Antoine Lejay
email: antoine.lejay@iecn.u-nancy.fr

Simulation of a stochastic process in a discontinuous, layered media ‡

Keywords: Skew Brownian motion, discontinuous media, occupation time, local time, last passage time, path decomposition, Brownian bridge, first hitting time, geophysics, Monte Carlo simulation AMS Subject classification: 60J60, 65C05

In this note, we provide a simulation algorithm for a diffusion process in a layered media. Our main tools are the properties of the Skew Brownian motion and a path decomposition technique for simulating occupation times.

Introduction

Simulation of diffusion processes in multi-dimensional discontinuous media is still a challenging problem, while recent progresses have been done for onedimensional media. The object of this note is to deal with the simulation of the stochastic process generated by the divergence form operator

L = 1 2 ∇(D∇•) + U ∇,
in a multi-dimensional layered media. By this, we mean that there is a direction n orthogonal to the discontinuities of a diffusion coefficient D and a convective term U that are constant in each layers. In geophysics, this could be used for example to model a solute in a vertically layered porous media submitted to a advective flow U and diffusion effects given by D [START_REF] Appuhamillage | Occupation and local times for Skew Brownian motion with application to dispersion accross an interface[END_REF][START_REF] Ramirez | Multi-skewed Brownian motion and diffusion in layered media[END_REF][START_REF] Ramirez | A generalized Taylor-Aris formula and skew diffusion[END_REF][START_REF] Salamon | A review and numerical assessment of the random walk particle tracking method[END_REF]. The concentration C(t, x) of the solute is then

∂C(t, x) ∂t = 1 2 ∇(D(x)∇C(t, x)) -∇(U (x)C(t, x)) (1)
1 with C(t, x) --→ t→0 δ y if the solute is initially at y. Eq. ( 1) is interpreted as a Fokker-Planck equation, and C(t, x) is then the density of the process Z at time t generated by L with starting point y.

Note also that our method may be locally used for media with such a geometry. In addition to geophysics, many domains face this kind of simulation problem: magneto-electro-encephalography [START_REF] Martinez | Interprétations probabilistes d'opérateurs sous forme divergence et analyse de méthodes numériques associées[END_REF], ecology [START_REF] Ovaskainen | Biased movement at a boundary and conditional occupancy times for diffusion processes[END_REF], astrophysics [START_REF] Zhang | Calculation of diffusive shock acceleration of charged particles by skew Brownian motion[END_REF], ... We simplify our analysis by considering that the spatial dimension is equal to 2 and (H 1 ) D(x, y) = D x (x) 0 0 D y (x) and U (x, y) = 0 U y (x) .

Here, n points toward the y-direction. There is no difficulty to generalize our results in any dimension d ≥ 3. We also assume that there exists a finite or countable set of separated points {x i } such D x , D y and U y are discontinuous at these points and constant elsewhere. The main idea is the following. Let Z be the two dimensional process generated by L. With the decomposition Z = (X, Y ), X does not depend on Y and is generated by the one-dimensional operator

L x = 1 2 ∇(D x (x)∇•),
while Y is solution to the equation

Y t = y + t 0 D y (X s ) dW y s + t 0 U y (X s ) ds,
where W y is a Brownian motion independent from X. The discontinuities of D may be seen as interfaces. From the heuristic point of view, these interfaces play the rôle of permeable barriers. However, this can only be said from a "mesoscopic" and not microscopic point of view, due to the irregularity of the paths of X. In particular, claims such that "the particle passes on one side with a given probability" makes no senses unless one says precisely where and when the particle goes. The simulation of one-dimensional stochastic processes generated by a divergence form operator with discontinuous coefficients has now been the subject of a large literature and several algorithms have been proposed (See [START_REF] Étoré | On random walk simulation of one-dimensional diffusion processes with discontinuous coefficients[END_REF][START_REF] Étoré | A Donsker theorem to simulate one-dimensional processes with measurable coefficients[END_REF][START_REF] Étoré | Exact simulation of one-dimensional stochastic differential equations involving the local time at zero of the unknown process[END_REF][START_REF] Hoteit | Three-dimensional modeling of mass transfer in porous media using the mixed hybrid finite elements and the random-walk methods[END_REF][START_REF] Labolle | Diffusion processes in composite porous media and their numerical integration by random walks: Generalized stochastic differential equations with discontinuous coefficients[END_REF][START_REF] Lejay | A scheme for simulating one-dimensional diffusion processes with discontinuous coefficients[END_REF][START_REF] Lejay | Simulating diffusion processes in discontinuous media: a numerical scheme with constant time steps[END_REF][START_REF] Martinez | Discrétisation d'équations différentielles stochastiques unidimensionnelles à générateur sous forme divergence avec coefficient discontinu[END_REF][START_REF]One-Dimensional Parabolic Diffraction Equations: Pointwise Estimates and Discretization of Related Stochastic Differential Equations with Weighted Local Times[END_REF][START_REF] Salamon | A review and numerical assessment of the random walk particle tracking method[END_REF][START_REF] Uffink | A random walk method for the simulation of macrodispersion in a stratified aquifer, Relation of Groundwater Quantity and Quality[END_REF] for a non-exhaustive list of possible algorithms). Some of these schemes generate random variates with the true distribution of X t . Others only provide some approximations. Yet the multi-dimensional case remains largely open and challenging (However, see [START_REF] Lejay | A method to assess the quality of Monte Carlo methods for multi-dimensional stochastic processes in discontinuous media[END_REF] for some possible schemes, in general for locally isotropic coefficients).

Here, we consider a Euler type scheme which computes X t+δt from X t for small values of δt. The probability that the particle leaves a ball of radius R √ δt centered on X t during [t, t + δt] is exponentially small as R increases. Hence, if √ δt is much smaller than the distance between two discontinuities, then the behavior of the particle is mostly influence by the closest interface. Following the previous discussion and for the sake of simplicity, we also assume:

(H 2 ) There is only one interface at x 1 = 0. At time t + δt, the y-component given (X s ) s∈ [t,t+δt] is then

Y t+δt = Y t + G + U ,
where U = t+δt t U y (X s ) ds and G is a Gaussian random variable independent from (X s ) s∈ [t,t+δt] with mean 0 and variance

σ 2 = Var(G|(X s ) s∈[t,t+δt] ) = t+δt t D y (X s ) ds.
The possible values of D y (X s ) (resp. V y (X s )) are D y + and D y -(resp. U y + and U y -) according to the sign of X s . Hence

σ 2 = D y + A + (t, t+δt)+D y -A -(t, t+δt) and U = U y + A + (t, t+δt)+U y -A -(t, t+δt)
where A + (t, t + δt) and A -(t, t + δt) are the occupation times:

A + (t, t + δt) = t+δt t 1 {Xs≥0} ds and A -(t, t + δt) = δt -A + (t, t + δt) = t+δt t 1 {Xs<0} ds.
Hence, the problem of simulating Y t+δt from Y t reduces to the problem of simulating the occupation times A + (t, t + δt). Following the seminal work of P. Lévy on the occupation time for the Brownian motion and the Arc-Sine distribution, the characterization of the occupation time distribution for general Markov processes has been the subject to a large literature since [START_REF] Lamperti | An occupation time theorem for a class of stochastic processes[END_REF] (See e.g. [START_REF] Watanabe | Generalized arc-sine laws for one-dimensional diffusion processes and random walks, Stochastic analysis[END_REF][START_REF] Watanabe | A density formula for the law of time spent on the positive side of one-dimensional diffusion processes[END_REF][START_REF] Yano | On the occupation time on the half line of pinned diffusion processes[END_REF]). A special branch of these researches concerns Brownian motions, Skew Brownian motions, Walsh Brownian motions (a generalization of the Skew Brownian motion by considering a Brownian diffusion on rays) and Skew Bessel processes. For this, the whole technology of excursions theory, scaling property, ... can be used (See e.g. [START_REF] Barlow | Une extension multidimensionnelle de la loi de l'arc sinus[END_REF][START_REF] Pitman | Arcsine laws and interval partitions derived from a stable subordinator[END_REF][START_REF]On the relative lengths of excursions derived from a stable subordinator[END_REF]).

