Sylvain Ervedoza
email: sylvain.ervedoza@math.u-bordeaux.fr

Zhiqiang Wang

Jiacheng Zhang

Recovering the velocity in a 1-d non-local transport equation *

Keywords: Transport equation, inverse problem, non-local velocity. AMS subject classifications 35Q49, 35R30, 93B52

In this article, we consider an inverse problem for the non-local system ∂tρ + λ(W (t))∂xρ = 0, in which

)dx is the total mass of the system. We propose an algorithm and derive a formula to reconstruct the velocity function λ(•), assumed to be strictly positive, in an interval [W-, W+] which contains the initial total mass W (0), by suitably choosing the influx condition u(t) = λ(W (t))ρ(0, t) and measuring the outflux y(t) = λ(W (t))ρ(1, t). Some numerical experiments are provided to illustrate the performance of our method.

Introduction 1.Main results

Our work is motivated by problems arising in the control of semiconductor manufacturing systems which are characterized by their highly re-entrant character, see [START_REF] Armbruster | A continuum model for a re-entrant factory[END_REF]. Such model is described by the following partial differential equation, corresponding to a 1-d non-local transport equation:

                  
∂ t ρ + λ(W (t))∂ x ρ = 0, (x, t) ∈ (0, 1) × (0, T ), W (t) = 1 0 ρ(x, t)dx, t ∈ (0, T ), u(t) = λ(W (t))ρ(0, t), t ∈ (0, T ), y(t) = λ(W (t))ρ(1, t), t ∈ (0, T ), ρ(x, 0) = ρ 0 (x), x ∈ (0, 1), (

Here, T > 0, λ : R → (0, ∞) is a C 1 continuous function, ρ 0 is the initial data which belongs to L p (0, 1)(for some p > 1) and ρ(•, •) is the corresponding weak solution of (1.1) which is defined in Section 2. In the manufacturing system the natural control input is the influx u(•), and the output is the outflux y(•).

The well-posedness of the open-loop system (1.1) with known influx u is considered in [START_REF] Coron | Analysis of a conservation law modeling a highly re-entrant manufacturing system[END_REF], see also [START_REF] Shang | Analysis and control of a scalar conservation law modeling a highly re-entrant manufacturing system[END_REF] for a generalization to the case of a velocity depending on x and W (t) of the form λ(x, W (t)), and [START_REF] Sylvain Ervedoza | Semi-global stabilization of a nonlinear transport equation with non-local velocity[END_REF] for global well-posedness for closed loop versions (that is with u given in terms of y) of (1.1).

In our work, we assume that λ is not known (but still satisfies the a priori condition λ : R → (0, ∞) and λ ∈ C 1 (R)), and our goal is to reconstruct it. In order to do that, we assume the following: Assumption 1.1. The initial mass W (0) = W 0 is known. Assumption 1.2. There exists p > 1 such that ρ 0 ∈ L p (0, 1) and there exists a positive constant m such that ρ 0 satisfies ρ 0 (x) m, a.e x ∈ (0, 1).

(1.2) Assumption 1.3. We can measure the output y, and design the input u in terms of y, so that u at time t depends only on y(t).

The main result of this paper is the following:

Theorem 1.1. Assume that λ : R → (0, ∞) is a C 1 (R) continuous function and that Assumptions 1.1-1.2-1.3 hold. Let W 1 > W 0 and α : [0, ∞) → [1, ∞) be a continuous non-decreasing function which is larger than 1 when t > 0.

Let us consider the following algorithm:

Step 1. While t > 0 is such that t 0 y(s) ds < W 0 , we impose u(t) = y(t) (1.3) in the equation (1.1).

We call T 0 the first time t > 0 in which t 0 y(s) ds = W 0 .

Step 2. For t > T 0 , we impose u(t) = α(t -T 0 )y(t), (1.4) in the equation (1.1), that we solve up to the time T 1 defined by T1 T0 (α(s -T 0 ) -1)y(s)ds = W 1 -W 0 .

(1.5)

This algorithm enjoys the following properties:

1. The times T 0 and T 1 are finite.

2. The function t → W (t) is known on the whole time interval [0, T 1 ] and is given by

W (t) = W 0 + t 0 (u(s) -y(s))ds, t ∈ [0, T 1 ]. (1.6)
Besides, for all t ∈ [0, T 1 ], W (t) ∈ [W 0 , W 1 ] and W is surjective on [W 0 , W 1 ]. The function W is constant on [0, T 0 ] and increasing on [T 0 , T 1 ].

3. For every t ∈ [T 0 , T 1 ], there exists a unique

f (t) ∈ [0, t) such that t f (t) y(s)ds = W (f (t)). (1.7)
This function f is C 1 and increasing on [T 0 , T 1 ].

The function

t → λ(W (t)) on [T 0 , T 1 ] satisfies    λ(W 0 ) = 1 T 0 , λ(W (t)) = f (t)λ(W (f (t))), t ∈ [T 0 , T 1 ]. (1.8)
Accordingly, we can recover the velocity λ on the whole interval [W 0 , W 1 ] from the information given by the measurement y on the whole time interval [0, T 1 ].

Remark 1.1. In Assumption 1.2, the condition (1.2) with m > 0 can be replaced by ρ 0 (x) 0, a.e.x ∈ (0, 1), see Section 4.1 by allowing to choose u under the form u(t) = y(t) + v(t) for some time, where v ∈ C 0 [0, ∞) ∩ L 1 (0, ∞) is a decreasing positive function.

Remark 1.2. If one wants to reconstruct the velocity λ in an interval I of the form I = [W -, W + ] with 0 < W -< W 0 < W + , we can put ourselves in the situation of Theorem 1.1 by doing as follows. For k ∈ (0, 1), we impose u(t) = ky(t) for t ∈ (0, T * ) in (1.1), where T * is the first time such that

W 0 + t 0 (u(s) -y(s)) ds = W -. It is easy to check that, if λ : R → (0, ∞) is a C 1 (R) function and Assumptions 1.1-1.2-1.3 hold, then T * is finite, W (T * ) = W -,

Related references

There is a wide range of literature on inverse problems for partial differential equations. We refer for instance to the textbooks [START_REF] Bellassoued | Carleman estimates and applications to inverse problems for hyperbolic systems[END_REF][START_REF] Isakov | Inverse problems for partial differential equations[END_REF][START_REF] Michael | Carleman estimates for coefficient inverse problems and numerical applications[END_REF] for an overview of the domain and presentation of some relevant results for inverse problems for hyperbolic equations, and in particular the wave equation.

In the context of the recovery of the velocity in a transport equation, we refer in particular to [START_REF] Connolly | On some inverse problems for a nonlinear transport equation[END_REF] and [START_REF] Connolly | Reconstruction of a nonlinear source term in a semi-linear wave equation[END_REF]. In [START_REF] Connolly | On some inverse problems for a nonlinear transport equation[END_REF], the authors consider the inverse problems of determining the velocity function c in the quasi-linear transport equation

u t + c(u)u x = 0. (1.9)
Under some assumptions on the monotonicity of the initial and boundary data, and the knowledge of the output boundary conditions, the inverse problem is proved to have a unique solution, which is well-posed or ill-posed depending on which side the additional measurement is performed. We also point out the discussion on the determination of source terms in quasilinear transport equations in [START_REF] Connolly | On some inverse problems for a nonlinear transport equation[END_REF][START_REF] Connolly | Reconstruction of a nonlinear source term in a semi-linear wave equation[END_REF].

