Luciano Campi
email: campi@ceremade.dauphine.fr.

Umut C ¸etin
email: u.cetin@lse.ac.uk.

Albina Danilova
email: a.danilova@lse.ac.uk

Equilibrium model with default and insider's dynamic information *

Keywords: Default time, defaultable claim, equilibrium, dynamic information, insider trading, dynamic Bessel bridge AMS classification (2000): 60G44, 60H05, 60H10, 93E11. JEL classification: D82, G14

We consider an equilibrium model à la Kyle-Back for a defaultable claim issued by a given firm. In such a market the insider observes continuously in time the value of firm, which is unobservable by the market maker. Using the construction of a dynamic Bessel bridge of dimension 3 in [5], we provide the equilibrium price and the optimal insider's strategy. As in [3], the information released by the insider while trading optimally makes the default time predictable in market's view at the equilibrium. We conclude the paper by comparing the insider's expected profits in the static and dynamic private information case. We also compute explicitly the value of insider's information in the special cases of a defaultable stock and a bond.

Introduction

Consider a defaultable claim issued by a company with no recovery and the payoff f (1 + β 1 ) in case of no-default. β denotes the fundamental value process and is assumed to be a standard Brownian motion with β 0 = 0. The default time, T 0 , is given by

T 0 := inf{t > 0 : 1 + β t = 0}.
Campi and C ¸etin [START_REF] Campi | Insider trading in an equilibrium model with default: a passage from reduced form to structural modelling[END_REF] study an equilibrium model à la Kyle-Back in the case when T 0 is known to the insider at time 0. The main result therein is that a risk-neutral insider who is thus maximizing her expected profit reveals part of her private information making the default time predictable in market's view, while that very default time was, by assumption, totally inaccessible before the trading started. Moreover, it is shown in [START_REF] Campi | Insider trading in an equilibrium model with default: a passage from reduced form to structural modelling[END_REF] that the equilibrium demand is a 3-dimensional

V (s)dB Z s t ≥ 0,
where V is the left derivative of V . Also observe that Z 1 = 1 + β 1 and T 0 = V (inf{t > 0 : Z t = 0}). A precise description of the market model, based on the latter observation, will be the content of the next section.

The main results of the present paper can be summarized as follows. We obtain even in this case the existence of an equilibrium where the insider maximizes her profit while the market maker sets a rational pricing rule based on the observation of the total demand. This is the content of the main Theorem 3.2. In particular, we compute explicitly both the equilibrium pricing rule and the optimal insider's strategy. The corresponding equilibrium total demand X * is a semimartingale behaving like a bridge hitting the default barrier 0 for the first time at V (τ ) on the default event [τ ≤ 1] while reaching the fundamental value Z 1 at maturity in case of no-default, i.e. on [τ > 1]. Moreover we obtain, as in the static information case in [START_REF] Campi | Insider trading in an equilibrium model with default: a passage from reduced form to structural modelling[END_REF], that the equilibrium total demand X * is a Brownian motion for the market maker, that is the insider hides her actions behind noise trading so that the "Inconspicuous trade theorem" holds true.

Using the characterization of the equilibrium we have just described, we can compare the static and dynamic information case. It turns out that an agent willing to pay a price (the value of information) for getting some private information is indifferent between knowing τ and Z 1 from the beginning or only at the end through a continuous-in-time monitoring of the fundamental value Z. This is due to the fact that from her viewpoint the expected profits in both cases are the same (Theorem 4.1 and the discussion below it). Finally, we computed explicitly the value of private information in the two important cases of a defaultable stock and a corporate bond, finding that the longer the defaultable claim is traded, the higher such a value is.

Since this paper deals with a Kyle-Back equilibrium model with default and gradually revealing information, it could also be viewed as a generalization of Back and Pedersen [START_REF] Back | Long-lived information and intraday patterns[END_REF] and Campi, C ¸etin and Danilova [START_REF] Campi | Dynamic Markov bridges motivated by models of insider trading[END_REF] to a financial market with default.

The structure of the paper is the following: Section 2 sets the model, while Section 3 gives the existence and the characterization of the equilibrium (Theorem 3.2) via an explicit computation of the optimal insider's expected profit (Proposition 3.1). In the proofs of that section a crucial role is played by the construction of a 3-dimensional dynamic Bessel bridge in [START_REF] Campi | An explicit construction of a dynamic Bessel bridge of dimension 3[END_REF]. Finally, Section 4 provides a comparison between the static and the dynamic information case.

Description of the market model

To formulate the model of the market precisely, let (Ω, F, (F t ) t≥0 , P) be a filtered probability space satisfying the usual conditions of right continuity and P-saturatedness. Assume that on this probability space there exist two independent standard Brownian motions, B and B Z .

Consider now a defaultable claim issued by a firm with no recovery and payoff f (Z 1 ) in case of no-default, where Z denotes the fundamental value and follows

Z t = 1 + t 0 σ(s)dB Z s t ≥ 0, (2.1)
where σ : [0, 1] → R + is a measurable deterministic function and V (t) := t 0 σ 2 (s) ds.

Assumption 2.1 V and f satisfy the following:

1. V (t) > t for all t ∈ [0, 1) and V (1) = 1;

2. there exists some ε > 0 such that

ε 0 1 (V (t)-t) 2 dt < ∞.
3. f : [0, ∞) → R + is a nondecreasing function which is not identically 0 and such that

|f (z)| ≤ k 1 exp (k 2 z) , ∀z ∈ [0, ∞),
for some constants k 1 and k 2 .

The firm's default time is given by the random time V (τ ) where

τ := inf{t > 0 : Z t = 0}. (2.2)
and the price of the defaultable claim is determined in equilibrium. We generalize, by incorporating dynamic information, the equilibrium framework of Campi and C ¸etin [START_REF] Campi | Insider trading in an equilibrium model with default: a passage from reduced form to structural modelling[END_REF], which in turn is an extension of that of Back [START_REF] Back | Insider trading in continuous time[END_REF] to a market with a defaultable bond. We refer the reader to Back [START_REF] Back | Insider trading in continuous time[END_REF] for motivation and details that are not explained in what follows.

Remark 1

The modeling of the default can also be interpreted in terms of economic, τ , and recorded, V (τ ), default. It is documented that these two notions of default do not necessarily coincide and the latter is typically later than the former (see Guo et al. [START_REF] Guo | Distressed Debt Prices and Recovery Rate Estimation[END_REF]).

The microstructure of the market, and the interaction of market participants, is modeled as follows.

There are three types of agents: noisy/liquidity traders, an informed trader (insider), and a market maker, all of whom are risk neutral. The agents differ in their information sets, and objectives, as follows.

• Noisy/liquidity traders trade for liquidity reasons, and their total demand at time t is given by a standard (F t )-Brownian motion B independent of B Z .

• The market maker sets the price of the defaultable claim using his information set, which consists of two parts. The first component is the total order of the noise traders and the insider, which is denoted with X and has the decomposition

X = Xθ = B V (τ ) + θ V (τ ) , (2.3)
where θ is the position of the insider in the defaultable claim so that the total demand right before the insider starts trading at time 0 equals 0. Note that we stop the market at time V (τ ) so that there is no trading in the defaultable claim once the default has occurred. Let X = 1 + X and observe that X and X generate the same filtration. We will denote the minimal right continuous and complete filtration generated by X with F X , where we suppress the dependency on θ in the notation.

The second part of the market maker's information comes from the observation of the recorded default event, i.e. the market maker also observes whether the recorded default has happened or not. In mathematical terminology, this makes V (τ ) a stopping time in his filtration. Therefore, the market maker's information is modeled by the filtration

F M = (F M t ) 0≤t≤1 where F M t := F X t ∨ σ(V (τ ) ∧ t)
. Similar to Back [START_REF] Back | Insider trading in continuous time[END_REF] and Campi and C ¸etin [START_REF] Campi | Insider trading in an equilibrium model with default: a passage from reduced form to structural modelling[END_REF], we assume that the market maker sets the price as a function of the total order process at time t, i.e. we assume that

S t = D t H (t, X t ) , ∀t ∈ [0, 1) (2.4)
where D is the non-default process, i.e. D t = 1 [V (τ )>t] . Moreover, a pricing rule H has to be admissible in the sense of Definition 2.1. In particular, H ∈ C 1,2 and, therefore, S is a semimartingale on [0, 1).

• The informed investor observes the price process S t = D t H (t, X t ) where X is given by (2.3), and the fundamental value Z t , i.e. her filtration, (F I t ) t∈[0,1] , is given by (F Z,S t

). She is risk-neutral, thus, her objective is to maximize the expected final wealth, i.e. sup θ∈A(H)

E W θ 1 = sup θ∈A(H) E (f (Z 1 )1 [V (τ )>1] -S 1∧V (τ )-)θ 1∧V (τ ) + V (τ )∧1- 0 θ s dS s (2.5)
where A(H) is the set of admissible trading strategies for the given pricing rule H, which will be defined in Definition 2.2. In particular, θ is an absolutely continuous process. That is, the insider maximizes the expected value of her final wealth W θ 1 , where the first term on the right hand side of equation (2.5) is the contribution to the final wealth due to a potential differential between price and fundamental value at time V (τ ) ∧ 1, when the market terminates, and the second term is the contribution to final wealth coming from the trading activity.

Note also that the above market structure implies that the insider's optimal trading strategy takes into account the feedback effect i.e. that prices react to her trading strategy according to (2.4). Our goal is to find the rational expectations equilibrium of this market, i.e. a pair consisting of a pricing rule and an admissible trading strategy such that: a) given the pricing rule the trading strategy is optimal, b) given the trading strategy, the pricing rule is rational in the following sense:

D t H(t, X t ) = S t = E f (Z 1 )1 [V (τ )>t] |F M t , t ∈ [0, 1], (2.6) with S 1∧V (τ ) = f (Z 1 )1 [V (τ )>1] .
To formalize this definition of equilibrium, we first need to define the sets of admissible pricing rules and trading strategies.

Definition 2.1 A measurable function H : [0, 1] × R → R is a pricing rule if 1. H ∈ C 1,2 ([0, 1) × R); 2. E[D 1 H(1, B 1 )] < ∞ and E[ 1 0 D t H(t, B t ) 2 dt] < ∞; 3.
x → H(t, x) is strictly increasing for every t ∈ [0, 1).