The occupation time of the Skew Brownian motion has already been used in relation with modelling in geophysics [START_REF] Appuhamillage | Occupation and local times for Skew Brownian motion with application to dispersion accross an interface[END_REF] or in population ecology [START_REF] Ovaskainen | Biased movement at a boundary and conditional occupancy times for diffusion processes[END_REF]. In particular, the trivariate density of the position of the Skew Brownian motion, its occupation time and its local time is given in [START_REF] Appuhamillage | Occupation and local times for Skew Brownian motion with application to dispersion accross an interface[END_REF]. However, it does not give rise to straightforward simulation algorithms. Hence, we combine techniques that consists in transforming X into a Skew Brownian motion as in [START_REF] Étoré | On random walk simulation of one-dimensional diffusion processes with discontinuous coefficients[END_REF][START_REF] Lejay | A scheme for simulating one-dimensional diffusion processes with discontinuous coefficients[END_REF], in using the Brownian bridge properties to determine the first hitting time of zero, and then use some known densities for the last passage of zero and the occupation time for the Skew Brownian motion. This technique may also be seen as a refinement of the one proposed in [START_REF] Lejay | On the decomposition of excursions measures of processes whose generators have diffusion coefficients discontinuous at one point, Markov Process[END_REF], which was a variant of [START_REF] Hausenblas | A numerical scheme using Itô excursions for simulating local time resp. stochastic differential equations with reflection[END_REF] for the simulation of Reflected Stochastic Differential Equations. Since our key point is the reduction to the Skew Brownian motion, all that is said in this article could also be applied to the process generated by a nondivergence form operator

1 2 d i,j=1 D i,j ∂ 2 ∂x i ∂x j + d i,j=1 U i ∂ ∂x i
under the same assumptions on D and U as above. In this case, θ given in Eq. ( 3) shall be changed into -θ in Eq. ( 4) (See [START_REF] Lejay | A scheme for simulating one-dimensional diffusion processes with discontinuous coefficients[END_REF]).

The Stochastic Differential Equation the process solves

The existence of a diffusion (Feller process) associated to the divergence form operator L follows from Gaussian estimates on the semi-group [START_REF] Stroock | Diffusion semigroups corresponding to uniformly elliptic divergence form operators[END_REF]. Proposition 1 gives some insights about the process under Hypotheses (H 1 ) and (H 2 ). See [START_REF] Bossy | Probabilistic interpretation and random walk on spheres algorithms for the Poisson-Boltzmann equation in Molecular Dynamics[END_REF] for a closely related representation of a diffusion process generated by a divergence form operator. For an alternative construction, see [START_REF] Ramirez | Multi-skewed Brownian motion and diffusion in layered media[END_REF][START_REF] Ramirez | A generalized Taylor-Aris formula and skew diffusion[END_REF].

Proposition 1. Under Hypotheses (H 1 ) and (H 2 ), L = 1 2 ∇(D∇•)+U ∇ is the infinitesimal generator of the family indexed by (x, y) ∈ R 2 of unique (strong) solutions (X, Y ) to the Stochastic Differential Equation (SDE) with local time        X t = x + t 0 D x (X s ) dW x s + D x + -D x - D x + + D x - L 0 t (X), Y t = y + t 0 D y (X s ) dW y s + t 0 U y (X s ) ds, ( 2
)
where

L 0 t (X) def = lim →0 (2 ) -1 t 0 1 Xs∈[-, ] ds is the symmetric local time of X and (W x , W y ) is a two-dimensional Brownian motion.
Proof. Since Y is a Gaussian process whose variance and mean at any time depend only on the couple (X, W y ), the problem of existence and uniqueness of (2) reduces to the problem of existence of the SDE solved by X. The xcomponent of the SDE may be interpreted as a particular case of SDE with local time whose theory is developed in [START_REF] Gall | One-Dimensional Stochastic Differential Equations Involving the Local Times of the Unknown Process, Stochastic Analysis and Applications[END_REF]. In particular, it follows from [17, Theorem 2.3, p. 61] that strong existence and uniqueness for X, and then for Y .

It remains to show that L is the infinitesimal generator of the two-dimensional process (X, Y ). For this, we use a regularization of the coefficients.

Let us consider a smooth family of approximations (D n , U n ) of the coefficients (D, U ). The divergence form operator L n = 1 2 ∇(D n ∇•) + U n ∇ may be transformed into a non-divergence form operator. The stochastic process (X n , Y n ) generated by L n is the the solution to

       X n t = x + t 0 D x n (X n s ) dW x s + 1 2 t 0 dD x n dx (X n s ) ds, Y n t = y + t 0 D y n(X n s ) dW y s + t 0 U y n (X n s ) ds,
where D x n , D y n and U y n are smooth approximations of D x , D y and U y . In particular, X n is generated by the one-dimensional divergence form operator

1 2 d dx D x n d dx .
As n goes to infinity, (X n , Y n ) converges to the process generated by L under P (x,y) for any starting point (x, y) ∈ R 2 (See [START_REF] Stroock | Diffusion semigroups corresponding to uniformly elliptic divergence form operators[END_REF]Theorem II.3.13] for the convergence of the corresponding semi-groups from which the convergence of the distribution is easily deduced thanks to the Markov property or [START_REF] Rozkosz | Weak Convergence of Diffusions Corresponding to Divergence Form Operator[END_REF]Theorem 7.3]).

It is also known that X n converges to X given by the first component of ( 2 The convergence of Y n to the second component of (2) follows from standard convergence results in this case (See for example [19, Lemmas 5 and 6] for similar computations). Hence, (X n , Y n ) converges in distribution to (X, Y ) under P (x,y) for any (x, y) ∈ R 2 and this allows one to identify (X, Y ) with the process generated by L.

The x-component and the Skew Brownian motion

Let us set

Ψ(x) = 1 D x + 1 x≥0 + 1 D x - 1 x<0 and θ = D x + -D x - D x + + D x - ∈ (-1, 1). ( 3
)
Under Hypotheses (H 1 ) and (H 2 ), let us denote as above by (X, Y ) the process generated by L and then solution to Eq. ( 2).

The main point of our analysis is the following result on the process V = Ψ(X). This transform has already been used for example in [START_REF] Étoré | On random walk simulation of one-dimensional diffusion processes with discontinuous coefficients[END_REF][START_REF] Lejay | A scheme for simulating one-dimensional diffusion processes with discontinuous coefficients[END_REF][START_REF] Ouknine | Skew-Brownian motion" et les processus qui en dérivent[END_REF][START_REF] Ramirez | Multi-skewed Brownian motion and diffusion in layered media[END_REF][START_REF] Ramirez | A generalized Taylor-Aris formula and skew diffusion[END_REF]. The proof of Proposition 2 below is a direct consequence of the Itô-Tanaka formula and some manipulations on the local time L 0 t (X) to the local time L 0 t (V ).

Proposition 2. Under Hypotheses (H 1 ) and (H 2 ), the process V t = Ψ(X t ) is a Skew Brownian motion of parameter θ, which means it is the unique strong solution to the SDE with local time

V t = Ψ(x) + W x t + θL 0 t (V ). (4)
The Skew Brownian motion (SBM) was introduced by K. Itô That is, with probability α, the excursion has a positive sign. Otherwise, it has a negative sign. Its relationship with the solution of ( 4) was proved by J. Harrison and L. Shepp [START_REF] Harrison | On skew Brownian motion[END_REF]. For a summary of the possible constructions of the SBM and its main property, see [START_REF]On the constructions of the Skew Brownian motion[END_REF].