We are not aware of many results in the same spirit, and we believe that Theorem 1.1 is the first one to address the recovery of the velocity in a non-local transport equation of the form (1.1).

In fact, so far, equation (1.1) has been mainly discussed from the controllability and stabilization points of view. Regarding controllability properties, we refer in particular to [START_REF] Coron | Analysis of a conservation law modeling a highly re-entrant manufacturing system[END_REF][START_REF] Shang | Analysis and control of a scalar conservation law modeling a highly re-entrant manufacturing system[END_REF] and in the context of networks, to [START_REF] Gugat | Analysis of a system of nonlocal conservation laws for multi-commodity flow on networks[END_REF].

The output feedback stabilization (i.e. with a closed loop control) for the system (1.1) under a suitable feedback law is another natural and interesting control problem, which has been studied in [START_REF] Coron | Output feedback stabilization for a scalar conservation law with a nonlocal velocity[END_REF], where it has been proved that if ρ is the target constant states around which we want to stabilize, for k ∈ (-1, 1), the proportional controller

u(t) -ρλ(ρ) = k(y(t) -ρλ(ρ)), t ∈ (0, ∞), (1.10)
stabilizes the closed-loop system (1.1) if and only if

d := ρλ (ρ) λ(ρ) satisfies d > -1. (1.11)
We also refer to [START_REF] Coron | Output feedback stabilization for a scalar conservation law with a nonlocal velocity[END_REF] for the spectral analysis of the linearized model, to [START_REF] Chen | Global feedback stabilization for a class of nonlocal transport equations: the continuous and discrete case[END_REF] for the global exponential stabilization result in L 2 (0, 1) for some classes of velocity functions, or to [START_REF] Sylvain Ervedoza | Semi-global stabilization of a nonlinear transport equation with non-local velocity[END_REF] for a time-varying feedback control to achieve a semi-global stabilization for (1.1) under the assumption that s → sλ(s) is injective (which, locally around ρ, only means d = -1).

Outline. The rest of the paper is organized as follows. In Section 2, we recall the well-posedness results for the problem (1.1). The proof of Theorem 1.1 is given in Section 3. Some extensions of Theorem 1.1 are discussed in Section 4. Finally in Section 5, we give some numerical experiments to illustrate the performances of the algorithm proposed in Theorem 1.1.

Preliminaries

Before considering the reconstructing problem, we need to recall the usual definition of a weak solution for (1.1) and give the corresponding well-posedness result for our closed-loop system. Definition 2.1. Let p > 1 and ρ 0 be a non-negative function in L p (0, 1). Let T > 0, α be a positive function in L ∞ (0, T ) and v be a non-negative function in L p (0, T ). A weak solution of the Cauchy problem (1.1) with boundary condition

u(t) = α(t)y(t) + v(t), t ∈ (0, T ) (2.1)
is a function ρ ∈ C 0 ([0, T ]; L p (0, 1)) such that for every s ∈ (0, T ] and every

ϕ ∈ C 1 ([0, 1] × [0, s]) satisfying ϕ(x, s) = 0, ∀x ∈ [0, 1],
one has:

s 0 1 0 ρ(x, t)(∂ t ϕ(x, t) + λ(W (t))∂ x ϕ(x, t)) dxdt + s 0 (α(t)y(t) + v(t))ϕ(0, t) dt - s 0 y(t)ϕ(1, t) dt + 1 0 ρ 0 (x)ϕ(x, 0) dx = 0. (2.2)
Applying a fixed point argument as in [8, Theorem 2.1], we can get the following well-posedness result:

Theorem 2.1. Assume that λ : R → R * + is a C 1 function. Let p > 1 and ρ 0 be a non-negative function in L p (0, 1). Let α be a positive function L ∞ loc (0, ∞) and v be a non-negative function in L p loc (0, ∞). There exists a unique time T * and a unique weak solution ρ ∈ C 0 ([0, T * ); L p (0, 1)) of system (1.1)-(2.1). Moreover, we have

W ∈ W 1,p loc ([0, T * )), y ∈ L p loc (0, T * ). (2.3)
In this case, T * is finite if and only if lim T →T * ρ(•, t) L p (0,1) = ∞.

Remark 2.1. In our setting, the initial data ρ 0 and the boundary data v are non-negative. Consequently, the solution ρ is non-negative in (0, L) × (0, T * ) and u, y are non-negative in (0, T * ).

Remark 2.2. In the following, we will check that the maps t → ρ(•, t) L p (0,1) and t → W (t) do not blow up in the reconstruction process of Theorem 1.1, which, according to Theorem 2.1, entails that the solution of (1.1) complemented with (1.3) on (0, T 0 ) and (1.4) on (T 0 , T 1 ) exist up to T 1 .

Remark 2.3. For later use, let us also point out that, for solutions ρ of (1.1)-(2.1), integrating in x the equation (1.1) (1) , we easily get

dW (t) dt = u(t) -y(t), in D (0, T ). (2.4)
3 Proof of Theorem 1.1

In this section, we give the proof of Theorem 1.1.

The time interval [0, T 0 ). We first focus on the time interval [0, T 0 ). During that time interval, the equations (1.1) are completed with the boundary conditions (1.3), so that from (2.4),

∀t ∈ [0, T 0 ), W (t) = W 0 ,
and the velocity λ(W (t)) is constant equal to λ(W 0 ) for all t ∈ [0, T 0 ).

Note that we also have that the norm ρ(•, t) L p (0,1) is constant for t ∈ [0, T 0 ) since the boundary conditions (1.3) amounts to solve a transport equation at constant speed on a torus, so that the solution is well-defined up to the time T 0 according to Theorem 2.1. In fact, for this argument to be made rigorous, we should set T0 as the supremum of the times τ > 0 such that sup t∈[0,τ ] ρ(•, t) L p (0,1) < ∞ and τ 0 y(s) ds W 0 , and check that this supremum T0 necessarily satisfies T0 = ∞ or T0 < ∞ and T0 0 y(s) ds = W 0 . We will check next that T0 is necessarily finite, so that it coincides with the time T 0 given in Theorem 1.1.

Using the characteristic method and Assumption 1.2, we easily get that ρ is bounded from below by m for all (x, t) ∈ [0, 1] × [0, T0 ), and thus that

∀t ∈ [0, T0 ), y(t) = λ(W (t))ρ(1, t) = λ(W 0 )ρ(1, t) mλ(W 0 ) > 0.
Accordingly, the map t → t 0 y(s) ds is strictly increasing and is bounded from below by t → mλ(W 0 )t. This implies that T0 = T 0 is finite and T0 0 y(s) ds = W 0 .

(3.1)

The time interval [T 0 , T 1 ). We start by noticing that ρ(•, T 0 ) is still bounded from below by m and belongs to L p (0, 1) with the same p as in Assumption 1.2.

We then introduce T1 as the supremum of the times τ > T 0 such that sup t∈[T0,τ ] ρ(•, t) L p (0,1) < ∞ and τ T0 (α(s -T 0 ) -1)y(s) ds W 1 -W 0 , and check that this supremum T1 necessarily satisfies T1 = ∞ or T1 < ∞ and

T1 T0 (α(s -T 0 ) -1)y(s) ds = W 1 -W 0 .
First, it is easy to check that for t ∈ (T 0 , T1 ), ρ(•, t) stays bounded from below by m, y stays positive and, thus, according to (2.4), W is increasing. Together with (2.4), the choice (1.5) imposes that W is necessarily smaller than W 1 on the time interval [T 0 , T1 ). Therefore, for all t ∈ [T 0 , T1 ),

W (t) ∈ [W 0 , W 1 ].
Since the velocity λ is bounded from below by inf s∈[W0,W1] {λ(s)} and from above by sup s∈[W0,W1] {λ(s)} on the time interval [T 0 , T1 ), the map t → ρ(•, t) L p (0,1) is bounded in any time interval of the form [T 0 , τ ] for τ finite and smaller than T1 , thus guaranteeing that T1 = ∞ or T1 < ∞ and can be alternatively defined by (1.5).