Moreover, let θ be a trading strategy of the insider. Given θ, a pricing rule H is said to be rational if it satisfies equation (2.6).

Remark 2

The strict monotonicity of H in the space variable implies x → H(t, x) is invertible for t ∈ [0, 1), thus, the filtration of the insider is generated by X and Z. This in turn implies that

F S,Z t = F B,Z t
, for all t ∈ [0, 1). Using the continuity of the processes involved we get (F S,Z t

) t∈[0,1] = (F B,Z t
) t∈[0,1] , i.e. the insider has full information about the market.

It is standard (see, e.g., [START_REF] Back | Long-lived information and intraday patterns[END_REF], [START_REF] Cho | Continuous auctions and insider trading: uniqueness and risk aversion[END_REF] or [START_REF] Wu | Construction of Brownian Motions in Enlarged Filtrations and Their Role in Mathematical Models of Insider Trading[END_REF]) in the insider trading literature to limit the set of admissible strategies to absolutely continuous ones motivated by the result in Back [START_REF] Back | Insider trading in continuous time[END_REF], and we do so. The formal definition of the set of admissible trading strategies is summarized in the following definition.

Definition 2.2 An F B,Z -adapted θ is said to be an admissible trading strategy for a given pricing rule H if 1. it is absolutely continuous with respect to the Lebesgue measure, i.e. θ t = t 0 α s ds; 2. (X, Z) is a Markov process adapted to (F t ); 3. and no doubling strategies are allowed i.e.

E 1 0 D t H 2 (t, X t ) dt < ∞.
(2.7)

The set of admissible trading strategies for a given H is denoted with A(H).

Given these definitions of admissible pricing rules and trading strategies, it is now possible to formally define the market equilibrium as follows.

Definition 2.3 A couple (H * , θ * ) is said to form an equilibrium if H * is an admissible pricing rule, θ * ∈ A(H * ), and the following conditions are satisfied:

1. Market efficiency condition: given θ * , H * is a rational pricing rule.

2. Insider optimality condition: given H * , θ * solves the insider optimization problem:

E[W θ * 1 ] = sup θ∈A(H * ) E[W θ 1 ].

Equilibrium with dynamic insider's information

In order to determine the conditions for equilibrium we start with the optimality conditions for the insider. The next proposition describes the optimal insider's strategy in terms of the behavior of the resulting optimal demand at maturity. Proposition 3.1 Assume that a pricing rule H is a classical solution to

H t (t, x) + 1 2 H xx (t, x) = 0, H(1, x) = h(x), (3.8)
where h is nondecreasing right continuous function on R with at most exponential growth, the range of h contains that of f and 0

∈ (inf x h(x), sup x h(x)). If θ * ∈ A(H) satisfies lim t↑1 H(V (τ ) ∧ t, X * V (τ )∧t ) = f (Z 1 )1 [τ >1] a.s.
, where X * = B + θ * , then θ * is an optimal strategy, i.e., for all θ ∈ A(H),

E[W θ 1 ] ≤ E[W θ * 1 ] = E 1 ξ(0,a * ) (H(0, u) -a * )du + 1 2 1∧V (τ ) 0 H x (s, ξ(s, a * ))ds , (3.9)
where

a * = 1 [τ >1] f (Z 1
) and ξ(t, a) is the unique solution of H(t, ξ(t, a)) = a for all a in the range of h or in the interval (inf x h(x), sup x h(x)) and t < 1.

Proof. We will adapt Wu's proof of his Lemma 4.2 in [START_REF] Wu | Construction of Brownian Motions in Enlarged Filtrations and Their Role in Mathematical Models of Insider Trading[END_REF]. This proof can be splitted into two main steps.

Step 1. Fix an a in the range of h or in the interval (inf x h(x), sup x h(x)). Suppose that a is the maximum of h, then ξ(t, a) = ∞ for all t ∈ [0, 1) since H is strictly increasing with supremum being equal to a. Similarly if a is the minimum of h, then ξ(t, a) = -∞ for all t ∈ [0, 1). If a is neither the minimum nor the maximum, ξ(t, a) is uniformly bounded for all t ∈ [0, 1) due to Lemma A.1 and the continuity of H.

Next, consider the function

Φ a (t, x) := ∞ -∞ y+x X a min (h(u) -a) 1 2π(1 -t) exp - y 2 2(1 -t) du dy, (3.10)
where

X a min := inf{x : h(x) ≥ a} and X a max := sup{x : h(x) ≤ a}. Direct calculations show that Φ a t + 1 2 Φ a xx = 0, with the boundary condition Φ a (1, x) = x X a min (h(u) -a)du.
Therefore, Φ a is jointly continuous and nonnegative on [0, 1] × R and C 1,2 ([0, 1) × R). Moreover, for t < 1, Φ a x (t, x) = H(t, x) -a, so that the minimum of Φ a is achieved at ξ(t, a).

Suppose that |ξ(t, a)| = ∞ so that a is either the minimum or the maximum of h and that Φ a (1, x) is positive and decreasing as x → ξ(t, a). Then, Φ a (t, ξ(t, a)) := lim

x→ξ(t,a) Φ a (t, x) = lim x→ξ(t,a) ∞ -∞ Φ a (1, x + y) 1 2π(1 -t) exp - y 2 2(1 -t) dy = ∞ -∞ lim x→ξ(t,a) Φ a (1, x + y) 1 2π(1 -t) exp - y 2 2(1 -t) dy = 0. (3.11)
Moreover, when ξ(t, a) is finite,

0 ≤ lim t↑1 Φ a (t, ξ(t, a)) ≤ lim t↑1 Φ a (t, X a min ) = lim t↑1 ∞ -∞ y+X a min X a min (h(u) -a) 1 2π(1 -t) exp - y 2 2(1 -t) du dy = 0. (3.12) Next observe that Φ a (t, x) -Φ a (t, ξ(t, a)) = ∞ -∞ 1 2π(1 -t) exp - y 2 2(1 -t) y+x y+ξ(t,a) (h(u) -a)du dy = ∞ -∞ 1 2π(1 -t) exp - y 2 2(1 -t) x ξ(t,a) (h(u + y) -a)du dy = x ξ(t,a) (H(t, u) -a)du.
(3.13)

Step 2. Consider for all ν < 1

Ψ a,ν (t, x) := x ξ(t,a) (H(t, u) -a)du + 1 2 ν t H x (s, ξ(s, a))ds, t ≤ ν.
Notice that the both integrals in the RHS are well-defined: the first one is well-defined for all values of ξ(t, a) thanks to Step 1, and the second one is well-defined due to the fact that t → H x (t, ξ(t, a)) is uniformly bounded on [0, ν]. Direct differentiation with respect to x gives that

Ψ a,ν x (t, x) = H(t, x) -a. (3.14)
Differentiating above with respect to x gives

Ψ a,ν xx (t, x) = H x (t, x). (3.15) Direct differentiation of Ψ a,ν (t, x) with respect to t gives Ψ a,ν t (t, x) = x ξ(t,a) H t (t, u)du - 1 2 H x (t, ξ(t, a)) = - 1 2 H x (t, x)
where in order to obtain the last equality we used (3.8). Combining this and (3.15) gives

Ψ a,ν t + 1 2 Ψ a,ν xx = 0.
Therefore from (3.14) and Itô's formula it follows that

Ψ a,ν (ν, X ν ) -Ψ a,ν (0, 1) = ν 0 (H(u, X u ) -a)dX u ,
and in particular, when a :=

1 [τ >1] f (Z 1 ), ν(t) = t ∧ V (τ ), lim t↑1 Ψ a,ν(t) (t ∧ V (τ ), X t∧V (τ ) ) -Ψ a,ν(t) (0, 1) = 1∧V (τ ) 0 (H(t, X t ) -1 [τ >1] f (Z 1 ))dX t . (3.16)
By the admissibility properties of θ, in particular dθ t = α t dt, the insider's optimization problem becomes

sup θ∈A(H) E[W θ 1 ] = sup θ∈A(H) E (f (Z 1 )1 [τ >1] -S 1∧V (τ )-)θ 1∧V (τ ) + V (τ )∧1- 0 θ s dS s = sup θ∈A(H) E V (τ )∧1 0 (f (Z 1 )1 [τ >1] -H(t, X t ))α t dt
Due to above and (3.16), we have

E[W θ 1 ] = -E lim t↑1 Ψ a,ν(t) (t ∧ V (τ ), X t∧V (τ ) ) -Ψ a,ν(t) (0, 1) . (3.17)
Notice that all the Brownian integrals vanish due to (2.7) in Definition 2.2, and

E 1∧τ 0 f (Z 1 )dB t 2 ≤ E f (Z 1 ) 2 E[B 2 1 ] < ∞,
since Z and B are independent.

The conclusion follows from the fact that lim t↑1 Ψ a,ν(t

) (t ∧ V (τ ), X t∧V (τ ) ) is nonnegative and equals 0 if lim t↑1 H(V (τ ) ∧ t, X * V (τ )∧t ) = f (Z 1 )1 [τ >1] . Indeed, observe that lim t↑1 Ψ a,ν(t) (t ∧ V (τ ), X t∧V (τ ) ) = lim t↑1 X t∧V (τ ) ξ(t∧V (τ ),a) (H(t ∧ V (τ ), u) -a)du. On the set [V (τ ) ≥ 1] we have lim t↑1 Ψ a,ν(t) (t ∧ V (τ ), X t∧V (τ ) ) = lim t↑1 Xt ξ(t,a) (H(t, u) -a)du = lim t↑1 {Φ a (t, X t ) -Φ a (t, ξ(t, a))} = Φ a (1, X 1 ),
which is nonnegative and is 0 if lim t↑1 H(t, X * t ) = f (Z 1 ) in view of Lemma A.1. Observe that we used (3.13) for the second equality above while (3.11) and (3.12) for the third one.

On the set [V (τ

) < 1], lim t↑1 X t∧V (τ ) ξ(t∧V (τ ),a) (H(t ∧ V (τ ), u) -a)du = X V (τ ) ξ(V (τ ),a) (H(V (τ ), u) -a)du,
which is nonnegative and equals 0 if X V (τ ) = ξ(V (τ ), a) due to the invertibility of H. Therefore, an insider trading strategy which gives lim t↑1 H(V (τ

) ∧ t, X * V (τ )∧t ) = f (Z 1 )1 [τ >1] is optimal.
Remark 3 The same results as in Proposition 3.1 above apply when the initial insider's information F I 0 is not trivial provided one replaces expectations with conditional expectations given F I 0 in the statement as well as in its proof.

