data in Fig. 1. Curves 1 2 3 4 5 6 7 i n 16 18 17 18 18 18 18 i B max; ˆ 0.7373 0. 7417 0.7272 0.7424 0.7140 0. 7079 0.804 0 i d K ; ˆ 1.4203 1. 5819 1.3512 1.4821 1.2329 1. 3821 2.819 7 i σ ˆ 0.0389 0. 0159 0.0181 0.0250 0.0168 0. 0068 0.012 4