But, since ρ(•, t) stays bounded from below by m and W stays in the interval

[W 0 , W 1 ], ∀t ∈ [T 0 , T1 ), y(t) m inf s∈[W0,W1] {λ(s)} > 0. (3.2)
Accordingly, t → t T0 (α(s -T 0 ) -1)y(s) ds is strictly increasing for t ∈ [T 0 , T1 ] and, for all > 0, for all

t ∈ [T 0 + , T1 ), t T0 (α(s -T 0 ) -1)y(s) ds (α( ) -1)(t -(T 0 + ))m inf s∈[W0,W1]
{λ(s)}.

Since α( ) > 1, we conclude that necessarily T1 = T 1 is finite and can be alternatively defined by (1.5).

Properties of W . Note that, in the two above paragraphs, we also proved that W is non-decreasing, that it is surjective on [W 0 , W 1 ] and takes value in [W 0 , W 1 ] on [0, T 1 ]. The formula (1.6) directly comes from (2.4). The fact that then W is known at all times then comes from the fact that u is explicitly given in terms of y in the time interval [T 0 , T 1 ]. In particular, we have

W (t) =    W 0 , for t ∈ [0, T 0 ], W 0 + t T0 (α(s -T 0 ) -1)y(s) ds, for t ∈ [T 0 , T 1 ]. (3.3)
In fact, using (3.2), we also have that W is strictly increasing in [T 0 , T 1 ].

Existence and uniqueness of f (t) for t ∈ [T 0 , T 1 ] satisfying (1.7). We introduce the function

F (t, τ ) = t τ y(s) ds -W (τ ), (t, τ ) ∈ [0, T 1 ] 2 . (3.4)
Note that W is non-decreasing on [0, T 1 ], always larger than W 0 , and y is positive on [0, T 1 ]. Recalling the property (3.1), we easily get that for t ∈ [T 0 , T 1 ], F (t, 0) 0 > F (t, t). Also, since W ∈ W 1,p (0, T 1 ) (recall Theorem 2.1) and is non-decreasing on [0, T 1 ], and y is strictly positive on [0,

T 1 ], for t ∈ [0, T 1 ], τ → F (t, τ )
is continuous and strictly decreasing with respect to τ ∈ [0, t]. By continuity and strict monotonicity, we conclude that for any fixed t ∈ [T 0 , T 1 ], there exists a unique f (t) ∈ [0, t) such that F (t, f (t)) = 0, i.e., (1.7) holds.

Now we investigate the continuity and regularity of the function f on [T 0 , T 1 ]. In order to do that, we rewrite F as

F (t, τ ) = h(t) -g(τ ), with h(t) = t 0 y(s) ds and g(τ ) = τ 0 y(s) ds + W (τ ).
(3.5)

The above arguments show that g is continuous on [0, T 1 ], and strictly increasing. Consequently, g has a continuous strictly increasing inverse g -1 defined on [g(0), g(T

1 )]. Noting that h(t) ∈ [g(0), g(T 1 )], t ∈ [T 0 , T 1 ], f is simply given by f (t) = g -1 (h(t)), t ∈ [T 0 , T 1 ]. Since h ∈ W 1,p (0, T ), it is continuous, so that f is continuous on [T 0 , T 1 ]. Besides, since h is strictly increasing on [0, T 1 ] , f is also strictly increasing on [T 0 , T 1 ].
Differentiability of f . To get that f is in fact differentiable, it is convenient to characterize f as follows: for all t ∈ [T 0 , T 1 ], f (t) coincides with f (t) defined as the unique element of [0,

T 1 ] such that t f (t) λ(W (s)) ds = 1.
(3.6) Indeed, since s → λ(W (s)) is continuous and strictly positive, for all t ∈ [T 0 , T 1 ], there exists at most one element f (t) such that (3.6) holds. Now, by the characteristic method, for t

∈ [T 0 , T 1 ], setting t ∈ [0, T 1 ] such that t f (t) λ(W (s)) ds = 1, we have, if t > t, i.e. t f (t) λ(W (s)) ds < 1, W (f (t)) = t f (t) y(s) ds = t f (t) λ(W (s))ρ(1, s) ds = t f (t) λ(W (s))ρ 1 - s f (t) λ(W (θ))dθ, f (t) ds = 1 1-t f (t) λ(W (θ))dθ ρ(x, f (t)) dx,
and, with similar computations, if t < t, i.e.

t f (t) λ(W (s)) ds > 1, W (f (t)) = t f (t) y(s) ds = t f (t) λ(W (s))ρ(1, s)ds + t t λ(W (s))ρ(1, s) ds = 1 0 ρ(x, f (t)) dx + t t λ(W (s))ρ(1, s) ds. Since, for t ∈ [T 0 , T 1 ], W (f (t)) = 1 0 ρ(x, f (t)) dx,
by definition of W , and ρ and λ(W ) are strictly positive, we necessarily have that t = t, that is, from the definition of t,

t f (t) λ(W (s)) ds = 1.
Therefore, for all t ∈ [T 0 , T 1 ], we have f (t) = f (t), where f (t) is defined by (3.6).

It follows that, equivalently to (1.7), f can be defined by the implicit relation

k(t, f (t)) = 0, where k(t, τ ) = t τ λ(W (s)) ds -1, for t ∈ [T 0 , T 1 ] and τ ∈ [0, T 1 ]. (3.7) Since W is continuous on [0, T 1 ], k is C 1 on [T 0 , T 1 ] × [0, T 1 ]. Since we also have that ∂ τ k(t, τ ) = -λ(W (τ )) does not vanish for (t, τ ) ∈ [T 0 , T 1 ] × [0, T 1 ], f is thus C 1 on [T 0 , T 1 ]
by the implicit function theorem and satisfies the equation

λ(W (t)) -λ(W (f (t)))f (t) = 0, for t ∈ [T 0 , T 1 ].
This proves items 3 and 4 of Theorem 1.1, except for the relation λ(W 0 ) = 1/T 0 that we now prove. Indeed, we easily check that f (T 0 ) = 0, since W 0 satisfies (3.1). Accordingly, we have, from (3.7), that

T0 0 λ(W (s)) ds = 1.
But, for all s ∈ [0, T 0 ], W (s) = W 0 and λ(W (s)) = λ(W 0 ). Therefore, we obtain from the previous relation that T 0 λ(W 0 ) = 1, as announced. This concludes the proof of Theorem 1.1.

4 Extensions of Theorem 1.1

In this section, we give two extensions of Theorem 1.1. The first one provides a relaxation of the condition (1.2) in Assumption 1.2. The second one discusses the choice of the function α in the algorithm of Theorem 1.1.

Relaxation of Assumption 1.2

In Assumption 1.2, the condition ρ 0 (x) m > 0, a.e. x ∈ (0, 1) can be replaced by ρ 0 (x) 0, a.e. x ∈ (0, 1) according to the following proposition.