Combining Proposition 3.1 and the dynamic Bessel bridge construction performed in [START_REF] Campi | An explicit construction of a dynamic Bessel bridge of dimension 3[END_REF], we can finally state and prove the main result of this paper. Before that, we need some preliminary notation.

Recall that τ = inf{t > 0 : Z t = 0}. Since Z is a time-changed Brownian motion, one can characterize the distribution of τ using the well-known distributions of first hitting times of a standard Brownian motion. To this end let

P [T a > t] = ∞ t (u, a) du, (3.18)
for a > 0 where

T a := inf{t > 0 : B t = a}, and (t, a) := a √ 2πt 3 exp - a 2 2t .
Another formulation for the distribution of T a can be stated in terms of the transition density of a Brownian motion killed at 0. Recall that this transition density is given by

q(t, x, y) := 1 √ 2πt exp - (x -y) 2 2t -exp - (x + y) 2 2t ,
for x > 0 and y > 0 (see Exercise (1.15), Chapter III in [START_REF] Revuz | Continuous Martingales and Brownian Motion (3rd Revised Edition)[END_REF]). Then one has the identity

P [T a > t] = ∞ 0 q(t, a, y) dy. (3.19)
Moreover, if f is a function satisfying the conditions in Assumption 2.1, then one can define

P (t, z) := ∞ 0 f (y)q(1 -V (t), z, y)dy (3.20) so that 1 [τ >t] P (t, Z t )
is the value of the defaultable claim for the insider at time t and the process (1

[τ >t] P (t, Z t )) t∈[0,1]
is a martingale for the insider's filtration. For reader's convenience, we recall here the main result of the paper [START_REF] Campi | An explicit construction of a dynamic Bessel bridge of dimension 3[END_REF]. This is the key ingredient to solve our equilibrium model. Theorem 3.1 ( [START_REF] Campi | An explicit construction of a dynamic Bessel bridge of dimension 3[END_REF]) Suppose that V (t) > t for all t > 0 and satisfies Assumption 2.1.2. Then, there exists a unique strong solution to

X t = 1 + B t + τ ∧t 0 q x (V (s) -s, X s , Z s ) q(V (s) -s, X s , Z s ) ds + V (τ )∧t τ ∧t a (V (τ ) -s, X s ) (V (τ ) -s, X s ) ds. Moreover, i) Let F X t = N σ(X s ; s ≤ t)
, where N is the set of P-null sets. Then, X is a standard Brownian motion with respect to F X = (F X t );

ii) V (τ ) = inf{t > 0 : X t = 0}.

Notice that we cannot apply directly the previous result to get the solution to our equilibrium problem, since we impose V (1) = 1 for the time-change V (t). Nonetheless, Theorem 3.1 will be very useful to prove our main result, which is stated in the following theorem. Theorem 3.2 Under Assumption 2.1 there exists an equilibrium (H * , θ * ), where (i) H * (t, x) = P (V -1 (t), x) where P (t, x) is given by (3.20) for (t, x)

∈ [0, 1] × R. (ii) θ * t = t 0 α * s ds where α * s = q x (V (s) -s, X * s , Z s ) q(V (s) -s, X * s , Z s ) 1 [s≤τ ∧1] + a (V (τ ) -s, X * s ) (V (τ ) -s, X * s ) 1 [τ ∧1<s≤V (τ )∧1] (3.21)
where the process X * is the unique strong solution under insider's filtration F X,Z of the following SDE:

X t = 1 + B V (τ )∧t + τ ∧t 0 q x (V (s) -s, X s , Z s ) q(V (s) -s, X s , Z s ) ds + V (τ )∧t τ ∧t a (V (τ ) -s, X s ) (V (τ ) -s, X s ) ds. (3.22) Moreover, V (τ ) = inf{t ∈ [0, 1] : X * t = 0}
, where inf ∅ = 1 by convention, and one has lim t↑1 X * t = Z 1 on the set of non-default [τ > 1]. As a consequence, V (τ ) is a predictable stopping time in the market filtration F M . Furthermore, the expected profit of the insider is

E 1 ξ(0,a * ) (H * (0, u) -a * )du + 1 2 1∧V (τ ) 0 H * x (s, ξ(s, a * ))ds , (3.23)
where a * = 1 [τ >1] f (Z 1 ) and ξ(t, a) is the unique solution of H(t, ξ(t, a)) = a for all a ≥ 0.

Proof. Observe that

H * (t, x) = ∞ -∞ f (x + y) 1 2π(1 -t) exp - y 2 2(1 -t) dy, where f (y) = f (y), y ≥ 0; -f (-y), y < 0. Thus, clearly, H * ∈ C 1,2 ([0, 1) × R), E[D 1 H * (1, B 1 )] < ∞, and E[ 1 0 D t H * (t, B t ) 2 dt] < ∞.
To show that it is a pricing rule in the sense of Definition 2.1, it is enough to show that it is strictly increasing for any t ∈ [0, 1). We have, due Assumption 2.1.3, for x > z and t < 1,

H(t, x) -H(t, z) = ∞ -∞ ( f (x + y) -f (z + y)) 1 2π(1 -t) exp - y 2 2(1 -t) dy ≥ 0.
Note that the above holds in equality if and only if f (x + y) = f (z + y) for almost all y, which is not possible due to the construction of f and the assumption on f . Therefore H * is a pricing rule in the sense of Definition 2.1.

Moreover, direct calculations show that H * as defined above satisfies (3.8). Since the process (1 [τ >t] P (t, Z t )) t∈[0,1] is a martingale for the insider's filtration, (D t H * (t, X * t )) t∈[0,1] would be an F M -martingale as soon as we show that (X * t ) t∈[0,1] is a Brownian motion stopped at V (τ ) in its own filtration, where V (τ ) is the first time that it hits 0. This would imply that F X * = F M and that H * is a rational pricing rule in the sense of Definition 2.1 and that the proposed optimal strategy is admissible for the insider.

To do so, we prove that there exists a unique strong solution of (3.22) on [0, 1), X * , satisfying the following properties:

1) lim t↑1 X * t∧V (τ ) = 0 a.s. on [τ < 1], 2) lim t↑1 X * t = Z 1 a.s. on the set [τ > 1],
3) (X * t ) t∈[0,1] is a Brownian motion stopped at V (τ ) in its own filtration. This will establish (H * , θ * ) as equilibrium in view of Proposition 3.1 since, due to 1) and 2), we have lim t↑1 H * (t ∧ V (τ ), X t∧V (τ ) ) = 1 [τ >1] f (Z 1 ). Moreover, the expected profit of the insider is given by (3.23) due to (3.9).

Due to Theorem 3.1 there exists unique strong solution, X * , to (3.22) on [0, 1). Moreover, V (τ ) = inf{t > 0 : X * t = 0} on the set [τ < 1], so that property 1) above is satisfied. On the non-default set [τ > 1], which is the same as [V (τ ) > 1], the SDE (3.22) becomes

X t = 1 + B t + t 0 q x (V (s) -s, X s , Z s ) q(V (s) -s, X s , Z s ) ds, t ∈ [0, 1). (3.24)
The function qx(t,x,z) q(t,x,z) appearing in the drift above can be decomposed as follows

q x (t, x, z) q(t, x, z) = z -x t + b(t, x, z) := z -x t + exp -2xz t 1 -exp -2xz t 2z t .
We want to prove that lim t↑1 X * t = Z 1 a.s. on the set [τ > 1]. Consider the process

R t := X * t -λ(t) t 0 b(V (s) -s, X * s , Z s ) ds λ(s) ,
where

λ(t) = exp - t 0 ds V (s) -s .
Direct calculations give that, on [τ > 1], dR t = Zt-Rt V (t)-t dt + dB t , thus we can apply the results in Back and Pedersen [START_REF] Back | Long-lived information and intraday patterns[END_REF] (see also [START_REF] Campi | Dynamic Markov bridges motivated by models of insider trading[END_REF], Proposition 3.2) to conclude that R t goes to Z 1 as t ↑ 1 a.s. on the set of non-default [τ > 1]. To deduce from it that X t → Z 1 a.s. on [τ > 1] when t ↑ 1, we have to show

lim t↑1 λ(t) t 0 b(V (s) -s, X * s , Z s ) ds λ(s) = 0, a.s. on [τ > 1]. Observe that b(V (t) -t, X * t , Z t ) = g 2X * t Zt V (t)-t 1 X * t , with g(u) = e -u
1-e -u u, and g(u) ∈ [0, 1] for all u ∈ [0, +∞]. Since on the set [τ > 1] we have inf t∈[0,1] Z t > 0 and inf t∈[0,1] X * t ≥ 0 (as, due to Theorem 3.1, inf t∈[0,1)

X * t > 0), we obtain inf t∈[0,1] b(V (t) -t, X * t , Z t ) ≥ 0 on [τ > 1]
, and therefore the following two cases are possible:

• Case 1: lim t↑1 t 0 b(V (s) -s, X * s , Z s ) ds λ(s) < ∞. Then, since 0 ≤ lim t↑1 λ(t) ≤ lim t↑1 exp - t 0 ds 1 -s = 0,
we are done.

• Case 2: lim t↑1 t 0 b(V (s) -s, X * s , Z s ) ds λ(s) = ∞. Since both λ(t) and t 0 b(V (s) -s, X * s , Z s ) ds λ(s)
are differentiable for fixed ω in [τ > 1], we can use de l'Hôpital's rule to get:

lim t↑1 λ(t) t 0 b(V (s) -s, X * s , Z s ) ds λ(s) ds = lim t↑1 (V (t) -t)b(V (t) -t, X * t , Z t ) = 0 (3.25) a.s. on the set of non-default [τ > 1] provided lim sup t↑1 b(V (t) -t, X * t , Z t ) < ∞ a.s.. Since b(V (t)-t, X * t , Z t ) = g 2X * t Zt V (t)-t 1 X * t with g being a bounded function on [0, +∞], to show that lim sup t↑1 b(V (t) -t, X * t , Z t ) < ∞, it is sufficient to demonstrate that lim inf t↑1 X * t > 0 on [τ > 1].
To prove it, consider two processes X and Y which follow

d Xt = Z t -Xt V (t) -t + g 2 Xt Z t V (t) -t 1 Xt 1 [τ >t] dt + dB t , t ∈ [0, 1), dY t = Z t -Y t V (t) -t 1 [τ >t] dt + dB t , t ∈ [0, 1].
The process X is well defined and is strictly positive for all t ∈ [0, 1) due to the Theorem 3.1.