In the case of multiple discontinuities, the same kind of transform may be applied but it gives rise to a SDE with local time at different points. Let us note that the occupation time for the SBM V is the same as for the process X. Hence, we still denote by A ± (s, t) the occupation time of V .

The occupation time

Using the strong Markov property, simulating A ± (s, t) when X s = 0 reduces to the simulation of the occupation time A + (0, t) when X 0 = 0. Besides, using the scaling property for the SBM, A + (0, t) is equal in distribution to tA + (0, 1) for any t > 0.

Let us also note the trivial relation:

A + (0, 1) + A -(0, 1) = 1.
The density and the distribution of the occupation time for the Skew Brownian motion of parameter α is explicitly known. We will not use Theorem 1 below under its full form, yet we recall it for it contains the density of the Arc-Sine distribution that will be used for simulating the last-passage time.

Theorem 1 ([16, Theorem 1]). The density

f (α) (x) of A + (0, 1) is f (α) (x) = α(1 -α) π (x(1 -x)) -1/2 α 2 (1 -x) + (1 -α) 2 x , x ∈ (0, 1)
and its distribution function is

F (α) (x) = 2 π arcsin (1 -α) 2 x (1 -α) 2 x + α 2 (1 -x) , x ∈ [0, 1].
Note also that the triple (A + (0, 1), A -(0, 1), L 0 1 (V ) 2 ) may also be represented in term of independent stable positive random variables of indice 1/2 [2, Theorem 1]. Hence, A + (0, 1) is easily simulated by inverting its distribution function: If U ∼ U(0, 1), where U(0, 1) is the uniform distribution on [0, 1), then

A + (0, 1) law = (F (α) ) -1 (U ), which means that A + (0, 1) law = α 2 Γ α 2 Γ + (1 -α) 2 (1 -Γ) , Γ = sin 2 πU 2 .
When α = 1/2, then V is a Brownian motion. In this case, f (1/2) is the density of the Arc-Sine distribution and Theorem 1 yields the famous result of P. Lévy [START_REF] Lévy | Sur certains processus stochastiques homogènes[END_REF] that the occupation time of the Brownian motion follows the Arc-Sine distribution.

Decomposition of the path

We have seen that the distribution of (X t+δt , Y t+δt ) when (X t , Y t ) is known is deduced from the one of (X t+δt , A + (t, t + δt)) when (X t , Y t ) is known. Up to a simple transform, one may equivalently consider the couple (V t+δt , A + (t, t+δt)) when (V t , Y t ) is known. Using the Markov property, one may assume without loss of generalities that t = 0. Let us start with the simulation of V δt when V 0 is known. For this, we apply an idea already used to simulate a Brownian motion killed at a boundary (See e.g. [START_REF] Gobet | Weak approximation of killed diffusion using Euler schemes[END_REF][START_REF] Lerche | Approximate exit probabilities for a Brownian bridge on a short time interval, and applications[END_REF]). Regarding the hitting time of zero, see e.g. [3, pp. 2436-2438].

For this, let us simulate a guess V δt as if there is no interface, that is

V δt = V 0 + ξ with ξ ∼ N (0, δt).
If V δt V 0 < 0, then the path (V s ) s∈[0,δt] crosses zero at a stopping time g. Using the properties of the Brownian bridge, g = δt • ζ/(1 + ζ) where ζ follows an inverse Gaussian distribution IG(-x/y, x 2 /δt) with x = V 0 and y = V δt .

The inverse Gaussian distribution IG(µ, λ) has density

r(µ, λ, x) = λ 2πx 3 exp -λ(x -µ) 2 2µ 2 x .
Random variates following the inverse Gaussian distribution are easily simulated using the algorithm of [START_REF] Michael | Generating random variates using transformations with multiple roots[END_REF] (See also [6, p. 148]). If V δt V 0 > 0, then the path (V s ) s∈[0,δt] may cross zero. This happens with probability exp(-2xy/δt). If this happens, then the first hitting time of 0 is

g = δt • ζ/(1 + ζ) with ζ ∼ IG(x/y, x 2 /δt).
If no crossing occurs, then we keep V δt as the next position of the Skew Brownian motion. In this case, A + (0, δt) = δt (resp. A + (0, δt) = 0) if V 0 > 0 (resp. V δt < 0). Let us consider now the case where a crossing occurs at time g < δt. Using the strong Markov property of the Skew Brownian motion and up to changing δt to δt -g, one may assume that g = 0 and consider that V 0 = 0.

Hence, let us set d = sup{s ≤ δt | V s = 0} be the last passage time to 0 before δt. This is not a stopping time. Basically, d relies only on the infinite sequence of lengths of the excursions of the Skew Brownian motion occurring before t, but not on the signs. Hence, the distribution of d is the same as the one of the Brownian motion (See [33, p. 301] for a discussion about this approach) and the ratio d/δt is known to be Arc-Sine distributed on [0, 1] (See [START_REF] Barlow | Une extension multidimensionnelle de la loi de l'arc sinus[END_REF][START_REF] Lévy | Sur certains processus stochastiques homogènes[END_REF][START_REF]On the relative lengths of excursions derived from a stable subordinator[END_REF] or [START_REF] Borodin | Handbook of Brownian motion-facts and formulae[END_REF]IV.3.18,p. 62]). This means that d/δt has a density f (1/2) introduced in Theorem 1. If U is a uniform random variable on [0, 1), d law = δt cos 2 (2πU ), so that d is easily simulated. We are now in a situation similar to Williams' decomposition of the Brownian path [START_REF] Borodin | Handbook of Brownian motion-facts and formulae[END_REF][START_REF] Salminen | On the excursion theory for linear diffusions[END_REF][START_REF] Yor | Local Times and Excursions for Brownian motion: a concise introduction[END_REF].

As a Skew Brownian motion may be constructed from a Brownian motion by choosing their sign of the excursions independently, (d -1/2 V dt ) t∈[0,1] is independent from the σ-algebra generated by (V t ) t≥d and by d under P 0 [2, p. 297]. The process (d -1/2 V ds ) s∈[0,1] has the distribution of a Skew Brownian bridge.

The part (V s ) s∈ [d,δt] is called a meander and is a part of an excursion straddling δt. Again from the construction of the Skew Brownian motion, P[V δt > 0|d] = α and sgn(V δt ) and |V δt | are independent when d is given. In addition, (|V s |) s∈ [d,δt] is equal in distribution to (|W s |) s∈ [d,δt] for a Brownian motion W . The distribution of |W δt | given d is the same as the distribution of e κ with κ = δt -d, where e is a Brownian excursion whose length is greater than κ. It is also the distribution of a 3-dimensional Bessel process at time κ and is known to have the density (See [START_REF] Hausenblas | A numerical scheme using Itô excursions for simulating local time resp. stochastic differential equations with reflection[END_REF] or [4, IV.3.19, p. 63]):

x κ exp - x 2 2κ .
This density is called the entrance law for the excursion. Using the corresponding distribution function, V δt given d is distributed as To conclude, it remains to simulate A + (0, d) under P 0 . As already noted, this random variable is independent from V δt given d. Under P 0 , A + (0, d) is equal in distribution to dA br [33, Eq. ( 84)], where A br is the occupation time above 0 of the Skew Brownian bridge over [0, 1] (which means the path (V s ) s∈[0,1] conditioned to V 0 = V 1 = 0). The density f (α),br of A br with respect to the Lebesgue measure is known to be (See [START_REF] Watanabe | A density formula for the law of time spent on the positive side of one-dimensional diffusion processes[END_REF]Example 3.2] or [START_REF] Yano | On the occupation time on the half line of pinned diffusion processes[END_REF])

f (α),br (x) = 1 2
α(1 -α) (α 2 (1 -x) + (1 -α) 2 x) 3/2 , x ∈ (0, 1).