Proposition 4.1. Assume p > 1, λ : R → (0, ∞) is a C 1 (R) function and that the initial data ρ 0 ∈ L p (0, 1) is non-negative. Let u(t) = y(t) + v(t), t ∈ (0, ∞), (4.1)
where v is a positive function in C 0 ([0, ∞)) ∩ L 1 (0, ∞). Then define T * by

T * = sup τ 0 such that τ 0 y(s)ds = W 0 (4.2)
where y is the outflux corresponding to the closed-loop system (1.1)-(4.1). Moreover, for any T > T * , there exists a positive constant m T depending on T such that ρ(x, T ) m T , a.e x ∈ (0, 1), and

W (T ) = W 0 + T 0 v(t) dt. Remark 4.1. A simple example of function v in C 0 ([0, ∞))∩L 1 (0, ∞) satisfying the assumption of Proposition 4.1 is given by v(t) = 1 1+t 2 for t ∈ [0, ∞).
Proof. Below, we will not discuss the fact that the solution of (1.1)-(4.1) is well-defined for all times. The proof of this result can be done easily, using Theorem 2.1 and proving that the L p norm of the solution cannot blow up in finite time, and its proof is left to the reader. The fact that the solution ρ stays non-negative everywhere is also important and is easy to check.

Firstly, we prove that T * given by (4.2) is finite. Integrating (2.4) and using (4.1) yield that

W (t) = W 0 + t 0 v(s)ds, t ∈ [0, ∞).
Thus, for all t 0, 0 For any T > T * , for almost every x ∈ (0, 1), the characteristic curve passing through the point (x, T ) and going backwards in time intersects the left boundary x = 0 at a positive time τ (x, T ) > 0 (characterized by

W (t) M W 0 + v L 1 (0,∞) , t ∈ [0, ∞). ( 4
T τ (x,T ) λ(W (s))ds = x). Additionally, ρ(x, T ) = ρ(0, τ (x, T )).
By the choice of u in (4.1) and (4.3), and since ρ is non-negative everywhere, we deduce that

ρ(x, T ) = ρ(0, τ (x, T )) = u(τ (x, T ))) λ(W (τ (x, T )))) inf s∈[τ (1,T ),T ] {v(s)} sup s∈[0,M ] {λ(s)} > 0.
This also implies that for all t > T * , y(t) > 0, and thus, for all T > T * , T 0 y(s) ds > W 0 .

Accordingly, T * coincides with T * given by (4.2). This concludes the proof of Proposition 4.1.

Discussion on the choice of the function α in the reconstruction algorithm

The algorithm proposes in Theorem 1.1 provides a formula (1.8) to recover the velocity λ, which requires, to obtain the value of λ at W (t), to know the value of λ at an early time W (f (t)), where f (t) is computed by (1.7).

The discussion below aims at indicating how the choice of the function α in Theorem 1.

1 can guarantee that ∀t ∈ [T 0 , T 1 ], f (t) ∈ [0, T 0 ]. (4.6)
Indeed, in such case, for all t ∈ [T 0 , T 1 ], W (f (t)) is simply given by W 0 and the computations of f (t) in (1.7) and of λ(W (t)) in (1.8) become easier; at least from the numerical point, they will generate less numerical errors (see Section 5 for some examples). Discussion 1. The case of a constant α. In this case, we get the following result: Accordingly, condition (4.6) is equivalent to the condition (4.7). Discussion 2. When α is not constant. We find the following sufficient condition to (4.6). Proposition 4.3. Under the assumptions of Theorem (1.1), with α : [0, ∞) → [1, ∞) being a continuous non-decreasing function which is strictly larger than 1 when t > 0.

If the function α is such that there exists τ > 0 such that

T0+τ T0 (α(s -T 0 ) -1)y(s)ds W 1 -W 0 , (4.9
)

and τ y L ∞ (T0,T0+τ ) + 1 α(τ ) -1 (W 1 -W 0 ) W 0 , (4.10)
then condition (4.6) holds.

Proof. First, since (α(s -T 0 ) -1)y(s) > 0 for all s > T 0 , it follows from (4.9) and (1.5), that

τ T 1 -T 0 .
Moreover, by the monotonicity of α and the definition of T 1 in (1.5), we have Since y is positive, we thus necessarily have f (T 1 ) T 0 , which entails (4.6) since f is non-decreasing. This concludes the proof of Proposition 4.3.

W 1 -W 0 = T1 T0 (α(s -T 0 ) -1)
In practice, it is easy to construct α satisfying the conditions of Proposition 4.3 if we additionally have rough estimates on ρ 0 L ∞ (0,1) and λ C 0 ([W0,W1]) of the form ρ 0 L ∞ (0,1) M 0 and λ C 0 ([W0,W1]) L. Indeed, we can choose τ > 0 sufficiently small such that

τ M 0 L W 0 2 and τ L 1. (4.11)
The second condition guarantees, that for t ∈ [T 0 , T 0 + τ ], the characteristic going backwards in time and starting from (1, t) reaches x = 0 for a time t T 0 , and then, since u(s) = y(s) for s ∈ [0, T 0 ], we easily get

ρ(1, t) M 0 , t ∈ [T 0 , T 0 + τ ].
In particular, we get

τ y L ∞ (T0,T0+τ ) W 0 2 .
Choose then the value of α(τ ) such that

2W 1 W 0 -1 α(τ ) W 1 -W 0 τ M 0 L + 1,
Then (4.10) and (4.9) are both satisfied. Note that such choice is always possible due to the first condition in (4.11).

Here is an example of a continuous non-decreasing function α strictly larger than 1:

α c,τ (t) = 1 + (c -1) t τ , for t > 0, with c ∈ 2W 1 W 0 -1, W 1 -W 0 τ M 0 L + 1 .
It is then easy to check that for

β 2L(W 1 -W 0 ) W 0 min{1, W 0 /(2M 0 )} ,
the function α β given by α β (t) = 1 + βt, for t > 0, (4.12)

coincides with α c,τ for c = 2W 1 /W 0 -1 and τ = 2(W 1 -W 0 )/(W 0 β) satisfying (4.11).
Of course, this suggests that, when we do not know any a priori guess on ρ 0 L ∞ (0,1) and λ C 0 ([W0,W1]) , we might simply consider choosing α = α β in (4.12) for sufficiently large β.

Numerics

Numerical implementations

The numerical scheme is implemented as follows.

For N ∈ N * , the space interval (0, 1) is divided into N parts, corresponding to a mesh size ∆x = 1/N . The corresponding time discretization parameter is ∆t = ∆x/λ M , where λ M max{λ(s), s ∈ R}, or at least λ M max{λ(s), s ∈ [W 0 , W 1 ]}, in which [W 0 , W 1 ] is the interval in which we want to reconstruct λ (note that this requires an a priori information on the L ∞ bound of λ on the interval of interest).

In the following, we assume that the initial condition ρ 0 satisfies the assumption 1.2 and is continuous, so that we can approximate it by a discrete function in a reasonable sense, and that

W 0 = ∆x N k=1 ρ 0 (k∆x),
and we set ρ 0 k = ρ 0 (k∆x) for all k ∈ {0, • • • , N }. The function ρ is then approximated by a sequence of vectors (

ρ n ) n∈N = (ρ n 0 , • • • , ρ n N )
n∈N , in the sense that ρ n m denotes the discrete approximation of the value of ρ at (t n , x m ) where t n = n∆t and x m = m∆x. Similarly, (W n ) n∈N is a sequence of real number approximating W at time t n and given by

W n = ∆x N k=1 ρ n k , (n ∈ N).
Corresponding to the algorithm presented in Theorem 1.1, we also introduce a continuous non-decreasing function α : (0, ∞) → (1, ∞).