Moreover, for all t ∈ [0, 1) we have

Y t 1 [τ >1] = R t 1 [τ >1] and Xt 1 [τ >1] = X * t 1 [τ >1] and therefore it is sufficient to show that lim inf t↑1 Xt > 0 on [τ > 1]. Observe that d(Y t -Xt ) = Xt -Y t V (t) -t -g 2 Xt Z t V (t) -t 1 Xt 1 [τ >t] dt,
and g and X are strictly positive, so that by Tanaka's formula (see Theorem 1.2 in Chap. VI of [START_REF] Revuz | Continuous Martingales and Brownian Motion (3rd Revised Edition)[END_REF])

(Y t -Xt ) + = t 0 1 [Ys> Xs] Xs -Y s V (s) -s -g 2 Xs Z s V (s) -s 1 Xs 1 [τ >s] ds ≤ t 0 1 [Ys> Xs] Xs -Y s V (s) -s 1 [τ >s] ds ≤ 0
since the local time of Y -X at 0 is identically 0 (see Corollary 1.9 in Chap. VI of [START_REF] Revuz | Continuous Martingales and Brownian Motion (3rd Revised Edition)[END_REF]).

Thus, on the set [τ > 1] we have lim inf

t↑1 X * t = lim inf t↑1 Xt ≥ Y 1 = R 1 = Z 1 > 0 as required.
Recall that P[τ = 1] = 0, therefore, one has V (τ ) = inf{t ∈ [0, 1] : X * t = 0}, where inf ∅ = 1 by convention. This makes V (τ ) a stopping time with respect to F X * and yields that F X * = F M . To complete the proof we need to show that X * is a Brownian motion in its own filtration, stopped at V (τ ). Notice first that the construction of the dynamic Bessel bridge in [START_REF] Campi | An explicit construction of a dynamic Bessel bridge of dimension 3[END_REF] implies in particular that X * is a Brownian motion in its own filtration over each interval [0, T ] for every T < 1, i.e. (X * t∧V (τ ) ) t∈[0,1) is a Brownian motion. As we have seen, lim t↑1 X * t∧V (τ ) = Z 1∧τ ; thus, it follows from Fatou's lemma that (X * t∧V (τ ) ) t∈[0,1] is a supermartingale. In order to obtain the martingale property over the whole interval [0, 1], it suffices to show that E[X * 1∧V (τ ) ] = 1. However, since X * 1∧V (τ ) = Z 1∧τ , this follows from the fact that Z is a martingale. In view of Lévy's characterization, we conclude that X * is a Brownian motion in its own filtration, stopped at V (τ ).

Comparison of dynamic and static private information

In this section we compare the expected profits of the insider in the cases of dynamic and static, i.e. when the insider knows τ and Z 1 in advance, private information. In order to do so, we first need to obtain the equilibrium and the associated expected profit when the private information is static. The concepts of equilibrium, admissibility and the market microstructure are analogous to the definitions in Section 2.

Recall that the proof of Proposition 3.1 did not depend on the type of the private information, therefore, the optimality conditions for the insider with a static information are still described by it after replacing expectations with conditional expectations (see Remark 3). Theorem 4.1 Suppose that the insider observes τ and Z 1 at time 0. Then, under Assumption 2.1 there exists an equilibrium (H * , θ * ), where (i) H * (t, x) = P (V -1 (t), x) where P (t, x) is given by (3.20)

for (t, x) ∈ [0, 1] × R + . (ii) θ * t = t 0 α * s ds for t ∈ [0, 1 ∧ V (τ )]
,where

α * s = q x (1 -s, X * s , Z 1 ) q(1 -s, X * s , Z 1 ) 1 [τ >1] + a (V (τ ) -s, X * s ) (V (τ ) -s, X * s ) 1 [τ ≤1] (4.26)
where the process X * is the unique strong solution under insider's filtration F X,Z 1 ,τ of the following SDE:

X t = 1 + B V (τ )∧t + V (τ )∧t 0 q x (1 -s, X * s , Z 1 ) q(1 -s, X * s , Z 1 )
1 [τ >1] + a (V (τ ) -s, X * s ) (V (τ ) -s, X * s ) 1 [τ ≤1] ds. (4.27) Moreover, V (τ ) = inf{t ∈ [0, 1] : X * t = 0}
, where inf ∅ = 1 by convention, and one has lim t↑1 X * t = Z 1 on the set of non-default [τ > 1]. As a consequence, V (τ ) is a predictable stopping time in the market filtration F M . Furthermore, the expected profit of the insider is

1 ξ(0,a * ) (H * (0, u) -a * )du + 1 2 1∧V (τ ) 0 H * x (s, ξ(s, a * ))ds, (4.28)
where a * = 1 [τ >1] f (Z 1 ) and ξ(t, a) is the unique solution of H(t, ξ(t, a)) = a for all a ≥ 0.

Proof. Since the optimality conditions for the insider are still described by Proposition 3.1, the proof will follow the same lines as the proof of Theorem 3.2 once we show that there exist unique strong solution of (3.24) on [0, 1), X * , satisfying the following properties:

1) lim t↑1 X * t∧V (τ ) = 0 a.s. on [τ < 1],

2) lim t↑1 X * t = Z 1 a.s. on the set [τ > 1],

3) (X * t ) t∈[0,1] is a Brownian motion stopped at V (τ ) in its own filtration.

To see this consider a Brownian motion β, in a possibly different probability space, with β 0 = 1 and T 0 = inf{t > 0 : β t = 0}. Let (G t ) t≥0 be the minimal filtration satisfying usual conditions and to which β is adapted and G 0 ⊃ σ(β 1 , T 0 ). Direct calculations show that

P[T 0 ∈ du, T 0 > 1, β 1 ∈ dy|F β t ] = 1 [1∧T 0 >t] (u -1, y)q(1 -t, β t , y) dy du + 1 [T 0 >t≥1] (u -t, β t ) du P[T 0 ∈ du, T 0 ≤ 1, β 1 ∈ dy|F β t ] = 1 [T 0 >t] 1 2π(1 -u) exp - y 2 2(1 -u) (u -t, β t ) dy du +1 [T 0 ≤t] 1 2π(1 -t) exp - (y -β t ) 2 2(1 -t) dy
Thus, it follows from Theorem 1.6 in [START_REF] Mansuy | Random Times and Enlargements of Filtrations in a Brownian Setting[END_REF] that

β t = 1 + βt + t∧1∧T 0 0 q x (1 -s, β s , β 1 ) q(1 -s, β s , β 1 ) 1 [T 0 >1] + a (T 0 -s, β s ) (T 0 -s, β s ) 1 [T 0 ≤1] ds + t∧T 0 t∧1∧T 0 a (T 0 -s, β s ) (T 0 -s, β s ) ds + t∧1 t∧1∧T 0 β 1 -β s 1 -s ds,
where β is a G-Brownian motion independent of β 1 and T 0 . Observe that Z V -1 (t) is a standard Brownian motion starting at 1 with V (τ ) as its first hitting time of 0. Moreover, the SDE satisfied by β until T 0 ∧1 is the same as (3.24) until time 1 since (β 1 , T 0 , β) has the same law as (Z 1 , V (τ ), B) due to V (1) = 1. Therefore, the law of (X * t∧V (τ )∧1 ) t≥0 is the same as that of (β t∧T 0 ∧1 ) t≥0 since the solution of the SDE for β has strong uniqueness. In particular, properties 1), 2) and 3) above are satisfied. Now, we are in a position to compare the value of static and dynamic information. This comparison is relevant to an uninformed and risk-neutral investor at time 0 who is about to decide whether to purchase a particular private information at a given price. Obviously, as the investor is uninformed her information prior to making this decision is trivial. Thus, the decision will be based on the comparison of the expected profits resulting from the purchased information, and the expectation will be taken with respect to the trivial σ-algebra. Comparison of (3.23) and (4.28) leads to the immediate conclusion that this risk-neutral investor is indifferent between purchasing static or dynamic information, whose value is given by (3.23).

This indifference might appear counterintuitive at first. However, it is clear that a necessary condition for the optimality is that the insider drives the market price to the fundamental value of the asset at the termination of the market since otherwise she wouldn't have used all her informational advantage. On the other hand, Proposition 3.1 demonstrates that this is also sufficient.

Consequently, the only thing she strives to achieve is to make sure that the price converges to the fundamental value. This observation together with her risk-neutrality will lead her to value both types of information same since the variance of the signals does not affect her valuation.

The same phenomenon is also responsible for the fact that the price of information does not depend on V , which also manifests itself in expression (3.23) since the distribution of V (τ ) is the same as that of the first hitting time of 0 by a Brownian starting at 1. In fact, it is easy to observe that the static information is the limiting case of dynamic ones characterized by an increasing sequence of functions V n with lim n→∞ V n (t) = 1 for all t ∈ (0, 1].

The value of information, (3.23), can be computed more explicitly as the following proposition shows.

Proposition 4.1 Suppose f is invertible with f (0) = 0 and satisfies Assumption 2.1. Then, (3.23) becomes

E[W θ * 1 ] = E[F (|1 + β 1 |)] -E F |β 1 | 1 -V (τ ) 1 [τ <1] -F (1)P[τ > 1] +E {(Z 1 -1)f (Z 1 ) -(F (Z 1 ) -F (1))} 1 [τ >1] , (4.29)
where β is a standard Brownian motion independent of B Z with β 0 = 0, and F (z) := z 0 f (y) dy.

Proof. For any a ≥ 0 let

g(a) := 1 ξ(0,a) (H * (0, u) -a)du + 1 2 1∧V (τ ) 0 H * x (s, ξ(s, a))ds.
Since ξ(s, 0) = 0 for any s ≥ 0,

g(0) = 1 0 H * (0, u)du + 1 2 1∧V (τ ) 0 H * x (s, 0)ds.
We will first compute the first term in the equation above.