For α = 1/2, f (α),br is the identity and A br is uniformly distributed on [0, 1). If α = 1/2, the distribution function is

F (α),br (x) = α(1 -α) 1 -2α   1 α - 1 α 2 (1 -x) + (1 -α) 2 x   .
and A br is easily simulated by

A br law = α 2 1 -2α    1 1 -1-2α 1-α U 2 -1   
where U ∼ U(0, 1). The simulation method presented in Algorithm 1 relies then on a combination of all the previous facts. This scheme was tested using the method given in [START_REF] Lejay | A method to assess the quality of Monte Carlo methods for multi-dimensional stochastic processes in discontinuous media[END_REF] and gave good results.

) [ 22 ,

 22 Proposition 3, p. 116 and Proposition 5, p. 118] or [17, Theorem 3.1, p. 64].

- 2 (

 2 δt -d) log(U ),where U ∼ U(0, 1) and is an independent Bernoulli random variable withP[ = 1] = 1-P[ = -1] = α. Once the sign = sgn(V δt ) is known, A + (d, δt) = δt -d if = 1 and A + (d, δt) = 0 if = -1.

Input:

The position X t of the particle at time t and a time step δt. Output: The couple (X t+δt , A + (t, t + δt) of the position X t+δt of the particle at time t + δt and its occupation time above 0.

Algorithm 1: Simulation of the couple (X t+δt , A + (t, t + δt)) when X t is given, under the Hypotheses (H 1 ) and (H 2 ).


Khaled Gaaloul

Erik Proper
email: erik.proper@tudor.lu

François Charoy
email: charoy@loria.fr

Injecting Task Delegation Constraints into a Role-based Access Control Model

Keywords: Access control, delegation, constraints, privileges

In role-based access control models, delegation of authority involves delegating roles that a user can assume or the set of permissions that he can acquire, to other users. Several role-based delegation models have been proposed in the literature. However, these models consider only delegation in presence of the role type, which have some inherent limitations to task delegation in workflow systems. In this paper, we address task delegation in a workflow and elaborate a security model supporting delegation constraints. Delegation constraints express security requirements with regards to task's resources, user's assignment and privileges (delegation of authority). Further, we show how, using a role-based security model, we inject formalised delegation constraints to compute delegation principals with their respective privileges.

Introduction

With the broad adoption of workflow management systems to model and automate business processes cross organisations, security becomes a crucial and essential topic. Typically, activities that are part of a process are represented as tasks. Organisations establish a set of authorisation policies that regulate how business processes and resources should be managed within a workflow [START_REF] Atluri | Supporting conditional delegation in secure workflow management systems[END_REF]. Authorisation information is given which authorises users to perform tasks. Such authorisation information may be specified using a simple access control list or more complex role-based structures [START_REF] Crampton | On delegation and workflow execution models[END_REF].

In current workflow management systems, the role-based access control (RBAC) model is widely adopted, where system administrators assign roles to users. It is more convenient for administrators to manage roles than to manage users directly [START_REF] Sandhu | Role-based access control models[END_REF]. One important factor that affects access control (authorisation) distribution among users is delegation. Delegation involves a user passing its authority to other users. If delegation is allowed, a delegator delegates authority (a privilege) to another active entity, called the delegatee, to carry out a task on behalf of the former. In the context of workflow systems, delegation can be very useful for real-world situations where a user who has to perform a task is either unavailable or too overloaded [START_REF] Gaaloul | A Secure Framework for Dynamic Task Delegation in Workflow Management Systems[END_REF]. Hence, we define task delegation as a means for assigning a task and its access rights from a delegator to a delegatee.

The concept of delegation has been presented in [START_REF] Atluri | Supporting conditional delegation in secure workflow management systems[END_REF][START_REF] Crampton | Delegation in role-based access control[END_REF]. Significant contributions to role-based delegation can be found in [START_REF] Barka | Framework for role-based delegation models[END_REF][START_REF] Zhang | PBDM: a flexible delegation model in RBAC[END_REF]. While much of the work in the area of delegation is limited to role-based access control, the goal of our paper is to consider task delegation constraints in workflow systems. Delegation constraints needs to tackle several issues with regards to workflow's invariants in terms of users, tasks and resources. In doing so, we need to come up with an access control model supporting the assignment of task delegation. Delegation assignment deals with delegation principals (delegator, delegatee) their respective rights (privileges) and their availability (no conflicts during task assignment). In this paper, we extend the RBAC model of Sandhu et al. in two directions: (i) our formal security model defines a Task-oriented Access Control (TAC) model which is capable of supporting task assignment condition in workflows and (ii) we leverage TAC specifications to inject delegation constraints, thereby computing potential delegatees and their required privileges.

The remainder of this article is organised as follows. Section 2 defines workflow authorisation constraints during task execution. In section 3, we present a formal security model to reason about task assignment within a workflow. This model is used in Section 4 to integrate delegation constraints in order to compute delegatees with their respective privileges. Finally, we conclude and discuss future work.

Workflow Authorisation Constraints

A workflow comprises various activities that are involved in a business process. Activities that are part of a process are represented as tasks [START_REF] Wfmc | Workflow Management Coalition Terminology and Glossary[END_REF]. Authorisation information is given which authorises users to perform tasks. Such authorisation information may be specified using a simple access control list or more complex role-based structures [START_REF] Crampton | Delegation and satisfiability in workflow systems[END_REF].

We define a task execution model using an activity diagram composed of three main activities : Initialisation, Processing and Finalisation (see Fig. 1). During the initialisation of the task, a task instance is created and then assigned to a user. During task processing, the assigned user can start or delegate the task which gathers all operations and rights over the business objects related to task's resources (see Definition 1). Finally, the task finalisation would notice the workflow management system that the task is terminated, where termination defines completeness, failure or cancellation.

Seeing a task as a block that needs protection against undesired accesses, the activity diagram includes an access control transition which is in charge of granting access to a task. Access control defines a transition from the creation of a task to its assignment to a user. This assignment will lead to the processing or the cancellation of a task. Cancellation can be triggered when an assigned user does not fulfill the required authorisation to execute a task.

Definition 1 (Permission). P is a set of permissions. P defines the right to execute an operation on a resource type. A permission p is a pair (f,o) where f is a function and o is a business object. We note : P ∈ F × O where F is a set of functions and O is a set of business objects.

Authorisation information define the access control transition. We define a permission as an authorisation allowing a users to perform a task. Authorisation makes an explicit binding between a user, a task resource (business object) and his rights over it (fuction/action). In our work, we define a task oriented access control model based on the RBAC model. We focus on task's requirements to analyse and specify security constraints while accessing workflow's data. Data access defines permissions on business objects related to task's resources.

Task-oriented Access Control Model

We propose a Task-oriented Access Control (TAC) model to support authorisation requirements in workflow systems (see Fig. 2). Authorisation information will be inferred from access control data structures, such as user-role assignment (URA) and task-role assignment (TRA) relations. In addition, we model permission assignment relations for tasks and roles in order to support the task execution context. The remaining relations are generic relations based on the RBAC model [START_REF] Sandhu | Role-based access control models[END_REF].