We then use the following explicit solver, given for n ∈ N by

                         ρ n+1 m = ρ n m - ∆tλ(W n ) ∆x (ρ n m -ρ n m-1 ), m = 1, • • • , N, ρ n+1 0 = ρ n+1 N , if t n+1 T 0 , α(t n+1 -T 0 )ρ n+1 N if t n+1 > T 0 , ρ 0 m = ρ 0 (m∆x), for m = 0, • • • , N, with W n = ∆x N k=1 ρ n k , (5.1)
where T 0 is given by

T 0 = t n0 + W 0 -∆t n0-1 n=0 y(t n ) y(t n0 ) ,
and where t n0 is such that

∆t n0-1 n=0 y(t n ) < W 0 ∆t n0 n=0 y(t n ).
Note that this amounts to approximate the observation y by the step function which is equal to λ(W

(t n ))ρ n N +1 on [t n , t n+1 ).
We stop the algorithm (5.1) when

W n > W 1 ,
which guarantees that t n > T 1 in some sense (recall that, in the continuous setting, W (T 1 ) = W 1 and that W is strictly increasing on [T 0 , T 1 ]). Note that, since ρ 0 satisfies Assumption 1.2 and is continuous, it is easy to check that (ρ n m ) n∈N,m∈{0,••• ,N } solving (5.1) stays positive. This is crucial to guarantee the fact that T 0 and T 1 are finite, and the existence and uniqueness of the various quantities defined below and obtained from the direct simulation of (5.1) (which can be proved in the same lines as the proof of Theorem 1.1).

From now on, we assume that we have computed the solution (5.1) and that only (y(t n )) n∈N and W 0 are now known, where y(t n ) = λ(W n )ρ n m . Note that W n can be computed immediately by the formula

W n =      W 0 if t n < T 0 , W 0 + ∆t n-1 k=n0 (α(t k -T 0 ) -1)y(t k ) if t n T 0 .
and is non-decreasing.

In the following, since the natural class for y is rather L p (0, T 1 ) and for W is rather W 1,p (0, T ), we do our computations by thinking that y is a piecewise constant function taking value y(t k ) on the interval [t k , t k+1 ), and W as a continuous piecewise linear function which coincides with

t → W k + (t -t k )(W k+1 -W k )/∆t on the interval [t k , t k+1 ].
For any mesh point t k ∈ [T 0 , T 1 ], we then define f (t k ) as follows. We introduce n k as the unique positive integer satisfying

W n k ∆t k n=n k y(t n ),
and ∆t

k n=n k +1 y(t n ) W n k +1 ,
where the uniqueness is deduced from the positivity of y and the monotonicity of W , and we then set f (t k ) as the unique solution of

W n k + (f (t k ) -t n k ) W n k +1 -W n k ∆t = ∆t k n=n k y(t n ) -(f (t k ) -t n k )y(t n k ).
This yields

f (t k ) = t n k + ∆t ∆t t k n=n k y(t n ) -W n k (∆t)y(t n k ) + W n k +1 -W n k . (5.2)
Finally, considering the approximation of λ(W ) by a linear piecewise continuous function, we approximate λ(W (f (t k ))) as follows

λ(W (f (t k ))) := λ(W n k ) + (f (t k ) -t n k ) λ(W n k +1 ) -λ(W n k ) ∆t . (5.3)
Formula (1.8) is then approximated as follows: For any mesh point

t k ∈ [T 0 , T 1 ], λ(W k ) =              λ(W 0 ) if t k = T 0 , λ(W 0 ) f (t k ) -0 t k -T 0 if t k-1 T 0 < t k , λ(W (f (t k ))) f (t k ) -f (t k-1 ) ∆t if t k-1 > T 0 .
(5.4)

Numerical tests

In this part, we will use Matlab to test numerically the algorithm of Theorem 1.1 for several choices of λ.

We take the initial value ρ 0 given by ρ 0 (x) = sin(πx) + 1, whose corresponding total mass is W (0) = 2 π + 1 ≈ 1.6366. We choose W 1 = 2W 0 , which means our reconstructing interval is [W (0), 2W (0)], i.e about [1.6366, 3.2732]. We will consider the following choices of λ: (5.5)

λ 1 (s) = 3 + 2s,
Note that λ 1 , λ 2 belong to C 1 and satisfy the assumptions of Theorem 1.1, while λ 3 and λ 4 do not: indeed, λ 3 presents strong oscillations and λ 4 presents discontinuities.

Example 1:

Choosing a constant α in the algorithm of Theorem 1.1

In this part, we choose a constant α in the algorithm in Theorem 1.1: see Figure 1 in the case α = 3. For λ 1 , λ 2 and λ 4 , the red curve and the blue curve almost overlap except for a small error close to W 0 , probably due to some instability of the formula (5.4) close to T 0 .

For λ 3 , the algorithm does not manage to catch the strong oscillations near s = 2, which results in a significant error close to s = 2. It seems a bit surprising however, that the algorithm seems to be able to catch rather correctly the velocity after this singularity.

The case of λ 4 also performs well, even if λ 4 presents discontinuities which are in principle not compatible with the regularity properties needed for the formula (1.8) to make sense. This is probably due to the fact, that since λ 4 is piecewise smooth, the singularities on f (t) should be sparse, and the formula (1.8) makes sense almost everywhere.

We also consider the choice α = 2 in the algorithm of Theorem 1.1, see Figure 2 for the recovery of λ 2 . Note that α = 2 is the limiting case in the formula (4.7). Figure 2 exhibits a large error in the reconstruction of λ 2 near W 1 . A simple explanation could be that the computation of λ 2 at W 1 is done by considering the characteristic of the equation (1.1) started from (x, t) = (0, T 0 ), which is precisely a point in which the boundary conditions is multiplied by the factor α, and is thus discontinuous. Accordingly, the numerical scheme is very likely quite imprecise there, thus yielding this large error in the reconstruction process. In this section, we choose a linear function α α(t) = 1 + βt, t 0, (

where β is a positive constant. Figures 3 and4 show the numerical reconstructions corresponding to the choice α as in (5.6) for the choices β = 1 and β = 200 in the algorithm of Theorem 1.1, which yields rather similar results as the ones plotted in Figure 1. When α(t) = 1 + t, we see some kind of regular fluctuations in the plots (for instance on Top Right, in Figure 3). These are due to the error transported by the characteristic line starting from (x, t) = (0, T 0 ), similar to what we saw in the previous section in Figure 2.

As plotted in Figure 4, choosing β larger in (5.6), for instance β = 200, this shortcoming can be corrected. This is of course related to the discussion in Section 4.2.

Convergence rates

To conclude our study, we also provide a rough analysis on the convergence of the reconstructed velocities in the aforementioned cases. Firstly, we define the discrete L 2 error between the true velocity λ and the reconstructed one λ by the formula λ -λ 2 (W0,W1) = i2 i=i1 (λ(W i ) -λ(W i )| t=ti ) 2 (W (t i+1 ) -W (t i ))

(5.7)

where i 1 ∈ N and i 2 ∈ N are the indices such that {t i } i∈{i1,••• ,i2} are all the mesh points in [T 0 , T 1 ]. Taking λ 2 in (5.5) for example, we can compute the error for different N , which are given in Table 1. Figure 5 displays the plot of the convergence rates of the algorithm in logarithmic scales, underlining the fact that all choices of α have similar convergence rates, all of them being roughly of order 1.

  and assumptions 1.1-1.2-1.3 still hold at time T * . Therefore, shifting t = T * to t = 0, we can apply Theorem 1.1 to reconstruct λ on [W -, W + ].