1 0 H * (0, u)du = 1 0 ∞ 0 f (y)q(1, u, y) dy du = - ∞ 0 F (y) 1 0 q y (1, u, y) du dy = ∞ 0 F (y) 1 √ 2π e -(y-1) 2 2 + e -(y+1) 2 2 dy -2 ∞ 0 F (y) 1 √ 2π e -y 2 2 dy = E[F (|1 + β 1 |)] -E[F (|β 1 |)].
In particular, this implies

E[F (|1 + β 1 |)] -E[F (c|β 1 |)] > 0 for any constant 0 ≤ c ≤ 1 since F is increasing.
The second term is given by 1 2

1∧V (τ ) where in the one to the last equality, we used the reflection principle and the last equality follows from the scaling property of Brownian motion. Thus, Taking the expectation of above, it is easy to see that the conclusion holds.

g(0) = E[F (|1 + β 1 |)] -E[F ( 1 -1 ∧ V (τ )|β 1 |)|τ ]. ( 4
Below are some explicit examples where we can compute the value of information.

Example 1 In the case of defaultable stock, f (x) = x. Then, the value of information becomes

E[W θ * 1 ] = 1 2 E (1 + β 1 ) 2 -E β 2 1 E (1 -V (τ ))1 [τ <1] + E (Z 2 1 -2Z 1 )1 [τ >1] = P[V (τ ) ≥ 1] + E V (τ )1 [V (τ )<1] = E[V (τ ) ∧ 1].
According to the last equality above, the longer the defaultable stock is traded, the higher is the insider's expected profit. Such a result is to be expected since the insider can speculate on her private information only when the market operates.

Although Proposition 4.1 requires f is invertible, one can still calculate the value of the information even if f fails this condition.

Example 2 Consider the defaultable zero-coupon bond with payoff f ≡ 1. Then, observe that (4.30) is still valid as we did not use the conditions on f to obtain it. Moreover, in this case a * takes values in {0, 1}. Thus, it remains to calculate g [START_REF] Back | Insider trading in continuous time[END_REF]. First, observe that H is bounded by 1 and strictly increasing, thus, ξ(t, 1) = ∞ and H x (t, ξ(t, 1)) = 0. Therefore,

g(1) = 1 ∞ (H * (0, u) -1)du = ∞ 1 ∞ -∞ (1 -sgn(y + u)) 1 √ 2π e -y 2 2 dydu = 2 πe -2P[β 1 < -1].
Thus, the value of information is

E[W θ * 1 ] = E g(0)1 [τ <1] + g(1)1 [τ >1] = 2 πe - 2 π E 1 -V (τ )1 [V (τ )<1] .
The last expectation on the RHS is an indicator on how far is, on average, the default time from the defaultable bond's maturity in the case of default before maturity. As in the previous example, the larger is that expectation, i.e. the larger is the average distance between market's default time and the maturity, the lower the value of the private information is.

f 0 f 0 f

 00 (y)P[T y < 1] dy -∞ (y)P[T y < 1 -1 ∧ V (τ )|τ ] dy = ∞ (y) P[|β 1 | > y] -P[|β 1-1∧V (τ ) | > y|τ ] dy = E[F (|β 1 |)] -E[F ( 1 -1 ∧ V (τ )|β 1 |)|τ ],

0 (ξ( 1 ∧

 01 (s, ξ(s, a))ξ a (s, a) ds.Differentiating the equality H * (s, ξ(s, a)) = a with respect to s and a yieldsH * s + H * x ξ s = 0, and H * x = 1 ξ a ,and using the fact thatH * t + 1 2 H * xx = 0, we get 1 2 H * xx ξ a = ξ s . Therefore, g (a) = ξ(0, a) -1 + 1∧V (τ ) 0 ξ s (s, a) ds = ξ(1 ∧ V (τ ), a) -1,and thusg(a) = E[F (|1 + β 1 |)] -E[F ( 1 -1 ∧ V (τ )|β 1 |)|τ ] + a V (τ ), u) -1)du. Plugging a = 1 [τ >1] f (Z 1 ) into above yields g(a) = E[F (|1 + β 1 |)] -E[F ( 1 -1 ∧ V (τ )|β 1 |)|τ ] + 1 [τ >1] f (Z 1 ) 0 (f -1 (u) -1)du = E[F (|1 + β 1 |)] -E[F ( 1 -1 ∧ V (τ )|β 1 |)|τ ] + 1 [τ >1] {(Z 1 -1)f (Z 1 ) -F (Z 1 )} .

* This research benefited from the support of the "Chair Les Particuliers Face aux Risques", Fondation du Risque (Groupama-ENSAE-Dauphine), the GIP-ANR "Croyances" project as well as from the "Chaire Risque de crédit", Fédération Bancaire Française, of the Europlace Institute of Finance, for their support. Part of this research has been done when the second author was visiting Evry University during September 2008.

A Appendix

Lemma A.1 Suppose that h is a nondecreasing right-continuous function with at most an exponential growth. Let

and (ξ n ) n≥1 be a convergent sequence such that lim n→∞ H(t n , ξ n ) = a for some a in the range of h or in the interval (inf x h(x), sup x h(x)), and some sequence (t n ) n≥1 ⊆ [0, 1) converging to 1. Then,

where X a min := inf{x : h(x) ≥ a} and X a max := sup{x : h(x) ≤ a}.

Proof. Suppose lim n→∞ ξ n < X a min . Then, there exists some ξ such that lim n→∞ ξ n < ξ < X a min . Since H is nondecreasing in x, one has

which is a contradiction. Similarly, we have that lim n→∞ ξ n ≤ X a max .


Jorieke E H Bergman

Nicole Janssen

Lies H Hoefsloot

Marjolijn C J Jongmans

Robert M W Hofstra

Conny M A Van Ravenswaaij
email: c.m.a.van.ravenswaaij@medgen.umcg.nl

Jeh Bergman

Mcj Jongmans

Rmw Hofstra

Cma Van Ravenswaaij-Arts

C M A Van

CHD7 mutations and CHARGE syndrome: the clinical implications of an expanding phenotype

Keywords: CHARGE syndrome, CHD7, phenotypic spectrum

published or not. The documents may come   L'archive ouverte pluridisciplinaire

INTRODUCTION

The first patients with what later became known as CHARGE syndrome (OMIM #214800) were described in 1961. [START_REF] Angelman | Syndrome of coloboma with multiple congenital abnormalities in infancy[END_REF][START_REF] Edwards | Coloboma with Multiple Congenital Anomalies[END_REF] In 1979, two independent clinicians recognised that coloboma, choanal atresia and congenital heart defects clustered together in several patients. [START_REF] Hall | Choanal atresia and associated multiple anomalies[END_REF][START_REF] Hittner | Colobomatous microphthalmia, heart disease, hearing loss, and mental retardation--a syndrome[END_REF] The acronym CHARGE dates from 1981 and summarises some of the cardinal features: ocular coloboma, congenital heart defects, choanal atresia, retardation of growth and/or development, genital anomalies and ear anomalies associated with deafness. [START_REF] Pagon | Coloboma, congenital heart disease, and choanal atresia with multiple anomalies: CHARGE association[END_REF] In 2004, mutations in the CHD7 gene were identified as the major cause and 'CHARGE association' was changed to 'CHARGE syndrome'. [START_REF] Vissers | Mutations in a new member of the chromodomain gene family cause CHARGE syndrome[END_REF] CHARGE syndrome occurs in approximately 1 in 10,000 newborns. [START_REF] Issekutz | An epidemiological analysis of CHARGE syndrome: preliminary results from a Canadian study[END_REF] The inheritance pattern is autosomal dominant with variable expressivity. Almost all mutations occur de novo, but parent-to-child transmission has occasionally been reported. [START_REF] Sanlaville | CHARGE syndrome: an update[END_REF] In this review, we explore the phenotypic spectrum of CHD7 mutations with special focus on the mild end of the spectrum. In the light of this expanding phenotype, we discuss whether CHARGE syndrome is a clinical or a molecular diagnosis, we propose guidelines for CHD7 analysis, and give updated recommendations for the clinical surveillance of CHD7-positive patients.

BACKGROUND Clinical diagnosis

Before discovery of the causative gene, CHARGE syndrome was a clinical diagnosis (Figure 1). Pagon was the first to introduce diagnostic criteria for CHARGE syndrome in 1981, [START_REF] Pagon | Coloboma, congenital heart disease, and choanal atresia with multiple anomalies: CHARGE association[END_REF] but these criteria are no longer in use. At present, the clinical criteria by Blake and Verloes are used in conjunction (Table 1). [START_REF] Blake | CHARGE association: an update and review for the primary pediatrician[END_REF][START_REF] Verloes | Updated diagnostic criteria for CHARGE syndrome: a proposal[END_REF] * Cleft palate can be substituted for choanal atresia, since these anomalies rarely occur together. [START_REF] Blake | CHARGE syndrome[END_REF] The Blake criteria [START_REF] Blake | CHARGE association: an update and review for the primary pediatrician[END_REF] were slightly adjusted by a consortium and last updated in 2009. [START_REF] Lalani | CHARGE Syndrome[END_REF] These criteria encompass four major and seven minor criteria. The four major criteria are coloboma, choanal atresia, cranial nerve dysfunction and abnormalities of the inner, middle, or external ear. At least four major, or three major and three minor, criteria must be present in order to diagnose CHARGE syndrome. In 2005, Verloes proposed renewed criteria. [START_REF] Verloes | Updated diagnostic criteria for CHARGE syndrome: a proposal[END_REF] He included semicircular canal defects as a major criterion, as these defects were shown to be a very specific and consistent feature in CHARGE syndrome. [START_REF] Admiraal | Temporal bone CT findings in the CHARGE association[END_REF] Verloes also anticipated broadening of the phenotypic spectrum and reduced the number of features necessary for CHARGE diagnosis (to only three major, or two major and two minor, criteria) and he made his criteria less age-and sexdependent. A common feature of both sets of criteria is that either coloboma or choanal atresia (which can sometimes be replaced by cleft palate, Table 1 [START_REF] Blake | CHARGE syndrome[END_REF]) must be present in order to diagnose CHARGE syndrome.