Formally, we define sets U, R, OU, T, P, S and TI as a set of users, roles, organisations units, tasks, permissions, subjects and task instances, respectively. We use a subject to denote the time a user selects roles for a session. During the task instantiation assignment, we create a user's current active role set and define it as a subject (see Fig. 2). For example, the user Alice with the role clerk defines a subject to execute the instance of a task "Check credit" in a bank loan process.

We define RH (Role Hierarchy), where RH is a partial order on R, r i and r j ∈ R. RH denotes that r i is a role superior to r j , as a result, r i automatically inherits the permissions of r j . We define RM (Role Mapping), where RM ⊆ OU i × OU j with OU i and OU j two organisations units. RM defines external roles accessing distributed resources cross-organisations. It provides a decentralised access control mechanism where externally known roles are publicly available : r k ∈ OU i and r l ∈ OU j , RM denotes that r l is a role mapped to r k , as a result, r l shares the permissions of r k .

Definitions of map relations

Formally, we define sets of relations as follows:

-U RA ⊆ U × R, the user role assignment relation mapping users to roles they are member of. -RP A ⊆ R × P , the permission role assignment relation mapping roles to permissions they are authorised to. -T P A ⊆ T × P , the task permission assignment relation mapping tasks to permissions. This defines the set of permission required to execute a task (see Definition 2). -T RA ⊆ T × R the task role assignment relation mapping roles to tasks they are assigned to.

Definitions of functions

Formally, we define sets of functions as follows:

-SU : S → U a function mapping a subject to the corresponding user.

-SR: S → R, a function mapping each subject to a role, where SR(s) = r, (SU (s), r) ∈ U RA} with a subject s having a permission p|(r, p) ∈ RP A}. -instance of : T I → T , a function mapping a task instance to its task type.

-claimed by : T I → S, a function mapping a task instance to a subject to execute it. It defines the user-task assignment condition s = claimed by (t i1 ) where :

{t i = instance of (t i1 ), (r, u) ∈ U RA|(SR(s) = r SU (s) = u), (t i , r) ∈ T RA}.

Definitions of constraints

Here we discuss Separation of duty (SoD) and Binding of duty (BoD) constraints. It defines security constraints between two tasks that compose a business process [START_REF] Botha | Separation of duties for access control enforcement in workflow environments[END_REF]. Such constraints help to verify whether a user is not allowed to execute a task due to some conflicts (e.g., conflict of interest). We define an exclusive relation between tasks for SoD, and a binding relation between tasks for BoD :

T T SOD : {(t i , t j ) ∈ T × T | t i is exclusive with t j } T T BOD : {(t i , t j ) ∈ T × T | t i is binding with t j }
If (t i , t j ) ∈ T T SOD , then t i and t j cannot be assigned to the same user. If (t i , t j ) ∈ T T BOD , then t i and t j must be assigned to the same user which defines a binding relation between two tasks.

Model contributions

The main contribution of the TAC model is to specify the task assignment relation where two conditions have to be verified: (1) the first condition is related to task's resources requirements. The role's permissions defined in RPA (rolepermission assignment) needs to satisfy the permissions defined in TPA (taskpermission assignment). (2) the task is executed if and only if the user/role is assigned to it. Basically, having a permission to execute a task but not being assigned to it will not satisfy the outlined conditions and, therefore, will deny the access to task resources. Definition 2 (Task Assignment). A task instance t i is assigned to a user u with an active subject s if and only if : (t, r) ∈ T RA ⇒ {p ∈ P |(t, p) ∈ T P A} ⊆ {p|(r, p) ∈ RP A} claimed by (t i ) = s, where (SR(s) = r SU (s) = u).

The user-task assignment requires the claimed by function. For instance, a task t i is assigned a set of permissions based on the TPA relation in order to carry out this task. A user u 1 with a role r j is assigned to t i if and only if u 1 verifies the TRA and claimed by conditions. However, if we consider another user u 2 member of same role r j having the same permissions based on the RPA relation but u 2 is not defined in claimed by (t i ), which means not assigned to this task. In this case, u 2 is not allowed to execute t i since he does not fulfill the user-task assignment relation (see condition 2). In the loan process example, let user Bob a member of role Clerk but not from the same bank agency. Bob is not allowed to perform the task "Check credit" since he is not assigned by the system to execute it. Within organisations, users can share different roles but are not assigned to the same tasks. This is due to privacy and security constraints such as the separation of duty. Therefore, we leverage condition 2 as an additional constraint when claiming a task instance by a user.

In the next section, we leverage the user-task assignment conditions to support task delegation assignment with regards to the delegatees and its required privileges.

Access Control Over Delegation using TAC

Delegation is a mechanism that permits a user to assign a subset of his assigned authorisations (privileges) to other users who currently do not possess it.

Definition 3 (Delegation Relation). We define a delegation relation DR ⊆ T ×U ×U ×2 DC where T a set of tasks, U a set of users and DC a set of delegation constraints. A task delegation relation is defined as DR = (t,u 1 ,u 2 ,{DC}), t is the delegated task and t ∈ T, u 1 the delegator and u 2 the delegatee ∈ U.

For instance, delegation constraints (DC) can be related to time or evidence specifications [START_REF] Gaaloul | A Secure Framework for Dynamic Task Delegation in Workflow Management Systems[END_REF]. In addition, organisational constraints regarding roles mapping cross organisations or role hierarchies within an organisation define user-to-user delegation constraints (see RM and RH relations of the TAC model in Fig. 2). For instance, a subordinate in an organisation hierarchy can act on behalf of his superior where the latter is the delegator and the former is the delegatee.

Here, a delegation relation defines the main constraints to be considered when delegating privileges with regards to users/roles, task and resources. Our focus is to integrate such constraints in a secure manner. In doing so, we leverage the TAC (task-oriented access control) model specifications to compute delegatees and privileges. The TAC model allows to compute the list of potential delegatees using the RPA (role-permission assignment) relation that may satisfy the delegated task requirements based on the TPA (task-permission assignment) relation. In doing so, we define a method for access control over task delegation using TAC. In the following, we detail our method and describe how valid delegatees are checked and whether they need delegated privileges grant.

Input: u 1 , u 2 ∈ U ; r 1 , r 2 ∈ R; t i , t j ∈ T.
1. Defining the role and permission assignments for each user (URA and RPA); 2. Instantiating the task t i1 and assigning it to the delegator s 1 who is the current user u 1 ; 3. Checking security constraints before delegation (SoD and BoD); 4. Computing the delegatee s 2 , who is the current user u 2 , based on his permissions assignment ((t i , p r2 ) ∈ T P A) or;

5. Granting privileges for s 2 based on the task instance permissions assignment (p r2 ← p r2 ∪ p ti ) which is defined in the claimed by function;

Output: Delegation relation instance : dr 1 = (t i1 ,s 1 ,s 2 ,{DC});

The main contribution of this method is to specify the delegated task assignment conditions based on Definition 2. If the two conditions are satisfied, then the task t i is delegated to the delegatee u 2 . However, if u 2 does not have the permission required and there is no conflicts (BoD or SoD) to execute t i . Then the delegated privileges are granted for u 2 based on the claimed by function.

The computation of the privileges is based on the TRA and claimed by specifications defined in our TAC model (see claimed by condition for permissions). Basically, we provide a method to compute the least privileges to delegate based on the current requirements of the task instances t i1 which is generated from the delegated task. At this stage, delegated privileges are done manually supporting a user-to-user delegation. However, the administration of new access rights has to be specified later on into authorisation policies in a compliant and dynamic manner. Authorisation policies will regulate how the business process and resources should be managed when delegating a task within a workflow. Delegation policies are not discussed in this paper due to space restrictions.