. 3 )

 3 Then λ(W (t)) is bounded below by inf s∈[0,M ] {λ(s)} > 0. Therefore the characteristic curve starting from the point (0, 0) reaches the right boundary x = 1 in finite time T * , i.e.,

Proposition 4 . 2 .

 42 Under the assumptions of Theorem 1.1. If α is a constant function, condition (4.6) holds if and only if Recall from Theorem 1.1 that f is increasing on [T 0 , T 1 ]. Therefore, condition (4.6) holds if and only if f (T 1 ) T 0 . (4.8) Recalling the definition (1.5) of T 1 , we get T1 T0 y(s)ds = W 1 -W 0 α -1 On the other hand, since W is constant equal to W 0 on [0, T 0 ] and strictly increasing on [T 0 , T 1 ], y is strictly positive, and T1 f (T1) y(s)ds = W (f (T 1 )), the condition (4.8) holds if and only if T1 T0 y(s)ds W 0 .

  ds -τ y L ∞ (T0,T0+τ ) . Therefore, if (4.10) holds, we have T1 T0 y(s)ds W 0 , Since for all t ∈ [0, T 1 ], W (t) ∈ [W 0 , W 1 ] by Theorem 1.1, we necessarily have T1 f (T1) y(s)ds = W (f (T 1 )) W 0 T1 T0 y(s)ds.

λ 2 (

 2 s) = 2 + sin(10s),λ 3 (s) = 2 + sin 10 2 -s (2 -s), λ 4 (s) = ([s] mod 2) + 2.

Figure 1 :

 1 Figure 1: Plots of the velocities reconstructed using the Algorithm presented in Theorem 1.1 discretized as explained in Section 5.1, with the choice α = 3, for the velocities λ in (5.5). Top left corresponds to λ 1 , top right to λ 2 , bottom left to λ 3 and bottom right to λ 4 . The reconstructed velocity is plotted in blue, and the true velocity in red.

Figure 2 :

 2 Figure 2: Plots of the velocities reconstructed using the Algorithm presented in Theorem 1.1 discretized as explained in Section 5.1, with the choice α = 2, for the velocity λ 2 in (5.5). The reconstructed velocity is plotted in blue, and the true velocity in red.

5. 2 . 2

 22 Example 2: Choosing a linear α in the algorithm of Theorem 1.1

Figure 3 :

 3 Figure 3: Plots of the velocities reconstructed using the Algorithm presented in Theorem 1.1 discretized as explained in Section 5.1, with the choice α(t) = 1 + t, for the velocities λ in (5.5). Top left corresponds to λ 1 , top right to λ 2 , bottom left to λ 3 and bottom right to λ 4 . The reconstructed velocity is plotted in blue, and the true velocity in red.

Figure 4 :

 4 Figure 4: Plots of the velocities reconstructed using the Algorithm presented in Theorem 1.1 discretized as explained in Section 5.1, with the choice α(t) = 1 + 200t, for the velocities λ in (5.5). Top left corresponds to λ 1 , top right to λ 2 , bottom left to λ 3 and bottom right to λ 4 . The reconstructed velocity is plotted in blue, and the true velocity in red.

  0.1835 0.0753 0.0383 0.0195 0.0081 0.0042 error of α = 1 + t 0.2201 0.1143 0.0482 0.0252 0.0133 0.0059 0.0032 error of α = 1 + 200t 1.2027 0.6824 0.2847 0.1435 0.0721 0.0288 0.0144 Table 1: Tables of the errors λ -λ 2 (W0,W1) with different N for the choices α(t) = 3, α(t) = 1 + t, and α(t) = 1 + 200t, in the case λ = λ 2 in (5.5).

Figure 5 :

 5 Figure 5: Plot of the errors λ -λ 2 (W0,W1) versus N in logarithmic scales for the choices α(t) = 3 (left, the linear regression has slope=-0.9663), α(t) = 1 + t (middle, the linear regression has slope=-0.9199), and α(t) = 1 + 200t (right, the linear regression has slope=-0.9695), in the case λ = λ 2 in (5.5).

* S.E. is partially supported by the Agence Nationale de la Recherche, Project TRECOS, under grant ANR-20-CE40-0009. Z.W. is partially supported by the Natural Sciences Foundation of China, under grant 11971119. J.Z. is supported by China Scholarship Council, under grant CSC-202006100152.


Jun-Ichi Yano

The Pressure Problem of the Mass-Flux Convection Parameterization

The Pressure Problem of the Mass-Flux Convection Parameterization

Jun-ichi

Introduction

Standard Mass-Flux Parameterization Formulation

A full set of equations describing a system of mass-flux convection parameterization may be given by: Here, w and u are the vertical and the radial components of the velocity with the two directions defined by the coordinates, z and r, respectively. An axisymmetry of the system is assumed. The other variables are: ρ = ρ(z) the references density, σ the fractional area occupied by convection, M = σρw c , The convective mass flux, b the buoyancy, p the pressuyre. The subscripts, c and e, are added whenever necessary to distinguish between the convective and the environmental components. Entrainment, E, and detrainment, D, may furthermore be defined in terms of the fractional rates, ε and δ, as:

E = εM, D = δM (2.2) 1
Eqs. (2.1a, c) are derived from Eq. (6.11) from Eq. (6.11) of [START_REF] Yano | Formulation structure of the mass-flux convection parameterization[END_REF] by setting the forcing F = ρb -∂p/∂z and F = 0, Eq. (2.1b) corresponds to Eq. (6.13) of [START_REF] Yano | Formulation structure of the mass-flux convection parameterization[END_REF].

Eq. (2.1a) can more directly be compared with Eq. (2.2) of [START_REF] Yano | Convective kinetic energy equation under the mass-flux subgrid-scale parameterization[END_REF]. Eq. (2.1d) is a standard form of a momentum equation, neglecting possible contributions of turbulent eddies.

The entraining-plume system considered in a companion paper is recovered by setting:

ρ = 1, σ = R 2 , M = R 2 w c , E = R 2 w d , D = 0, b e = w e = 0.
An equation to be solved for the pressure can be derived by constraining the momentum equation by the mass continuity. For this purpose as the latter, it is more convenient to adopt the form:

1 ρ ∂ ∂r ru + 1 ρ ∂ρw c ∂z = 0. (2.3)
Furthermore, we re-write the vertical momentum equation (2.1a) into an davective form:

∂w c ∂t + w c ∂w c ∂z = b c - 1 ρ ∂p ∂z + E ρσ (w e -w c ) (2.4)
with the help of the mass continuity (2.1b).

A (2.6a)

where ∂u ∂r = - 1 2 1 ρ ∂ρw c ∂z (2.6b)
is a constant of r. This fact turns out to be useful for further deduction of Eq. (2.5) in the following.