Molecular diagnosis

Nowadays, CHARGE syndrome can also be diagnosed by a molecular genetic test. The CHD7 gene, mutated in the majority of patients with CHARGE syndrome, consists of 37 coding exons and one non-coding exon. [START_REF] Vissers | Mutations in a new member of the chromodomain gene family cause CHARGE syndrome[END_REF] The gene encodes for a 2997 amino-acidlong protein that belongs to the Chromodomain Helicase DNA binding (CHD) family. [START_REF] Woodage | Characterization of the CHD family of proteins[END_REF] CHD7 can form complexes with different proteins thereby ensuring specific binding to different enhancer regions leading to time-and tissue-specific regulation of gene expression. [START_REF] Layman | Chromodomain proteins in development: lessons from CHARGE syndrome[END_REF] One example is the association of CHD7 with BPAF (polybromo-and BRG1-associated factor containing complex) that is essential for neural crest gene expression and cell migration. [START_REF] Bajpai | CHD7 cooperates with PBAF to control multipotent neural crest formation[END_REF] This is in line with previous assumptions that many of the congenital defects seen in CHARGE syndrome may be neural crest related. [START_REF] Siebert | Pathologic features of the CHARGE association: support for involvement of the neural crest[END_REF] CHD7 was also shown to associate with rDNA and was therefore suggested to play a role as positive regulator of rRNA synthesis. [START_REF] Zentner | CHD7 functions in the nucleolus as a positive regulator of ribosomal RNA biogenesis[END_REF] Haploinsufficiency of the CHD7 gene leads to CHARGE syndrome and, as expected, most patients are found to have truncating CHD7 mutations. [START_REF] Aramaki | Phenotypic spectrum of CHARGE syndrome with CHD7 mutations[END_REF][START_REF] Jongmans | CHARGE syndrome: the phenotypic spectrum of mutations in the CHD7 gene[END_REF][START_REF] Lalani | Spectrum of CHD7 mutations in 110 individuals with CHARGE syndrome and genotypephenotype correlation[END_REF][START_REF] Lee | Clinical and genetic analysis of the CHD7 gene in Korean patients with CHARGE syndrome[END_REF][START_REF] Wincent | CHD7 mutation spectrum in 28 Swedish patients diagnosed with CHARGE syndrome[END_REF][START_REF] Zentner | Molecular and phenotypic aspects of CHD7 mutation in CHARGE syndrome[END_REF] Missense mutations occur in a minority of patients and partial or full deletions of the CHD7 gene are rare events. [START_REF] Vissers | Mutations in a new member of the chromodomain gene family cause CHARGE syndrome[END_REF][START_REF] Aramaki | Phenotypic spectrum of CHARGE syndrome with CHD7 mutations[END_REF][START_REF] Wincent | CHD7 mutation spectrum in 28 Swedish patients diagnosed with CHARGE syndrome[END_REF][START_REF] Arrington | Interstitial deletion 8q11.2-q13 with congenital anomalies of CHARGE association[END_REF][START_REF] Bergman | Exon copy number alterations of the CHD7 gene are not a major cause of CHARGE and CHARGE-like syndrome[END_REF][START_REF] Hurst | Balanced t(6;8)(6p8p;6q8q) and the CHARGE association[END_REF][START_REF] Johnson | Confirmation of CHD7 as a cause of CHARGE association identified by mapping a balanced chromosome translocation in affected monozygotic twins[END_REF][START_REF] Udaka | An Alu retrotransposition-mediated deletion of CHD7 in a patient with CHARGE syndrome[END_REF][START_REF] Vuorela | Molecular analysis of the CHD7 gene in CHARGE syndrome: identification of 22 novel mutations and evidence for a low contribution of large CHD7 deletions[END_REF][START_REF] Wincent | Detection of CHD7 deletions by MLPA in CHARGE syndrome patients with a less typical phenotype[END_REF] Most CHD7 mutations occur de novo. There are no mutational hotspots and recurrent mutations are rare. [START_REF] Jongmans | CHARGE syndrome: the phenotypic spectrum of mutations in the CHD7 gene[END_REF] No clear genotype-phenotype correlation exists, although it seems that missense mutations in general are associated with a milder phenotype. [START_REF] Jongmans | CHARGE syndrome: the phenotypic spectrum of mutations in the CHD7 gene[END_REF] CHD7 analysis detects mutations in more than 90% of patients fulfilling the clinical criteria for CHARGE syndrome. The lack of mutation detection in the remaining 5-10% of patients suggests genetic heterogeneity. The SEMA3E gene was proposed as a candidate gene, but it seems to play a minor role as only two SEMA3E alterations have been described in patients with CHARGE syndrome. [START_REF] Lalani | SEMA3E mutation in a patient with CHARGE syndrome[END_REF] Besides genetic heterogeneity, it is also possible that mutations in intronic regions, 5' or 3' untranslated regions, or in regulatory elements of CHD7 underlie the CHD7-negative cases.

Phenocopies of CHARGE or CHARGE-like syndrome can be due to teratogen exposure (e.g., thalidomide, retinoic acid and maternal diabetes) or chromosomal aberrations. [ Missense mutations and splice site mutations occurred in 13% and 17%, respectively and deletions were rarely present (<1%). The phenotypic spectrum of the missense mutations was more variable and on average milder when compared to the truncating mutations.

In Table 2 we present an overview of the clinical features of 280 of our CHD7-positive patients, the CHD7-positive cohort reported in the literature (reviewed by Zentner n=254 [START_REF] Zentner | Molecular and phenotypic aspects of CHD7 mutation in CHARGE syndrome[END_REF]) and a cohort of patients clinically diagnosed with CHARGE syndrome, but of whom the CHD7 status is unknown (n=124 [START_REF] Issekutz | An epidemiological analysis of CHARGE syndrome: preliminary results from a Canadian study[END_REF][START_REF] Tellier | CHARGE syndrome: report of 47 cases and review[END_REF]). We only included 280 of our 360 CHD7-positive patients, because clinical data were lacking in the other 80 patients. The phenotypes of 64 of the 280 patients have been published previously (Table 2). [START_REF] Jongmans | CHARGE syndrome: the phenotypic spectrum of mutations in the CHD7 gene[END_REF][START_REF] Bergman | Exon copy number alterations of the CHD7 gene are not a major cause of CHARGE and CHARGE-like syndrome[END_REF][START_REF] Bech | Isolation of the left subclavian artery from the pulmonary artery in a patient with CHARGE association[END_REF][START_REF] Douglas | Extending the phenotypic spectrum of CHARGE syndrome: a case with preaxial polydactyly[END_REF][START_REF] Jongmans | Familial CHARGE syndrome and the CHD7 gene: a recurrent missense mutation, intrafamilial recurrence and variability[END_REF][START_REF] Van De Laar | Limb anomalies in patients with CHARGE syndrome: an expansion of the phenotype[END_REF][START_REF] Pedersen | Molecular diagnosis of CHARGE syndrom[END_REF][START_REF] Writzl | Immunological abnormalities in CHARGE syndrome[END_REF] ^ Frequencies are represented as the number of patients with a particular feature / the total number of patients that were tested for that particular feature † The range of percentages presented between brackets was calculated as: (positive / total) x 100% -(positive + unknown / total) x 100% (for further explanation see text)

« Outside the frequency range of patients with a CHD7 mutation

The clinical features of the CHD7-positive patients, previously reported or presented here, are rarely completely known. When calculating the percentage of patients who exhibit a certain feature, the incompleteness of the clinical data will have a major effect on the accuracy of the percentage. In order to compensate for this inaccuracy, we also calculated the frequency range. The minimum frequency is defined as the number of patients with a particular feature divided by the total number of patients in the cohort. The maximum frequency is defined as the number of patients with a particular feature plus patients for whom it is unknown whether they have the feature, divided by the total number of patients in the cohort.

Four features are almost always present in patients with a CHD7 mutation:

external ear anomalies, cranial nerve dysfunction, semicircular canal hypoplasia, and delayed attainment of motor milestones (Table 2). The characteristic external ear anomaly consists of triangular conchae or cup-shaped ears (Figure 1) and occurs in more than 90% of patients with a CHD7 mutation. The second feature, cranial nerve dysfunction, is present in more than 95% of patients. The seventh and eighth cranial nerves are most often affected leading to facial palsy and sensorineural hearing loss, respectively. Dysfunction of other cranial nerves can also occur. The third feature, semicircular canal hypoplasia, is not always assessed, but when investigated it is found to be present in over 90% of patients. The high frequency of semicircular canal hypoplasia is reflected in the delayed attainment of motor milestones (often scored as developmental delay in previous papers), that is almost universally present in patients with CHARGE syndrome. A delay in speech development is also common in these patients who suffer from multiple sensory impairment (e.g. blindness and/or deafness). [START_REF] Goldson | The CHARGE association. How well can they do?[END_REF][START_REF] Money | Pedagogical handicap associated with micropenis and other CHARGE syndrome anomalies of embryogenesis: four 46, XY cases reared as girls[END_REF] In our cohort, approximately 75% of patients had intellectual disability, indicating that one quarter had a normal intelligence.

Two features seem to occur more frequently since CHD7 analysis has become available as a diagnostic tool in CHARGE syndrome (Table 2). These are cleft lip and/or palate and genital hypoplasia; in the study by Tellier [START_REF] Tellier | CHARGE syndrome: report of 47 cases and review[END_REF], the percentages of these two features were below our frequency range. The most likely explanation is that in the past, patients with cleft palate, and thus often without choanal atresia were not recognised as having CHARGE syndrome. Mutation analysis enables a diagnosis in these clinically less typical patients. The higher prevalence of genital hypoplasia in patients with a CHD7 mutation can be explained by a higher mean age in the patients for whom molecular studies have been performed, but it may also be due to an increased awareness that genital hypoplasia is a frequent feature in patients with a CHD7 mutation.