Related Work

Barka et al. proposed a role-based delegation model based on the RBAC model. Their unit of delegation is a role. Authors focused also on role-based models supporting role hierarchies when studying delegation in the context of both RBAC0 model (flat roles) and RBAC1 model (hierarchical roles) of the RBAC96 family [START_REF] Barka | Framework for role-based delegation models[END_REF]. However, users may want to delegate a piece of permission which is not supported in such models. This is the case when computing delegated privileges.

Task-based access control (TBAC) aims to provide a task context during permission assignments [START_REF] Thomas | Task-based authorization controls (tbac): A family of models for active and enterprise-oriented autorization management[END_REF]. A workflow system consisting of tasks is assumed. Each of these tasks is then assigned a "protection state", providing information as to who gets to have which permission on a task basis. According to the current state of the workflow system moving through the process instance, different permission assignments are activated or deactivated as ordered by the protection state. The TBAC design is process oriented, however, ignoring human-centric interactions such as user-to-user delegation.

Team based access control (TMAC) is an access control scheme similar to RBAC, but it provides the assignment of both users and permissions to teams [START_REF] Thomas | Team-based access control (tmac): a primitive for applying rolebased access controls in collaborative environments[END_REF]. Each team then is bound to the task it was created for. At runtime, more than one team can be created out of the same template, but each team will be working on a different task instance and accordingly will need access to different object instances. TMAC model is out of the scope of this paper where we consider constraints on tasks and users rather than a team.

Conclusion

In this paper, we integrated task delegation constraints into a formal security model. In doing so, we analysed task authorisation constraints to support security requirements for delegation. We defined a Task-oriented Access Control (TAC) model to support access control over task delegation in workflow systems. Moreover, we presented a method to compute potential delegatees and their delegated privileges. In future work we plan to examine the XACML (eXtensible Access Control Markup Language) standard for the TAC model to support task delegation constraints in particular and authorisation policies in general.

Fig. 1 .

 1 Fig. 1. Task execution model.

Fig. 2 .

 2 Fig. 2. Task-oriented access control (TAC) model.


Abou Rami
email: rami.aboujamra@uk-erlangen.de

Sigrun Jamra

Markus Wohlfart

Steffen Zweier

Lutz Uebe

Arif B Priebe

Susanne Ekici

Ahmad Giesebrecht

Abboud

Ayman Muhammad

Mahmoud Al-Khateeb

Fakher

M.D. Jamra

Sigrun Wohlfart

Markus Zweier

Steffen Uebe

Lutz Priebe

Arif Ekici

Susanne Giesebrecht

Ahmad Abboud

Ayman Al-Khateeb

Mahmoud Fakher

Saber Hamdan

Safia Muhammad

Markus M Nöthen

Johannes Schumacher

André Reis

Homozygosity mapping in 64 Syrian consanguineous families with non-syndromic intellectual disability reveals 11 novel loci and high heterogeneity

Keywords: non-specific, mental retardation, microcephaly, epilepsy, MRT

come

INTRODUCTION

Intellectual disability (ID) is the most common reason for referral to clinical genetic centers and is an unsolved problem in health care with important impact on the individuals, their families and the health system. While mild forms of ID (intelligence quotient (IQ) 70 -50) may in some cases be of genetic complex nature, severe forms (IQ < 50, incidence of 0.4 %) are thought to be caused by striking environmental or genetic factors. 1 In addition to the classification based on severity, patients may be classified into syndromic and non-syndromic forms. In syndromic forms, dysmorphisms, biochemical parameters, imaging findings, and organ anomalies exist and may help identifying a syndrome of known etiology. In the non-syndromic form, further non-specific symptoms such as epilepsy, ataxia, mild microcephaly, and behavioral disorders may exist. This means that a characteristic pattern of subtle symptoms may be overlooked in the first patient presenting and will be recognized only when examining several patients with mutations in the same gene. Non-specific intellectual disability may thus be considered as an exclusion diagnosis of distinguishable, syndromic intellectual disability, with a considerable grey zone between both forms.

In the last few years it became clear, that genetic factors, i.e. chromosomal anomalies and point mutations, play a central role in the etiology of ID. In a representative study Rauch and colleagues showed that 15 % of the cases of intellectual disability are due to cytogenetic anomalies, and that further 15 % are due to submicroscopic aberrations. 2 X chromosomal mutations are possibly the cause in about 10 % of the cases. 3 This means that in more than half of cases the etiology of ID remains unsolved. The majority of these have probably an autosomal form of ID. In industrial countries, both dominant and recessive forms usually appear as sporadic cases due to small family size. This hampers genetic approaches for identifying the underlying genetic cause. Genetically distinct subtypes are often clinically indistinguishable, precluding pooling of data from unrelated families and rendering their elucidation even more difficult. It is therefore not surprising, that to date only few genes have been identified (recently reviewed by Ropers). 1 The frequency of autosomal recessive, non-specific intellectual disability is unknown.

Although functional considerations and empirical data from mouse models suggest that most gene defects are inherited as recessive disorders, 1, 4, 5 a recent study suggests that autosomal dominant de novo mutations are most prevalent. 6 Until now 26 significantly linked NS-ARID loci were described 7-10 and only ten genes (six of them located in described regions) were identified: PRSS12 (OMIM#606709, on 4q26, former MRT1), CRBN (OMIM#609262, on 3p26, former MRT2), CC2D1A (OMIM#610055, on 19p13.12, former MRT3), ST3GAL3 (OMIM#606494, on 1p34.3), GRIK2

(OMIM#138277, on 6q16, former MRT6), TUSC3 (OMIM#601385, on 8p22, former MRT7), ZNF26 (no OMIM#, on 19q13.2), TRAPPC9 (OMIM#613192, on 8q24, former MRT13), ZCH14 (OMIM#613279, on 14q31.3), and TECR (MIM*610057 on 19p13.12) (Figure 2). 1,[11][12][13][14][15][16][17][18][19] Taking into account that over 90 genes are responsible for only about 40 % of X chromosomal recessive intellectual disability cases and that about a half of the estimated 22,000 human genes are expressed in the brain, the total number of ARID genes may run into the hundreds. 1 Considering the heterogeneity observed in previous linkage studies and the fact that the so far identified genes do not account for a significant fraction of NS-ARID cases, systematic approaches are the most promising strategy to identify further genes. 7,8,10 Large consanguineous families provide enough information to map loci based on analysis in a single family and thus represent the best starting point. We present here the results of the homozygosity mapping in a series of 64 Syrian multiplex consanguineous families with NS-ARID.

PATIENTS AND METHODS

Affected families were identified with the help of local pediatricians. After obtaining written informed consent, all affected individuals were clinically evaluated by a pediatrician (AAb, AAl, MF, SH, AI, or SM) and by a geneticist (RAJ), with special attention to neurological, morphological, ophthalmological, dermatological, and skeletal symptoms. The mental status of affected and unaffected individuals was estimated by investigation of verbal and motor aptitudes at the time of visit. Parents were interviewed to discriminate facts about prenatal, perinatal and neonatal medical history of all children either affected or unaffected. Since formal assessment of intellectual disability was not possible we used a description of developmental milestones and abilities to evaluate severity of intellectual disability as previously described. 20 A detailed pedigree was constructed and the genealogical relationship within families was crosschecked by interviewing different family members. Blood probes were drawn and DNA was isolated from all available individuals. For one patient of each nuclear family, testing for Fragile X syndrome was carried out by PCR. In addition, all mothers were checked for X-inactivation status.

From the 85 families initially ascertained, 11 were excluded either for not providing sufficient blood samples, non-random X-inactivation in the mother of the index case suggesting X-linked inheritance, or non-informativeness of clinical and familial history.