Pressure problem (1): Without r-Dependence

The pressure equation (2.5) may be further re-written as:

1 r ∂ ∂r r ∂ ∂r + ∂ 2 ∂z 2 p = S, (3.1)
whre the source term, S, in the right-hand side is given by

S = ρ r ∂ ∂r rF r + ∂ ∂z zF z (3.2a) with F r = -(u ∂u ∂r + w ∂u ∂z ) (3.2b) F z = b c -w c ∂w c ∂z + E ρσ (w e -w c ) (3.2c)
In solving Eq. (3.1), first note that the source, S, is independent of r. It can be shown followingly: with the help of Eq. (2.6a, b), we find:

F r = -r ∂u ∂r 2 + rw c ∂ ∂z ∂u ∂r (3.2d)
is linear in r, thus rF r is proportional to r 2 , and (ρ/r)∂rF r /∂r is independent of r; F z does not depend on r with all the dependent variables under SCA. Since S does not depend on r, we may also drop z-dependence from the left-hand side of Eq. (3.1), thus:

d 2 dz 2 p = S.
We may separate the pressure into two components, corresponding to the two components of the source (3.2b, c). Thus, p = p (r) + p (z) , and

d 2 dz 2 p (r) = ρ r ∂ ∂r rF r ≡ S r (3.3a) d 2 dz 2 p (z) = ∂ ∂z zF z (3.3b)
By substituting (3.2) into its definition, we find

S r = -2ρ ∂u ∂r 2 + w c ∂ ∂z ∂u ∂r (3.3c)
Note that Eq. (3.3b) can be integrated immediately, and we find

d dz p (z) = ρF z (3.4)
which can be considered a generalized hydrostatic balance. More specifically, as a result, buoyancy, b c , is perfectly in balance with the pressure gradient force, thus it never accelerates the vertical velocity. This would be a very unrealistic situation. Especially, when the mass-flux parameterization is formulated prognostically as presented here, convection would be never initiated even under a presence of buoyancy anomaly.

However, when a diagnostic formulation is adopted, as the case with majority of current operational parameterizations (cf., Plant and Yano 2015), the above constraint (3.4) may not be that serious, because the dynamic pressure, p (r) , plays a similar role as the buoyancy in this case.

The point can be seen in the following manner. Let us consider a convective circulation schematically given in Fig. 1(a): convergence and divergence at the lower and the upper levels, respectively, which induces an updraft at the center by continuity. Since F r is a rate that the radial velocity, u, is accelerated by advection, thus F r < 0 at the lower level. This tendency is enhanced towards the center, thus ∂F r /∂r < 0, and presumably also S r < 0 by its definition (3.3a). A similar considering at the upper level suggests S r > 0, as summarized in Fig. 1(b). Finally, by solving Eq. (3.3a), we expect the tendency of p (r) ∼ -S r for the dynamic pressure, as schematically shown in Fig. 1(c), thus the resulting pressure gradient drives the updraft at the center, as expected for the buoyancy.

The tendency inferred above must still be demonstrated quantitatively that this configuration leads to a reasonable result.

The last point is more formally shown by directly substituting the hydrostatic balance (3.4) into Eq. (2.1a), also assuming the steadiness:

1 ρ ∂ ∂z M w c = - σ ρ ∂p (r) ∂z .
Let us assume, for the sake of the purpose of making an immediate analytical progress, that σ is constant with height. In that case, the above equation reduces to:

∂ ∂z [ρw 2 c + p (r) ] = 0.
It can be immediately integrated vertically, and:

ρw 2 c = -p (r) + [ρw 2 c + p (r) ] z=0 . (3.5)
Thus, we find that in association of a decrease of the dynamic pressure with height, as suggested in Fig. 1(c), the vertical velocity increases with height.

However, the main remaining problem is whether we can actually arrive at such a steady solution after a certain iteration proceduire. Such a procedure becomes necessary under a traditional diagnostic formulation, because Eq. (3.3a) cannot be solved without knowing the source term, S r . To know the source term, S r , in turn, the vertical momentum equation (2.1a) must already be solved. For this purpose, in turn, the pressure must already be known.

To feel a degree of this difficulty, as an initial Ansatz, let us set:

w c = w 0 sin πz D (3.6)
with D a constant height. Its substitution into Eq. (3.3c) leads to

S = ρ 4 πw 0 D 2 [cos 2πz D -3] (3.7)
By further substituting this result (3.7) into Eq. (3.3a), and solving it for p (r) , we find:

p (r) = 2ρw 2 0 [sin 2 πz D - 1 2 ] - 3 8 ρ πw 0 D 2 z 2 (3.8)
using the condition dp/dz = 0 at z = 0 and also setting an arbitrary constant to be zero.

The result with the first iteration is, thus, obtained by substituting the solution (3.8) into

Eq. (3.5). It is clearly qualitatively different from the initial Ansazt (3.6), and it is hardly obvious how we get to a converging solution by repeating this procedure.

4. Pressure problem (2): with r-Dependence

Contribution of a Homogeneous Solution to the Pressure Problem

The analysis of the last section is, however, still incomplete, because a possible contribution of a homogeneous solution is neglected, which may still contains a radial dependence. This fact further suggests that the whole problem

1 r ∂ ∂r r ∂ ∂r + ∂ 2 ∂z 2 p = S (3.1)
may be solved by retaining the radial dependence explicitly. Rather unintuitively, this approach leads to a different result as seen in the following.

Eq. (3.1) can be solved by setting

p = ∞ i=1 pi (z, t)J 0 (ξ i r) (4.1a)
where J 0 (x) is the Bessel function of the 0-th order, ξ i (i = 1, . . . , ∞) are roots for J 0 (ξ i R) = 0 given in increasing order, and R is a radius of the convective updraft, which is assumed to be constant with height for now. Note that a general solution (4.1a) is set to satisfy the boundary condition of p| r=R = 0.

The expansion coefficient, pi , can be defined by

pi = 1 λ i R 0 pJ 0 (ξ i r)rdr, (4.1b)
where

λ i = R 2 2 dJ 0 (ξ i R) d(ξ i R) 2 = R 2 J 2 0 (ξ i R) 2 . (4.1c)
As partially used above, the following formulas for the Bessel functions become useful:

dJ 0 (x) dx = -J 1 (x), d dx xJ 1 (x) = xJ 0 (x).
Note that the series of Bessel functions adopted here for representing the pressure constitutes an orthogonal complete set. That means that any function defined for the range of (0, R), that also includes a constant function. Note more specifically that

∞ i=1 2 ξ i RJ 1 (ξ i R) J 0 (ξ i r) = 1.
Using this relation, the source term is represented as

S = ∞ i=1 Ŝi J 0 (ξ i r) (4.2a) with Ŝi = 2S ξ i RJ 1 (ξ i R) . (4.2b)
The problem is normalized by setting Ŝi = S Si and pi = pi Si . Since Si does not depend on z, the problem of solving for pi reduces to that of pi , and the latter is obtained by solving:

( d 2 dz 2 -ξ 2 i )p i = S (

=

  4.3) 4.2. Pressure due to a Localized Buoyancy As a simple example of a solution obtained under the formulation presented in the last subsection, let us consider a pressure due to a localized buoyancy given by b = b c -D/2 ≤ z ≤ D/2, r ≤ R 0 otherwise (4.4) with b c a constant. The resulting pressure source term is: r ≤ R, and otherwise S = 0, where δ designates the Diracs delta. By integrating Eq. (4.3) vertically, thus, we obtain ρb c , (4.5b)Since apart from two levels at z = ±D/2, there is no source term in the right hand side of Eq. (4.3). The resulting solution to this homogeneous solution for the three segments are:ξ i z + b i e -ξ i z |z| ≤ D/2 p i+ e -ξ i z z ≥ D/2 p i-e ξ i z z ≤ -D/2 (4.6)assuming pi → 0 as z → ±∞. Here, the constants, a i , b i , and p i± are determined from the boundary conditions (4.5a, b), and we finally find:pi = ρb c 2 [e ξ i (z-D/2) -e -ξ i (z+D/2) ] (4.7)for |z| ≤ D/2, and the pressure averaged over the convective updraft segment (i.e., r ≤ R) [e ξ i (z-D/2) -e -ξ i (z+D/2) ],And an average pressure gradient over the buoyancy anomaly (-D/2 ≤ z ≤ D/2) is: ξ 2 i R 2 = 1/4.Fig. 2 (convection/pressure/plot factor. . f) plots the factor, D < d p dz > /ρb c as a function of D/R: The factor approaches to zero and unity towards D/R → 0 and → ∞, respectively, as expected.