One

Familial CHARGE syndrome

Very mildly affected patients with CHARGE syndrome can be identified by studying familial CHARGE syndrome. In the literature, only 16 families have been described with recurrence of molecularly confirmed CHARGE syndrome. [START_REF] Jongmans | CHARGE syndrome: the phenotypic spectrum of mutations in the CHD7 gene[END_REF][START_REF] Lalani | Spectrum of CHD7 mutations in 110 individuals with CHARGE syndrome and genotypephenotype correlation[END_REF][START_REF] Wincent | CHD7 mutation spectrum in 28 Swedish patients diagnosed with CHARGE syndrome[END_REF][START_REF] Jongmans | Familial CHARGE syndrome and the CHD7 gene: a recurrent missense mutation, intrafamilial recurrence and variability[END_REF][START_REF] Delahaye | Familial CHARGE syndrome because of CHD7 mutation: clinical intra-and interfamilial variability[END_REF][START_REF] Pauli | Proven germline mosaicism in a father of two children with CHARGE syndrome[END_REF][START_REF] Vuorela | A familial CHARGE syndrome with a CHD7 nonsense mutation and new clinical features[END_REF] These families include seven sib-pairs, three monozygotic twin-pairs, and six two-generation families. In this review, we describe another two-generation family from our CHD7positive cohort, making a total of 17 families (Table 3). +, fulfilling the criteria, -, not fulfilling the clinical criteria of Blake and/or Verloes. [START_REF] Blake | CHARGE association: an update and review for the primary pediatrician[END_REF][START_REF] Verloes | Updated diagnostic criteria for CHARGE syndrome: a proposal[END_REF] Of the 39 CHD7-positive individuals, only 24 (62%) fulfilled the clinical criteria for CHARGE syndrome as defined by either Blake or Verloes. [START_REF] Blake | CHARGE association: an update and review for the primary pediatrician[END_REF][START_REF] Verloes | Updated diagnostic criteria for CHARGE syndrome: a proposal[END_REF] Atypical CHARGE patients are most frequently seen in the two-generation families. Often, the mildly affected individuals were recognised only after a CHD7 mutation was found in a more severely affected family member. The most mildly affected patients described in the literature had dysmorphic ears and balance disturbance as the only manifestations of CHARGE syndrome. Somatic mosaicism was considered unlikely in two of the very mildly affected parents, because the CHD7 mutation was found in different tissues. [START_REF] Jongmans | Familial CHARGE syndrome and the CHD7 gene: a recurrent missense mutation, intrafamilial recurrence and variability[END_REF][START_REF] Vuorela | A familial CHARGE syndrome with a CHD7 nonsense mutation and new clinical features[END_REF] The monozygotic twin-pairs showed strikingly discordant features and underscore the great intra-familial variability seen in CHARGE syndrome. [START_REF] Jongmans | CHARGE syndrome: the phenotypic spectrum of mutations in the CHD7 gene[END_REF][START_REF] Lalani | Spectrum of CHD7 mutations in 110 individuals with CHARGE syndrome and genotypephenotype correlation[END_REF][START_REF] Wincent | CHD7 mutation spectrum in 28 Swedish patients diagnosed with CHARGE syndrome[END_REF] This variability might be explained by differential epigenetic regulation or fluctuating embryonic CHD7 levels in relation to a time-and tissue-dependent critical threshold during embryonic development.

Mildly affected patients from our CHD7-positive cohort

The most widely used criteria are those of Blake/Lalani. [ Kallmann syndrome usually presents as the combination of hypogonadotropic hypogonadism (HH) and anosmia. Both features also occur in the majority of patients with CHARGE syndrome. [START_REF] Asakura | Endocrine and radiological studies in patients with molecularly confirmed CHARGE syndrome[END_REF][START_REF] Blustajn | Olfactory anomalies in CHARGE syndrome: imaging findings of a potential major diagnostic criterion[END_REF][START_REF] Chalouhi | Olfactory evaluation in children: application to the CHARGE syndrome[END_REF][START_REF] Pinto | CHARGE syndrome includes hypogonadotropic hypogonadism and abnormal olfactory bulb development[END_REF] Other features that can be present in both syndromes are hearing loss, cleft lip/palate and renal malformations. Two studies have been performed in which patients with normosmic HH or Kallmann syndrome were screened for CHD7 mutations. CHD7 mutations were reported in seven out of 197 patients with normosmic HH or Kallmann syndrome, [START_REF] Kim | Mutations in CHD7, encoding a chromatin-remodeling protein, cause idiopathic hypogonadotropic hypogonadism and Kallmann syndrome[END_REF] in three out of 36 patients with Kallmann syndrome (confirmed by a smell test), but in none of 20 patients with normosmic HH. [START_REF] Jongmans | CHD7 mutations in patients initially diagnosed with Kallmann syndrome--the clinical overlap with CHARGE syndrome[END_REF] The second study showed that after thorough clinical examination of the CHD7-positive Kallmann patients, other CHARGE features were universally present. The authors concluded that these patients represent the mild end of the CHARGE phenotypic spectrum, as we also demonstrated in our patient who was referred with Kallmann syndrome (see the section "Mildly affected patients from our own CHD7 positive cohort").

CHD7 AND CHARGE SYNDROME: THE CLINICAL IMPLICATIONS

Based on the studies conducted after the identification of CHD7 and summarised above, we discuss whether CHARGE syndrome is a clinical or molecular diagnosis, propose a new guideline for CHD7 analysis, and give recommendations for clinical surveillance of CHD7-positive patients.

CHARGE syndrome, a clinical or molecular diagnosis?

In our opinion, CHARGE syndrome is primarily a clinical diagnosis. If patients fulfil the clinical criteria of Blake or Verloes and chromosomal aberrations and teratogenic exposure effects fully explaining the clinical features have been ruled out, then they have CHARGE syndrome, irrespective of the results of CHD7 analysis. On the other hand, patients who do not completely fulfil the clinical criteria should not be excluded from CHD7 analysis. If a mutation is found in these patients, clinical follow-up and genetic counselling should be performed as in clinically diagnosed patients with CHARGE syndrome.

Guideline for CHD7 analysis

Considering the broad phenotypic spectrum, it is evident that CHD7 analysis should not be restricted to patients fulfilling the clinical criteria for CHARGE syndrome. Coloboma and choanal atresia (or cleft palate) are not always present in CHARGE syndrome.

Therefore patients with other CHARGE features, but without those cardinal features, should not be excluded from CHD7 analysis. When a patient is suspected of CHARGE syndrome, the external ears, cranial nerve function and semicircular canals should be thoroughly examined, as these features occur in the great majority of patients with a CHD7 mutation (Table 2).

We propose a guideline for CHD7 analysis in Figure 3. In our experience, imaging of the semicircular canals is not an easy routine in daily clinical practice, especially in children in whom sedation can be complicated (see clinical surveillance and Table 4). Therefore, in our guideline we have indicated when imaging of the semicircular canals is needed to support the decision for CHD7 analysis. We based our guideline on the clinical features that were present in our CHD7-positive patients (n=280). When applying our guideline, CHD7 analysis would not have been recommended in one of our patients. This patient is the first patient described in "Mildly affected patients from our own CHD7 positive cohort" and is extremely mildly affected. A prospective study is needed to evaluate the usefulness of this guideline in clinical practice.

Clinical surveillance of patients with a CHD7 mutation or typical CHARGE syndrome

Ideally, follow-up of patients with a CHD7 mutation or typical CHARGE syndrome should be done by an expert multidisciplinary team, because this approach will ensure optimal treatment of this very complex patient group. In the Netherlands, several specialities are involved in the CHARGE outpatient clinic of the University Medical Centre Groningen: clinical genetics, paediatric endocrinology, ENT, speech and occupational therapy, ophthalmology, child and youth psychiatry, social paediatrics, gynaecology, endocrinology, paediatric cardiology, neuroradiology and dentistry. In Table 4, we show updated recommendations for clinical surveillance of patients with a CHD7 mutation based on the experiences of our CHARGE outpatient clinic, on the clinical features in our CHD7-positive cohort (Table 2), and on a literature review.

5 #

 5 Updated by a consortium in 2006 and 2009.[START_REF] Lalani | CHARGE Syndrome[END_REF]

Figure 1 ••• 2 *Figure 2

 122 Figure 1 Overview of features occurring in CHARGE syndrome #

Table 1

 1 Clinical criteria for CHARGE syndrome

		Major criteria	Minor criteria	Inclusion rule
	Blake # [9]	1. Coloboma,
		microphthalmia
		2. Choanal atresia or
		stenosis*
		3. Characteristic external
		ear anomaly,
		middle/inner ear
		malformations, mixed

8] PHENOTYPIC SPECTRUM OF PATIENTS WITH A MUTATION IN THE CHD7 GENE Phenotypic spectrum in our CHD7-positive cohort compared to two other cohorts

	suspected of CHARGE syndrome and 360 CHD7 mutations were found (360/863 =
	42%). The mutations were scattered throughout the entire coding region and splice sites
	of the CHD7 gene. One third of the mutations were found in exon 2, 3, 30 and 31 (34%
	of mutations, 33% of genomic size). However, exons 8, 12, 26, 30 and 36 showed a
	remarkably high number of mutations relative to their genomic size (19% of mutations,
	9% genomic size). No mutations were found in exon 6, 7, 20 and 28, but these exons
	comprise only 3% of the coding genome of CHD7. Apart from the high number of
	mutations in exon 2 (the largest exon), this is not in accordance with a previous report
	(n=91).[33] Most mutations were nonsense (38%) or frameshift mutations (32%).
	Our CHD7-positive cohort consists of patients who had CHD7 analysis done in
	Nijmegen, the Netherlands. In Nijmegen, CHD7 analysis was performed in 863 patients

Table 2

 2

	Clinical features of patients with a CHD7 mutation compared to clinically
	diagnosed patients with CHARGE syndrome

Feature Our CHD7-positive cohort (n=280) CHD7-positive cohort from the CHARGE patients before CHD7 literature (n=254) # discovery (n=124) *

		76% (68-78%)	77%	85% «
	External ear anomaly Tracheo-oesophageal	224/231 ^ 42/146	214/235 35/185	74/77 22/124
	anomaly	97% (80-98%) † 29% (15-63%)	91% 19%	96% 18%
	Cranial nerve dysfunction Developmental delay	173/174 Delayed motor	? Developmental delay	107/124 Developmental delay
	(VII, VIII and others)	99% (62-100%) milestones	107/141	86% 47/47
	Semicircular canal anomaly 110/117 147/149	94/96 76%	12/12 100%
		94% (39-98%) 99% (53-99%)	98%	100%
		Intellectual disability
	Coloboma	189/234	190/253	96/124
		108/134
		81% (68-84%)	75%	77%
		74% (39-91%)
	Choanal atresia	99/179	95/247	76/124
	Growth retardation	35/94	101/141	80/124
		55% (35-71%)	38%	61%
		37% (13-79%)	72%	65%
	Cleft lip and/or palate	79/163	79/242	22/124
		48% (28-70%)	33%	18% «
	Feeding difficulties	90/110	?	40/47
	necessitating tube feeding	82% (32-93%)		85%
	Facial palsy	80/121	72/187	17/47
		66% (29-85%)	39%	36%
	Anosmia on formal smell	24/30	?	?
	testing	80%
	Genital hypoplasia	118/145	116/187	45/124
		81% (42-90%)	62%	36% «
	Congenital heart defect	191/252	193/250	105/124

# CHD7-positive cohort from the literature as reviewed by Zentner 2010.

[START_REF] Zentner | Molecular and phenotypic aspects of CHD7 mutation in CHARGE syndrome[END_REF]

This cohort partially overlaps with our CHD7-positive cohort because the phenotypes of 64 of our patients were published previously.

