Résumé

A model of two-type (or two-color) interacting random balls is introduced. Each colored random set is a union of random balls and the interaction relies on the volume of the intersection between the two random sets. This model is motivated by the detection and quantification of co-localization between two proteins. Simulation and inference are discussed. Since all individual balls cannot been identified, e.g. a ball may contain another one, standard methods of inference as likelihood or pseudolikelihood are not available and we apply the Takacs-Fiksel method with a specific choice of test functions.

A two-color interacting random balls model for co-localization analysis of proteins

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        <identifier identifierType="DOI">10.23723/11784/14259</identifier><creators><creator><creatorName>Frederic Lavancier</creatorName></creator><creator><creatorName>Charles Kervrann</creatorName></creator></creators><titles>
            <title>A two-color interacting random balls model for co-localization analysis of proteins</title></titles>
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A model of two-type (or two-color) interacting random balls is introduced. Each colored random set is a union of random balls and the interaction relies on the volume of the intersection between the two random sets. This model is motivated by the detection and quantification of co-localization between two proteins. Simulation and inference are discussed. Since all individual balls cannot been identified, e.g. a ball may contain another one, standard methods of inference as likelihood or pseudolikelihood are not available and we apply the Takacs-Fiksel method with a specific choice of test functions.

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A testing procedure A model for co-localization Estimation A two-color interacting random balls model for co-localization analysis of proteins. Frédéric Lavancier, Laboratoire de Mathématiques Jean Leray, Nantes INRIA Rennes, Serpico team Joint work with C. Kervrann (INRIA Rennes, Serpico team). GSI’15, 28-30 October 2015. A testing procedure A model for co-localization Estimation Introduction : some data Vesicular trafficking analysis and colocalization quantification by TIRF microscopy (1px = 100 nanometer) [SERPICO team, INRIA] ? =⇒ Langerin proteins (left) and Rab11 GTPase proteins (right). Is there colocalization ? ⇔ Is there some spatial dependencies between the two types of proteins ? A testing procedure A model for co-localization Estimation Image pre-processing After segmentation Superposition : ? ⇒ After a Gaussian weights thresholding Superposition : ? ⇒ A testing procedure A model for co-localization Estimation The problem of co-localization can be described as follows : We observe two binary images in a domain Ω : First image (green) : realization of a random set Γ1 ∩ Ω Second image (red) : realization of a random set Γ2 ∩ Ω −→ Is there some dependencies between Γ1 and Γ2 ? −→ If so, can we quantify/model this dependency ? A testing procedure A model for co-localization Estimation 1 A testing procedure 2 A model for co-localization 3 Estimation problem A testing procedure A model for co-localization Estimation 1 A testing procedure 2 A model for co-localization 3 Estimation problem A testing procedure A model for co-localization Estimation Testing procedure Let a generic point o ∈ Rd and p1 = P(o ∈ Γ1), p2 = P(o ∈ Γ2), p12 = P(o ∈ Γ1 ∩ Γ2). If Γ1 and Γ2 are independent, then p12 = p1p2. A testing procedure A model for co-localization Estimation Testing procedure Let a generic point o ∈ Rd and p1 = P(o ∈ Γ1), p2 = P(o ∈ Γ2), p12 = P(o ∈ Γ1 ∩ Γ2). If Γ1 and Γ2 are independent, then p12 = p1p2. A natural measure of departure from independency is ˆp12 − ˆp1 ˆp2 where ˆp1 = |Ω|−1 x∈Ω 1Γ1 (x), ˆp2 = |Ω|−1 x∈Ω 1Γ2 (x), ˆp12 = |Ω|−1 x∈Ω 1Γ1∩Γ2 (x). A testing procedure A model for co-localization Estimation Testing procedure Assume Γ1 and Γ2 are m-dependent stationary random sets. If Γ1 is independent of Γ2, then as |Ω| tends to infinity, T := |Ω| ˆp12 − ˆp1 ˆp2 x∈Ω y∈Ω ˆC1(x − y) ˆC2(x − y) → N(0, 1) where ˆC1 and ˆC2 are the empirical covariance functions of Γ1 ∩ Ω and Γ2 ∩ Ω respectively. Hence to test the null hypothesis of independence between Γ1 and Γ2 p-value = 2(1 − Φ(|T|)) where Φ is the c.d.f. of the standard normal distribution. A testing procedure A model for co-localization Estimation Some simulations Simulations when Γ1 and Γ2 are union of random balls A testing procedure A model for co-localization Estimation Some simulations Simulations when Γ1 and Γ2 are union of random balls Independent case (and each color ∼ Poisson) Number of p−values < 0.05 over 100 realizations : 4. A testing procedure A model for co-localization Estimation Some simulations Dependent case (see later for the model) Number of p−values < 0.05 over 100 realizations : 100. A testing procedure A model for co-localization Estimation Some simulations Independent case, larger radii Number of p−values < 0.05 over 100 realizations : 5. A testing procedure A model for co-localization Estimation Some simulations Dependent case, larger radii and "small" dependence Number of p−values < 0.05 over 100 realizations : 97. A testing procedure A model for co-localization Estimation Real Data Depending on the pre-processing : T = 9.9 T = 17 p − value = 0 p − value = 0 A testing procedure A model for co-localization Estimation 1 A testing procedure 2 A model for co-localization 3 Estimation problem A testing procedure A model for co-localization Estimation We view each set Γ1 and Γ2 as a union of random balls. We model the superposition of the two images, i.e. Γ1 ∪ Γ2. A testing procedure A model for co-localization Estimation We view each set Γ1 and Γ2 as a union of random balls. We model the superposition of the two images, i.e. Γ1 ∪ Γ2. The reference model is a two-type (two colors) Boolean model with equiprobable marks, where the radii follow some distribution µ on [Rmin, Rmax]. A testing procedure A model for co-localization Estimation We view each set Γ1 and Γ2 as a union of random balls. We model the superposition of the two images, i.e. Γ1 ∪ Γ2. The reference model is a two-type (two colors) Boolean model with equiprobable marks, where the radii follow some distribution µ on [Rmin, Rmax]. Notation : (ξ, R)i : ball centered at ξ with radius R and color i ∈ {1, 2}. → viewed as a marked point, marked by R and i. xi : collection of all marked points with color i. Hence Γi = (ξ,R)i∈xi (ξ, R)i x = x1 ∪ x2 : collection of all marked points. A testing procedure A model for co-localization Estimation Example : three realizations of the reference process A testing procedure A model for co-localization Estimation The model We consider a density on any bounded domain Ω with respect to the reference model f(x) ∝ zn1 1 zn2 2 eθ |Γ1∩ Γ2| where n1 : number of green balls and n2 : number of red balls. This density depends on 3 parameters z1 : rules the mean number of green balls z2 : rules the mean number of red balls θ : interaction parameter. If θ > 0 : attraction (co-localization) between Γ1 and Γ2 If θ = 0 : back to the reference model, up to the intensities (independence between Γ1 and Γ2). A testing procedure A model for co-localization Estimation Simulation Realizations can be generated by a standard birth-death Metropolis-Hastings algorithm. Examples : A testing procedure A model for co-localization Estimation 1 A testing procedure 2 A model for co-localization 3 Estimation problem A testing procedure A model for co-localization Estimation Estimation problem Aim : Assume that the law µ of the radii is known. Given a realization of Γ1 ∪ Γ2 on Ω, estimate z1, z2 and θ in f(x) = 1 c(z1, z2, θ) zn1 1 zn2 2 eθ |Γ1∩ Γ2| , where c(z1, z2, θ) is the normalizing constant. A testing procedure A model for co-localization Estimation Estimation problem Aim : Assume that the law µ of the radii is known. Given a realization of Γ1 ∪ Γ2 on Ω, estimate z1, z2 and θ in f(x) = 1 c(z1, z2, θ) zn1 1 zn2 2 eθ |Γ1∩ Γ2| , where c(z1, z2, θ) is the normalizing constant. Issue : The number of balls n1 and n2 is not observed. ⇒ likelihood or pseudo-likelihood based inference is not feasible. = A testing procedure A model for co-localization Estimation An equilibrium equation Consider, for any non-negative function h, C(z1, z2, θ; h) = S(h) − z1I1(θ; h) − z2I2(θ; h) where