Supplementary Information for On characterizing protein spatial clusters with correlation approaches

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1 Supplementay Infomation fo On chaacteizing potein spatial clustes with coelation appoaches A. Shivananan, J. Unnikishnan, A. Raenovic Supplementay Notes Contents Deivation of expessions fo p = a t % aius fo iffeent moels Poofs egaing lowe boun fo aius of maximal aggegation Raius of maximal aggegation in the case of K, n of Lagache et al Bias in paamete estimation base on exponential PCF appoximation Case of powe law PCF Supplementay Figues List S Maximal aggegation: Compaison of p = a / t fom theoy an simulations, with eobas 6 S Scaling in exponential PCF appoach: /D fo Ising moel S3 Scaling in exponential PCF appoach: a/a fo Ising moel, its epenency on D S4 Exponential PCF appoach: compaison between theoy an simulations, with eobas.. 7 S5 Exponential PCF appoach, with a powe law tue PCF Supplementay Tables List S Cluste moels use fo analysis S Exact expessions fo the aius of maximal aggegation a fo iffeent cluste moels... 9

2 Supplementay Notes Deivation of expessions fo p = a t Hee we eive the elation in the case of Neyman-Scott pocess with Gaussian shape clustes. The eivation in the case of othe istibutions ae simila, stating fom the expessions in Supplementay Table S. We stat fom the K-function fo Gaussian shape clustes: In the fom K = π + A h = K = π + κ σ exp 4σ. Substituting in the equation exp. 4σ H as in Main Text, this coespons to A = κ,h = exp 4σ an A = h a 4πH a a h a fom Main Text an eaangement will give the elation as in Supplementay Table S. 95% aius fo iffeent moels These wee foun by solving the CDF 0 f pf =.95 fo, whee f pf is the aial pobability ensity function fo each moel 3. In the case of Cauchy an vagamma moels, maginal PDFs of in pola cooinates wee obtaine fom the bivaiate PDFs in catesian cooinates by stana tansfomationmultiplication by π. The esults ae given in the following table, along with the 95% limits. K ν. enotes the moifie Bessel function of the secon kin. Moel f pf.95 = u.95 t Lowe boun fo p.95 Gaussian exp.448σ.94 σ σ isk.975r.39 R Cauchy VaGamma + ω 4 3/4 K η 4 η 7/4 Γ 3 4 ω 3/ 4.469ω η Poofs egaing lowe boun fo aius of maximal aggegation Lemma.. Let h : R + R + be a unimoal iffeentiable function with a unique maximum at m > 0 an a eivative satisfying h > 0 fo 0 < m, an h < 0 fo > m. Note: this is satisfie by all the moels in Supplementay Table S. Futhe assume that thee exists > 0 that satisfies H h = 0. Then the aius of maximal aggegation a whee a is obtaine as a solution to fo some A > 0. Futhemoe as A, we have a. Poof. Define w = H h. Clealy w0 = 0 an the eivative satisfies w = h. Fom the popeties of h we have w 0 fo 0 < m, with stict inequality fo 0 < < m, an w > 0 fo > m. Hence w < 0 fo 0 < m. 3

3 Since w = 0 it follows that > m. Moeove since w is stictly positive fo m, ], it follows that w < 0 fo m,. Combining with 3 it follows that w < 0 fo 0,. Now, we know that a satisfies fo some A > 0. Thus we must have w a > 0 an hence it follows that a. Now consie the situation in which A. Define z = h H h to enote the expession on the ight han sie of without the facto of 4π inclue. Since z = h w we know fom the ealie analysis of w that z 0 fo < an z 0 fo <. Now consie the eivative of z. We have z = H hhh + h h H h = hh H h h H h = hh H h H h 4 Now consie the function q = H h fo. At = we have q = H h = H > 0. Moeove the eivative of this function is q = h h which is non-negative fo > because h < 0. Thus q > 0 fo >. This obsevation combine with the fact that h < 0 fo > an 4 implies that z < 0 fo >. Thus we have that z is stictly eceasing in the inteval,. Moeove z as appoaches fom above. Hence as A the left han sie of an thus by vitue of we must have a. 4 Raius of maximal aggegation in the case of K, n of Lagache et al = 0 fo isk clustes as iscusse in Main Text, followe by outine manipulations lea us to the elation: Setting K,n p 4p cos 0.5p + p 4 p p m m p mp + np 3 p p.4563m.663mp + np 4 p p + p 8p cos p m 3 +.m p 3.459mp np 3 4 p p + p 8p cos p 8 sin p 8 sin p = 0, 5 whee p = a /R, m = sie/r whee A = sie, P = 4.sie. The contou plot of p vs m, base on this expession, is shown in the Main Text, fo iffeent values of n. In the case of Gaussian clustes, the elation is simple: m 3 p e p m p.p 3.66e p mp 3.459p e p np p e p = 0, 6 an the coesponing contou plot is povie in Main Text. 3