[START_REF] Jongmans | CHARGE syndrome: the phenotypic spectrum of mutations in the CHD7 gene[END_REF][START_REF] Bergman | Exon copy number alterations of the CHD7 gene are not a major cause of CHARGE and CHARGE-like syndrome[END_REF][START_REF] Bech | Isolation of the left subclavian artery from the pulmonary artery in a patient with CHARGE association[END_REF][START_REF] Douglas | Extending the phenotypic spectrum of CHARGE syndrome: a case with preaxial polydactyly[END_REF][START_REF] Jongmans | Familial CHARGE syndrome and the CHD7 gene: a recurrent missense mutation, intrafamilial recurrence and variability[END_REF][START_REF] Van De Laar | Limb anomalies in patients with CHARGE syndrome: an expansion of the phenotype[END_REF][START_REF] Pedersen | Molecular diagnosis of CHARGE syndrom[END_REF][START_REF] Writzl | Immunological abnormalities in CHARGE syndrome[END_REF]

* Cohort of patients with clinically diagnosed CHARGE syndrome reported by Tellier in 1998 and Issekutz in 2005, before CHD7 analysis was possible.

[START_REF] Issekutz | An epidemiological analysis of CHARGE syndrome: preliminary results from a Canadian study[END_REF][START_REF] Tellier | CHARGE syndrome: report of 47 cases and review[END_REF]

Exploration of the mild end of the phenotypic spectrum of CHD7 mutations Patients

feature seems to occur less frequently since CHD7 analysis became available: congenital heart defects were present in 76% of CHD7-positive patients and in 85% of patients with a clinical diagnosis of CHARGE syndrome. The most likely explanation is that the clinical diagnosis was more readily made in hospitalised children with a heart defect and that, like children with cleft palate, children without a heart defect were more likely to remain unrecognised as having CHARGE syndrome prior to CHD7 analysis. with a typical presentation of CHARGE syndrome are easily clinically recognised, but those who are mildly affected can be missed, as the mild end of the CHARGE spectrum is only recently starting to emerge. Several studies have shown that an increasing number of patients with a CHD7 mutation do not fulfil the clinical criteria, as they do not have coloboma or choanal atresia or cleft palate.

[START_REF] Jongmans | CHARGE syndrome: the phenotypic spectrum of mutations in the CHD7 gene[END_REF]

Exploration of the mild end of the CHARGE spectrum can be undertaken in four ways: by studying familial CHARGE syndrome, by evaluating very mildly affected patients who are picked up with CHD7 analysis, by performing CHD7 analysis in cohorts of patients with only one CHARGE feature, and finally by studying syndromes that show clinical overlap with CHARGE syndrome (e.g., 22q11 deletion syndrome and Kallmann syndrome).

Table 3

 3 Familial CHARGE syndrome

	Reference		Fulfilling clinical criteria	Segregation
	Sib-pairs	CHD7	Sib 1	Sib 2
		mutation
	1. Wincent	c.4015C>T;	+ (case 11a) + (case 11b)	Father no
	2008[23]	p.R1339X			mutation
	2. Pauli	c.7302dupA	+ (girl)	+ (boy)	Germline
	2009[44]				mosaicism in
					father
	3. Lalani	p.W2332X	+ (died)	-(case	Parents no
	2006[21]			CHA76)	mutation
	4. Jongmans	c.2442+5G>C	-(case 1)	+ (case 2)	Mother no

* Somatic mosaicism was excluded (the CHD7 mutation was present in both peripheral blood lymphocytes and buccal cells).

CHD7 analysis in cohorts of patients with only one CHARGE feature

  and low-set, over-folded ear helices. The patient did not have coloboma, choanal atresia or cleft palate, but did have typical CHARGE ears with triangular conchae. To our knowledge, CHD7 sequence analysis has not yet been performed in a cohort of VCFS patients without deletion or mutation of TBX1. In Figure2we illustrate how difficult it can be to distinguish between CHARGE syndrome and 22q11 deletion syndrome. The phenotypic similarity between VCFS and CHARGE syndrome is also apparent in mice with haploinsufficiency of Tbx1 and Chd7.[START_REF] Randall | Great vessel development requires biallelic expression of Chd7 and Tbx1 in pharyngeal ectoderm in mice[END_REF] Both genes are required in pharyngeal ectoderm for fourth pharyngeal artery development. In addition, both genes are important in development of the thymus and semicircular canals. The Tbx1 and Chd7 gene were shown to interact in mice, but a direct regulatory effect of Chd7 on Tbx1 expression could not been demonstrated.[START_REF] Randall | Great vessel development requires biallelic expression of Chd7 and Tbx1 in pharyngeal ectoderm in mice[END_REF]

	Some authors have undertaken CHD7 screening in patients with only one CHARGE
	syndrome feature, e.g., cleft lip and/or palate,[46] congenital heart disease,[47] or
	scoliosis.[48] These studies did not identify pathogenic CHD7 mutations. The general
	impression is that in the absence of other CHARGE features, the chance of finding a
	CHD7 mutation is very low.
	Velocardiofacial or 22q11 deletion syndrome, shares many features with
	CHARGE syndrome, including congenital heart defects, cleft palate, developmental
	delay, renal anomalies, growth retardation, ear anomalies, hearing loss, hypoglycaemia
	and lymphopenia.[49] Especially thymus aplasia and hypoparathyroidism are
	increasingly recognised in CHARGE syndrome and mark the clinical overlap with the
	DiGeorge phenotype of 22q11 deletions.[50, 51] In approximately 85% of VCFS
	The second patient had mild semicircular canal anomalies and a mild hearing patients, a common 3 Mb heterozygous deletion of 22q11.2 is present, resulting in TBX1
	loss. His external ears were normal. He was only recognised as having CHARGE haploinsufficiency. Mutations in the TBX1 gene are present in a minority of VCFS
	syndrome after a CHD7 splice site mutation was found in his more severely affected patients. Array CGH in a cohort of VCFS patients without 22q11 deletion or TBX1
	children (Table 3, two-generation family from this study). mutation revealed one heterozygous deletion encompassing the CHD7 gene in a patient
	The third patient was diagnosed with Kallmann syndrome and had sensorineural with features typical of VCFS.[52] This patient had a learning difficulty with speech delay,

[START_REF] Blake | CHARGE association: an update and review for the primary pediatrician[END_REF][START_REF] Verloes | Updated diagnostic criteria for CHARGE syndrome: a proposal[END_REF]

Interestingly, 18 out of the 131 (14%) CHD7-positive patients that could be scored for these criteria had only one or two major Blake features and thus could not be clinically diagnosed as having CHARGE syndrome. Based on the presence of none, or only one major Verloes feature, as many as 17% (22/124 patients) could not be clinically diagnosed with CHARGE syndrome using the Verloes criteria. The phenotypes of the three most mildly affected (previously unpublished) patients are presented below.

The first patient had abnormal external ears and a congenital heart defect but no other features of CHARGE syndrome. She had normal semicircular canals, no cranial nerve dysfunction and a normal pubertal development. She had a de novo pathogenic missense mutation in the CHD7 gene that had not been described before (c.4406A>G, p.Tyr1469Cys in exon 19). hearing loss. After a de novo pathogenic missense mutation in the CHD7 gene (c.6322G>A, p.Gly2108Arg in exon 31) was identified, a CT scan of his temporal bone was re-evaluated and semicircular canal hypoplasia was seen. He had normal external ears.

Studies in syndromes that overlap with CHARGE syndrome

Thus far, two clinically overlapping syndromes have been studied in relation to CHD7 mutations: velocardiofacial syndrome (VCFS) and Kallmann syndrome.

severe feeding difficulties, a congenital heart defect (interruption of the aortic arch, coarctation of the aorta, bicuspid aortic valve, ventricular and atrial septal defect), long slender fingers

Table 4

 4 Clinical surveillance of patients with a CHD7 mutation Patients themselves, when fertile with or without appropriate hormone replacement therapy, have a 50% chance of transmitting the CHD7 mutation to their offspring. The severity of CHARGE syndrome in offspring can not be predicted, because intra-familial variability is large. Prenatal diagnosis, either by molecular analysis or ultrasound, and pre-implantation genetic diagnosis, when appropriate, should be discussed with parents and patients.CONCLUSIONSCHARGE syndrome is extremely variable, an observation that has been strongly underscored since the discovery of the CHD7 gene. The phenotype cannot be predicted from the genotype, as exemplified by intra-familial variability. CHARGE syndrome remains primarily a clinical diagnosis, but molecular testing can confirm the diagnosis in mildly affected patients. Guidelines for CHD7 analysis in individuals suspected of having CHARGE syndrome are proposed in Figure3. In addition, updated guidelines for the surveillance of patients with a CHD7 mutation or typical CHARGE syndrome are presented in Table4.

	Evaluation	Tests

Acknowledgements:

We are indebted to the patients for cooperating in this study. We thank the Netherlands Organisation for Health Research (ZonMW 92003460 to JEHB) for financial support and thank Jackie Senior for editing the manuscript.

Competing interests: None declared.

Exclusive licence statement:

The Corresponding Author has the right to grant on behalf of all authors and does grant on behalf of all authors, an exclusive licence (or non-exclusive for government employees) on a worldwide basis to the BMJ Publishing Group Ltd and its Licensees to permit this article (if accepted) to be published in Journal of Medical Genetics and any other BMJPGL products to exploit all subsidiary rights, as Ethics committee approval: not necessary. Clinical information was collected with informed consent and subsequently used strictly anonymously according to local ethical regulations, except for some individual patients who might be identifiable in the paper, but who all gave their consent.

Proposal for lay out of Figure 1