Four further families were excluded because a diagnosis could be made (Phenylketonuria, Miller-Dieker-Syndrome, van der Knaap Leukoencephalopathy, Cohen syndrome). 6 further families with only one affected child each were also not analyzed further, as an autosomal dominant cause could not be excluded. Of the remaining 64 families, 6 nuclear families could be aggregated into two complex families with three branches of affected cousins each, and further 16 families into 8 complex families with two branches each. In total, 41 nuclear and 10 complex families consisted of 305 children (140 affected) and 128 parents were subject to molecular genetics analysis. Of those, 31 families had 2 affected children and 20 families had 3 or more affected children. Kinship co-efficiencies varied between 0.0039 and 0.1563. 12 families had only affected males and 11 only affected females. Though X chromosomal recessive inheritance cannot be definitely excluded, this is unlikely since mothers of affected males were healthy, had several healthy brothers, nephews, and uncles on the maternal side, fragile X syndrome was excluded, and X-inactivation was unremarkable.

Affected children presented either with moderate ID in 45 or severe ID in 19 nuclear families. All cases were non-specific, i.e. no peculiar or distinguishable combination of symptoms was found. Further unspecific symptoms we documented were epilepsy in 17 families, growth retardation in 11 families, muscular hypertonia in 5 families and hypotonia in 17 families, microcephaly in 14 families, ataxia in 4 families, and developmental regression in 3 families. The symptoms and severity were comparable in sibs. Since many syndromes have variable phenotypes, we cannot exclude syndromic forms with certainty.

All available affected individuals and their healthy siblings were genotyped, healthy parents only when grandparents were also consanguineous. In total, 154 healthy and Human SNP Array 6.0 (Affymetrix, Santa Clara, CA, USA) microarrays. The platform used was consistent within each family, with the exception of three individuals recruited at a later stage. Marker density was approx. 600,000 SNPS, 300,000 SNPs, and 900,000, respectively. The minimal coverage of approx. 300.000 SNPs is more than sufficient to identify homozygosities of at least 500kb (average of 50 markers).

To detect the presence of copy-number variants, molecular karyotyping was performed using either the Affymetrix Genotyping Console Software (Version 3.0.2) or the PennCNV software 21 for the Illumina arrays. In IBD regions we set filter criteria to ≥10kb and ≥5 markers (Affymetrix), and ≥20kb and a confidence value ≥10 (Illumina).

To discriminate between genomic polymorphisms and aberrations containing MRassociated genes or regions, we compared findings within each family, with the database of Genomic Variants (http://projects.tcag.ca/variation/) and with 820 (Affymetrix) and 750 (Illumina) internal controls.

Mendelian segregation was calculated using PedCheck and the EasyLinkage interface software and was confirmed in all instances. [22][23][24] Genome-wide analysis was performed using Homozygosity Mapper and re-checked manually. 25 In addition, linkage analysis was performed using ALLEGRO or GeneHunter under an autosomal recessive mode of inheritance with 99% penetrance and a disease allele frequency of 0.001 using the EasyLinkage interface software. 23,26,27

RESULTS AND DISCUSSION

Novel ARID gene loci

Twelve families showed only one identical by descent (IBD) region each, with significant LOD scores above 3. Eleven of these were not included in published regions and were thus designated MRT numbers (HUGO Gene Nomenclature Committee, http://www.genenames.org/index.html, see table 2). In the twelfth family, MR055, we identified a mutation in TRAPPC9. The linked regions vary in length between 1.2 and 45.6Mb and include between 9 and 625 RefSeq genes (table 2).

In the era of Next Generation Sequencing, families with chromosomal regions IBD are of great value, even if they have more than one IBD region. In addition to the 11 families with a single locus, further 17 families showed IBD regions with a total length of less than 1 % of the genome (< 31Mb), distributed over two (6 families), three (8 families), four (1 family), or five (2 families) IBD regions. These IBD segments will be highly valuable for identifying further mutations in candidate genes. CNV analysis in IBD regions identified no deletion or duplication in candidate genes. In one family with 7 IBD regions we found a relatively common, recently described deletion over the gene OTOA at 16p12.2 explaining their hearing impairment but not the ID. 28

Identifying the causing mutation in TRAPPC9 in Family MR055

Family MR055 consists of two branches (a and b). In the first branch parents were cousins I° and there were three affected females (two are true twins) and one healthy male. In the second branch parents were cousins III° once removed and there were three affected males (two are true twins) and one healthy female. Both families are distantly related since great-grand parents were cousins.

At the time of examination, the three affected boys of family MR055a were 13 and 7 (twins) years old, 141, 116, and 116cm tall, and had head circumferences of 47, 45.5, and 46cm, respectively. Pregnancies and births were unremarkable, in the neonatal period parents reported muscular hypotonia. The affected boys learned walking at age of 5 years, they can understand simple phrases, but cannot speak. All have stereotypic movements and hand-flapping behavior. The elder brother had few epileptic seizures.

The affected brothers resemble each other with low frontal hairline, synophrys, and microcephaly. At the time of examination, the three affected women of family MR055b were 33 and 23 (twins) years old, 132, 133, and 139cm tall, and had head circumferences of 43.5, 42, and 42cm, respectively. Pregnancies and births were unremarkable. They learned walking at age of seven years, never learned speaking, and they do not understand simple phrases. All have stereotypic movements and handflapping behavior. None of the affected had epileptic seizures. The elder affected woman is spontaneously losing her teeth and losing weight. The parents reported that this started at age of 20, which can be observed by the younger two affected sisters (23 years at the time of examination). As in the first branch, the affected sisters resemble each other with low frontal hairline, synophrys, and microcephaly.

Family MR055 showed significant linkage on chromosome 8 (table 2) including TRAPPC9, which is a perfect candidate gene. [16][17][18] We sequenced the gene and identified the mutation c.1423C>T;p.R475X recently reported in a Palestinian family. 17 Because of the geographical and ethnical proximity (our family MR055 is from the south of Syria) and assuming an extremely low frequency of this mutation, we postulate that this is a founder mutation of this specific population and/or geographical area. Symptoms of the Palestinian and Syrian patients were comparable; moderate to severe intellectual disability, microcephaly, hand-flapping movements, no regression, and no epilepsy. In contrast to our patients, Mir and colleagues reported normal height and less severe intellectual disability, 16 and Philippe and colleagues reported normal height and obesity. 18 Thus we conclude that mutations in TRAPPC9 cause intellectual disabilities with tendency to severe forms and microcephaly. Further symptoms such as short stature, stereotypic movements, epilepsy, and remarkable MRI findings seem to be not specific. Because of the relative similarity of our and the Palestinian patients, we do not exclude a genotype-phenotype correlation.

Loci overlapping known ARID loci

Apart from family MR055 with the TRAPPC9 mutation, none of the families with one linkage locus overlapped any of the 10 to date described ARID genes. Nonetheless, five families overlap seven already known loci but with yet undetected causative gene (Fig. 2). Overlapping findings are of importance since these might facilitate the finding of a second allele. Further, we checked for overlapping IBD regions across all families studied, i.e. also families with several IBDs. We identified five regions, each with overlapping IBDs in five families on chromosomes 3 (175.5-177.5Mb and 178.5-180.6Mb), 19 (7.6-7.9Mb and 8.3-9.2Mb), and 22 (20.2-21.1Mb, all hg19) and 13 regions with four overlapping IBDs, each. These regions harbor interesting candidates for further analysis. Nevertheless, in the 51 families we identified a total of 275 IBD segments encompassing 3,196Mb, thus any overlap may also be coincidental.