S(h) = (ξ,R)∈x,ξ∈Ω h((ξ, R), x\(ξ, R)) and for i = 1, 2, Ii(θ; h) = Rmax Rmin Ω h((ξ, R)i, x) λ((ξ, R)i, x) 2zi dξ µ(dR). Denoting by z∗ 1 , z∗ 2 and θ∗ the true unknown values of the parameters, we know from the Georgii-Nguyen-Zessin equation that for any h E(C(z∗ 1 , z∗ 2 , θ∗ ; h)) = 0. A testing procedure A model for co-localization Estimation Takacs Fiksel estimator Given K test functions (hk)1≤k≤K, the Takacs-Fiksel estimator is defined by (ˆz1, ˆz2, ˆθ) := arg min z1,z2,θ K k=1 C(z1, z2, θ; hk)2 . (1) A testing procedure A model for co-localization Estimation Takacs Fiksel estimator Given K test functions (hk)1≤k≤K, the Takacs-Fiksel estimator is defined by (ˆz1, ˆz2, ˆθ) := arg min z1,z2,θ K k=1 C(z1, z2, θ; hk)2 . (1) Consistency and asymptotic normality studied in Coeurjolly et al. 2012. A testing procedure A model for co-localization Estimation Takacs Fiksel estimator Given K test functions (hk)1≤k≤K, the Takacs-Fiksel estimator is defined by (ˆz1, ˆz2, ˆθ) := arg min z1,z2,θ K k=1 C(z1, z2, θ; hk)2 . (1) Consistency and asymptotic normality studied in Coeurjolly et al. 2012. Recall that C(z1, z2, θ; h) = S(h) − z1I1(θ; h) − z2I2(θ; h) where S(h) = (ξ,R)∈x,ξ∈Ω h((ξ, R), x\(ξ, R)) To be able to compute (1), we must find test functions hk such that S(h) is computable A testing procedure A model for co-localization Estimation Takacs Fiksel estimator Given K test functions (hk)1≤k≤K, the Takacs-Fiksel estimator is defined by (ˆz1, ˆz2, ˆθ) := arg min z1,z2,θ K k=1 C(z1, z2, θ; hk)2 . (1) Consistency and asymptotic normality studied in Coeurjolly et al. 2012. Recall that C(z1, z2, θ; h) = S(h) − z1I1(θ; h) − z2I2(θ; h) where S(h) = (ξ,R)∈x,ξ∈Ω h((ξ, R), x\(ξ, R)) To be able to compute (1), we must find test functions hk such that S(h) is computable How many ? At least K = 3 because 3 parameters to estimate. A testing procedure A model for co-localization Estimation A first possibility : h1((ξ, R)i, x) = Length S(ξ, R) ∩ (Γ1)c 1{i=1} where S(ξ, R) is the sphere {y, ||y − ξ|| = R}. ⇓ ⇓ ⇓ ⇓ A testing procedure A model for co-localization Estimation What about S(h1) = (ξ,R)∈x,ξ∈Ω h1((ξ, R), x\(ξ, R)) ? A testing procedure A model for co-localization Estimation What about S(h1) = (ξ,R)∈x,ξ∈Ω h1((ξ, R), x\(ξ, R)) ? = A testing procedure A model for co-localization Estimation What about S(h1) = (ξ,R)∈x,ξ∈Ω h1((ξ, R), x\(ξ, R)) ? = ⇒ S(h1) = P(Γ1) (the perimeter of Γ1) A testing procedure A model for co-localization Estimation So, for h1((ξ, R)i, x) = Length S(ξ, R) ∩ (Γ1)c 1{i=1} S(h1) = P(Γ1) and the Takacs-Fiksel contrast function C(z1, z2, θ; h1) is computable. A testing procedure A model for co-localization Estimation So, for h1((ξ, R)i, x) = Length S(ξ, R) ∩ (Γ1)c 1{i=1} S(h1) = P(Γ1) and the Takacs-Fiksel contrast function C(z1, z2, θ; h1) is computable. Similarly, Let h2((ξ, R)i, x) = Length S(ξ, R) ∩ (Γ2)c 1{i=2} then S(h2) = P(Γ2). A testing procedure A model for co-localization Estimation So, for h1((ξ, R)i, x) = Length S(ξ, R) ∩ (Γ1)c 1{i=1} S(h1) = P(Γ1) and the Takacs-Fiksel contrast function C(z1, z2, θ; h1) is computable. Similarly, Let h2((ξ, R)i, x) = Length S(ξ, R) ∩ (Γ2)c 1{i=2} then S(h2) = P(Γ2). Let h3((ξ, R)i, x) = Length S(ξ, R) ∩ (Γ1 ∪ Γ2)c then S(h3) = P(Γ1 ∪ Γ2). A testing procedure A model for co-localization Estimation Simulations with test functions h1, h2 and h3 over 100 realizations θ = 0.2 (and small radii) θ = 0.05 (and large radii) Frequency 0.15 0.20 0.25 0.30 05101520 Frequency 0.00 0.01 0.02 0.03 0.04 0.05 0.06 0.07 010203040 A testing procedure A model for co-localization Estimation Real Data We assume the law of the radii is uniform on [Rmin, Rmax]. (each image is embedded in [0, 250] × [0, 280]) Rmin = 0.5, Rmax = 2.5 Rmin = 0.5, Rmax = 10 ˆθ = 0.45 ˆθ = 0.03 A testing procedure A model for co-localization Estimation Conclusion The testing procedure allows to detect co-localization between two binary images is easy and fast to implement does not depend too much on the image pre-processing The model for co-localization relies on geometric features (area of intersection) can be fitted by the Takacs-Fiksel method allows to compare the degree of co-localization θ between two pairs of images if the laws of radii are similar