4 5 Bias in paamete estimation base on exponential PCF appoximation We simply show the case fo Ising moel. Deivation fo othe moels follow the same poceue. Fo g a = + a exp / an f = + A /4 exp /D, the Least Squae Eo citeia gives: E a E We obtain: E = a = 0 = E a = a = 0 = E = a â, ˆ = ag min a, + e m + e m + e m m + 3aA3/4 m +Dm D E = ag min a, m + A π mef[ m D ] md A3/4 m 0 Γ[ 4] Γ[ 3 3 4, +Dm 3/4 =0 +Dm a e m m +Dm Solving both equations sepaately fo a = â, we obtain: m ] 3/4 Ae m Γ[ 3 3 4] Γ[ 4 â =, +Dm â = an, +e m +Dm 3/4 4ADe +Dm m 3/4 3A 7/4 +D m Γ [ 3 4] +Dm +e m m e + 3A 7/4 [ m Γ 3 4, +Dm ] 7/4 + +Dm 7/4 m aae f g a. 7 ] ] Γ[ 4] Γ[ 3 3 4, +Dm 7/4 =0 aa3/4 m +Dm m +D /4 m Γ[ 3 4] Γ[ 3 4, +Dm +Dm +Dm D 3/4 Equating both the above expessions of â, simplifying, an setting m = /D an k = m /D, we get: e k m Γ 3 4 Γ 3 4,k+ m + mek m 4 k m +k3/4 3Γ 3 m 4e k +k +3e m k +k Γ 3 4,k+ m e k m m+ m e k = 0 m k Note that this equation oes not contain the amplitue paametes a an A. A contou plot of this equation is shown in Supplementay Figue S. Fo easonably lage values of m i.e., m > D, m = ˆ/D =.5. That is, the coelation length paamete estimate by the appoximate moel is half of the coelation length of the tue moel. Fom these esults, the paamete values k = 4, m =.5 o any k > can be substitute in the expession fo â, to obtain: n = a A =.503D /4 That is, the amplitue paamete of the appoximate moel is epenent on both the tue amplitue paamete as well as the coelation length. The elationship is shown in Supplementay Figue S3. This paamete coul be n = scale fom the tue amplitue paamete fo D = 5 000nm, elevant scales fo membane potein clustes. Now, the aveage numbe of points pe cluste: 7 N I = + ρ f π πad.75 Γ 4 0 N a πa ρ = AD.75 = N I That is, the appoximate moel uneestimates the aveage numbe of points pe cluste by ove 40%. 4 ]

5 6 Case of powe law PCF In the case of the PCF g = + c 0 s, assuming s, K = π + πc s 0 s fo s <. A in 0 of Main Text will be A = s πc. Using 0, we get: 8 p = a 0 = c s /s. 9 s A plot of this equation fo iffeent s is shown in Supplementay Figue S5. It can be seen that p vaies acoss oes of magnitue base on values of s an c. 5

6 Supplementay Figues p= a / t Theoetical Simulations κ t p= a / t 3 4 Theoetical Simulations κ t Supplementay Figue S: Compaison of p = a / t fom theoy an simulations. Figue in Main Text with eo basσ. k= m /D Supplementay Figue S: Contou plot of k = m /D vs m = /D fo Ising moel. m is the istance value to which the Least Squaes sum is taken, whee D is the tue size paamete of the Ising moel, an that of the exponential appoximation of PCF. Afte m > D, the m value is fixe at.5. 6

7 Supplementay Figue S3: Plot of D vs n = a/a fo Ising moel, at k = 4, m =.5. See Supplementay Figue S fo etails on paameteic values of k, m. l=n estim /N tue σ=.0 Gaussian fit Exp fit N tue No of molecules/cluste l=n estim /N tue σ=.05 Gaussian fit Exp fit N tue No of molecules/cluste Supplementay Figue S4: Compaison of fitting empiical PCF of Gaussian clustes to exponential PCF g a an theoetical PCF of Gaussian clustes, fo iffeent tue cluste σ. Figue 6b in Main Text shown with eo basσ. 7

8 Supplementay Figue S5: Ratio of aius of maximal aggegation to tue cluste size paamete p = a 0 fo powe law PCF, as a function of amplitue paamete c fo iffeent values of powe s. Depening on s, p coul be cucially epenent on c. 8

9 Supplementay Tables Moel t g K π Gaussian σ exp 4πκσ 4σ κ exp 4σ isk R π R κ cos R R Cauchy ω 8πω κ 4R + 3/ 4ω κ + vaiance Gamma ν = / η 3 πη κ exp /η κ e η 4ω + η Ising 4 a I /4 exp /ξ πa I ξ 7/4 Γ 7 4 Γ 7 4, ξ Supplementay Table S: Cluste moels use fo analysis. κπ cos R R 4R +R + 4R 3 sin R. Also, fo isk moel, the functions povie hee ae fo R, fo > R, it is 0. Note that fo isk, g = at R, which povies a simple estimato fo R. Cluste moel Expession fo p = a / t fo p to 5 igits Theoetical lowe boun Gaussian p = a /σ κσ = e p 4 p 8π p +e p 4.48 Disk p = a /R κr p p 4 p = 4 accos p π 4 p 3p p 8p accos p +8 acsin p.9564 Cauchyp = a /ω κω = p πp +4 3/ p +4 3/ 4p 8 p vagamma p = a /η κη = 4πexpp exppp +p Ising p = a /ξ π a I ξ /4 =.370 exp pp 3/ 4π exp pp 7/4 Γ 7 4,p+Γ 7 4 Supplementay Table S: Exact expessions fo the aius of maximal aggegation a fo iffeent cluste moels. 9

10 Refeences [] Illian, J., Penttinen, A., Stoyan, H. & Stoyan, D. Statistical analysis an moelling of spatial point pattens, vol. 70 John Wiley & Sons, 008. [] Ghobani, M. Cauchy cluste pocess. Metika 76, [3] Jalilian, A., Guan, Y. & Waagepetesen, R. Decomposition of Vaiance fo Spatial Cox Pocesses. Scan J Stat 40, [4] Veatch, S. L. et al. Coelation functions quantify supe-esolution images an estimate appaent clusteing ue to ove-counting. PLOS ONE 7, e