Intrafamilial heterogeneity in complex families

The two complex families MR061 and MR071, each consisting of three branches, a, b, and c, are an example of the impressive heterogeneity of the NS-ARID phenotype (figure 1). Both families did not show one common linkage region for all three branches. When analyzed together, families MR061a and MR061b showed one homozygous locus on chromosome 14q11-q12 (MRT26, 9.1Mb long, LOD-score of 3.85), whereas branch c does not map to this locus (figure 1). The nuclear families MR071a, MR071b, and MR071c also did not show a common homozygote region.

Depending on the analyzed combination of nuclear families, different linked loci are possible. While the combination of either MR071a and MR071b or MR071a and MR071c showed four linkage regions of up to 30Mb with LOD scores of ≥2.4 each, the combination MR071b and MR071c showed one homozygote region of about 33Mb on chromosome 15q21-q25 (LOD-score of 2.7). Though we would expect that genotyping further members of this family of extensive consanguinity (kinship co-efficient >0.15) might define the true linkage region, this family shows that even high LOD-scores must be considered cautiously. On the other hand, suggestive LOD scores may turn out to be true findings. An example has been shown by Najmabadi and colleagues who identified a LOD score of only two in family M001, and later identified a mutation in TRAPPC9 which is located in this region and proved that this locus is true, though the linkage analysis was not significant. 7,16 Eight complex families MR006, MR013, MR018, MR019, MR022, MR043, MR049, and MR055, have two branches each, a, and b. Families MR022a and MR022b, and MR049a and MR049b showed no common homozygous region (supplementary figure 1). Five of the other six families, MR006, MR013, MR019, MR043, and MR055 showed one common homozygous region with LOD scores >3, each, on 6q12-q15 (MRT18), 18p11.32-p11.31 (MRT19), 16p12.1-q12.1 (MRT20), 11p15.5-15-4 (MRT21), and over TRAPPC9, respectively (table 2). We conclude that large and complex families represent a promising template to identify candidate regions, but bare the risk of false positive results because of heterogeneity. At the extreme end, such heterogeneity may exist within the same nuclear family.

Conclusions and outlook

Our findings suggest that NS-ARID is extremely heterogeneous in the Syrian population which is in line with prior findings in other populations. 7,8 Autosomal recessive disorders are even a greater challenge in populations with low consanguineous marriages since those will often be compound heterozygote and thus not due to IBD mutations. In addition, it seems certain by now that mutations in the currently known NS-ARID genes (PRSS12, CRBN, CC2D1A, GRIK2, TUSC3, TRAPPC9, ZNF526, ZC3H14, ST3GAL3, and TECR) are responsible for only a small fraction of NS-ARID cases, while hundreds of additional genes are likely to be identified in the near future.

The most promising concept for this is the systematic analysis of a large number of consanguineous families, followed by ultra deep sequencing strategies. Identification of these genes will improve genetic counseling of affected families and advance our understanding of molecular networks involved in cognitive processes. Understanding these processes should allow the replacement of symptomatic and supportive therapies with pharmacotherapies based on a principled understanding of the causes of cognitive impairment as is already becoming evident for several neurodevelopmental disorders. 29 Though affected are closely related, there is genetic heterogeneity. A de novo happening in MR022b, MR049a, or MR049b cannot be excluded, since there is only one affected child. In family MR055 a nonsense mutation in TRAPPC9 was identified (for details see text).

Figure 1 :

 1 Figure 1: Overview of the 12 families which showed one linkage locus. The families were simplified as possible, important information such as healthy brothers of mothers or grandmothers are shown. Asterisks indicate genotyped DNAs.

Figure 2 :

 2 Figure 2: Overview of NS-ARID loci reported to date 7-10 (blue bars) and novel loci reported here (red bars). Positions of known NS-ARID genes are marked with an arrow next to the gene symbol. Diagram was drawn with help of the GenomeGraph tool of the UCSC genome browser. Supplementary figure 1: Pedigree of families MR022a and b and MR049a and b.

Table 1 :

 1 Structure and clinical description of the 12 families which showed one linkage locus Neonatal normal development, moderate ID, mild microcephaly, moderate motor delay, severe lingual delay, CT shows deficiencies of the white matter * Number of affected individuals and their gender, e.g. 5fm means 5 affected individuals of both sexes. ** Degree of relatedness between parents, e.g. 0.063 for I° cousins. In complex families, values for each nuclear family are given. These values are lower estimates since often relationships are more complex than could be reliably documented. *** If not otherwise indicated, pregnancy and birth were unremarkable, no facial gestalt, no epilepsy, no growth retardation, no microcephaly was observed.

	Family	# aff.* Kinship-co-	Age at presentation of	Symptoms***
			efficient**	affected individuals
	MR003	5fm	0.097	18, 14, 9, 6, 2yrs	Neonatal muscular hypotonia, mild motor delay, severe ID, no speech, understanding better, no autistic symptoms, MRI and
					further detailed examinations unremarkable
	MR006	4f	0.023, 0.094 15, 10, 7, 2yrs	Neonatal muscular hypotonia, severe motor delay, no walking, crawling, muscular hypotonia, severe ID, no speech, weak
					interaction with surrounding, macroglossia, growth retardation and mild microcephaly, scream loud and often
	MR013	3fm	0.133, 0.016 43, 23, 19yrs	Mild motor delay, moderate ID, can speak and interact with surrounding properly, arthrogryposis of several small joints,
					deformity of feet in two affected, no or discrete facial gestalt, amicable, cheerful
	MR019	4fm	0.004, 0.005 6, 2, 2 ,1yrs	Mild motor delay, moderate ID, can speak single words, understanding better, no or discrete facial gestalt, early onset of
					epilepsy by three patients
	MR043	3f	0.063, 0.078 8, 6, 4yrs	Severe motor delay, no walking, no sitting or sitting at age of 5yrs, severe ID, no or single words, absence epilepsy by one
					patient, mild microcephaly
	MR050	2m	0.063	9, 4yrs	Neonatal muscular hypotonia, later hypertonia of the upper extremities by one patient, moderate motor delay, severe ID,
					tremor, no speech, understanding better, CT unremarkable
	MR055ab 6fm	0.078, 0.008 33, 23, 13, 7yrs	Neonatal muscular hypotonia, severe motor delay, single words or no speech, simple understanding or no understanding,
					severe or mild growth retardation, severe or moderate microcephaly, stereotypic movements, hand-flapping, facial gestalt
	MR056	3fm	0.004	17, 12, 11yrs	Microcephaly and small size at birth, motor delay, delayed and simple speech, good understanding, moderate ID, growth
					retardation, mild microcephaly, MRI unremarkable
	MR059	3fm	0.078	11, 10, 8yrs	Mild motor delay, delay of speech, moderate ID, vision impairment (not defined), autistic symptoms, MRI unremarkable,
					anxiety
	MR061bc 5fm	0.016, 0.063 22, 20yrs	Severe motor delay, walked for few months and then lost ability to walk, atrophy and spastic of lower extremeties, severe ID,
					no speech, weak interaction with surrounding, normal or mild microcephaly, growth retardation
	MR068	5fm	0.063	19, 18, 15, 7, 6yrs	Normal motor development, delayed speech, moderate ID, epilepsy
	MR073	3m	0.094	10, 7, 1,5yrs

Table 2 :

 2 Genetic findings in 12 families showing a unique linkage locus

	Family

ACKNOWLEDGMENTS

We thank the patients and their families for their cooperation. We also thank Angelika Diem, Nadine Kluck, and Petra Rothe for the excellent technical support. We thank Dr.

Cornelia Kraus for the diagnostics of Fra X syndrome and X inactivation. R. Abou Jamra was supported by the Deutsche Forschungsgemeinschaft (DFG) (grant AB393). This study was supported by the German Intellectual disability Network (MRNET) through a grant from the German Ministry of Research and Education to A.

Reis (01GS08160).

CONFLICT OF INTEREST

The authors declare no conflict of interest.