domain: sdv chim

text:
Yi Shen

Yurong Wen

Silvia Sposini

Anjali Amrapali Vishwanath

Ahmed S Abdelfattah

Eric R Schreiter

M Joanne Lemieux

Jaime De Juan-Sanz

David Perrais

Robert E Campbell

Anjali Amrapali

Jaime De Juan

Rational Engineering of an Improved Genetically Encoded pH Sensor Based on Superecliptic pHluorin

Keywords: pHluorin, fluorescent proteins, genetically encoded sensors

Genetically encoded pH sensors based on fluorescent proteins are valuable tools for the imaging of cellular events that are associated with pH changes, such as exocytosis and endocytosis. Superecliptic pHluorin (SEP) is a pH-sensitive green fluorescent protein (GFP) variant widely used for such applications. Here, we report the rational design, development, structure, and applications of Lime, an improved SEP variant with higher fluorescence brightness and greater pH sensitivity. The X-ray crystal structure of Lime supports the mechanistic rationale that guided the introduction of beneficial mutations. Lime provides substantial improvements relative to SEP for imaging of endocytosis and exocytosis. Furthermore, Lime and its variants are advantageous for a broader range of applications including the detection of synaptic release and neuronal voltage changes.

G enetically encoded sensors based on fluorescent proteins (FPs) are essential tools for studying molecular and cellular physiology. 1,2 Over the past two decades, an expanded toolkit of FP-based sensors has been engineered to monitor numerous biochemical parameters including the Ca 2+ concentration, membrane potential, neurotransmitter dynamics, and pH. 3,4 pH changes are tightly associated with cellular processes such as endocytosis and exocytosis. Using the example of synaptic transmission, the lumen of synaptic vesicles is typically acidified to pH ∼5.5 by the proton pump activity of vacuolartype ATPase. Upon fusion of a synaptic vesicle to the plasma membrane, the contents of the vesicle are exposed to the neutral pH (∼7.0 to 7.5) of the extracellular environment. 5 This substantial change in pH allows the process of exocytosis to be visualized using a genetically encoded pH sensor fused with a vesicular membrane-associated protein. 6,7 Monitoring of endocytosis and exocytosis has provided valuable insights that have helped to elucidate the fundamental biology of neurotransmission and have aided investigations of the underlying mechanism of various neurological diseases. [8][9][10][11] Genetically encoded pH sensors are FPs that demonstrate pH-dependent fluorescence spectrum or intensity change within the physiologically relevant pH range. 6,[12][13][14][15][16][17] Most naturally occurring FPs exhibit pH-dependent fluorescence changes, but some are much more sensitive in the physiological range than others. For example, the wild-type Aequorea victoria green fluorescent protein (GFP) chromophore exists in a pHdependent equilibrium between protonated and deprotonated states, which absorb maximally near 400 and 488 nm, respectively. of the voltage-sensing domain from Ciona intestinalis voltagesensitive phosphatase to SEP-A227D. [START_REF] Jin | Single Action Potentials and Subthreshold Electrical Events Imaged in Neurons with a Fluorescent Protein Voltage Probe[END_REF] Although widely used, SEP has a few shortcomings such as relatively low brightness and suboptimal trafficking, which limits its utilities in some applications. 10,[START_REF] Ashby | Removal of AMPA Receptors (AMPARs) from Synapses Is Preceded by Transient Endocytosis of Extrasynaptic AMPARs[END_REF][START_REF] Rathje | AMPA Receptor PHluorin-GluA2 Reports NMDA Receptor-Induced Intracellular Acidification in Hippocampal Neurons[END_REF][START_REF] Ventimiglia | Diverse Modes of Synaptic Signaling, Regulation, and Plasticity Distinguish Two Classes of C. Elegans Glutamatergic Neurons[END_REF] Here, we report the design and characterization of Lime, a rationally engineered SEP variant with higher fluorescence brightness and improved performance. Lime takes advantage of the high brightness and expression of sfGFP [START_REF] Cabantous | In Vivo and in Vitro Protein Solubility Assays Using Split GFP[END_REF] and is cysteine-free for potentially better compatibility with membrane protein 79 fusion. [START_REF] Costantini | A Palette of Fluorescent Proteins Optimized for Diverse Cellular Environments[END_REF] We determined the X-ray crystal structure of Lime crystal structure of Lime at pH 10.0 revealed that the S147D mutation had induced a significant rearrangement. The side chain of S147D has become inward-facing and is directly interacting with the chromophore, which is certain to be influencing the pK a of the chromophore (Figure 2C). The direct interaction of the side chain of an aspartic acid with the GFP chromophore has been reported previously in GFP S65T/H148D. In this variant, H148D was found to interact with the chromophore via hydrogen bonding and resulted in an elevated pK a value of 7.9 (Figure 2D). [START_REF] Shu | Ultrafast Excited-State Dynamics in the Green Fluorescent Protein Variant S65T/H148D. 1[END_REF] The conformational rearrangement of the residue at position 147 has also been observed in deGFP1 (S65T/H148G/T203C). In this variant, the side chain of S147 is outward-facing at pH 5.6 but has flipped to inward-facing at pH 9.0 (Figure 2E,F). [START_REF] Hanson | Green Fluorescent Protein Variants as Ratiometric Dual Emission PH Sensors. 1. Structural Characterization and Preliminary Application[END_REF] To examine possible structure differences at different pH values, we solved structures of Lime at a lower pH of 6.0.

Superimposition of the two structures at pH 10 and 6 revealed no major structural rearrangements, with the side chain of S147D pointing inside the FP barrel at both the high and low pH values (Figure S2A). However, the superimposition revealed two relatively subtle structural differences (Figure S2B). First, the chromophore in the higher pH structure To further confirm the key role of S147D mutation in pK a modulation, we introduced this single mutation into GFP variants, EGFP (pK a of 5.5) [START_REF] Heim | Improved Green Fluorescence[END_REF] and mClover3 (pK a of 6.5). [START_REF] Bajar | Improving Brightness and Photostability of Green and Red Fluorescent Proteins for Live Cell Imaging and FRET Reporting[END_REF] The pH titrations of EGFP-S147D, and mClover3-S147D both revealed a significant increase in apparent pK a values to 7.0 and 7.3, respectively. Thus, the S147D mutation effectively makes EGFP and mClover3 into pH-sensitive GFPs similar to SEP t2 and Lime (Table 2), confirming its ability to modulate the pH sensitivity of GFP variants. It remains to be determined if a similar mutation at the aligned position of FPs from organisms other than Aequorea (e.g., W138 of mNeonGreen) [START_REF] Ong | Rational Design of the Β-bulge Gate in a Green Fluorescent Protein Accelerates the Kinetics of Sulfate Sensing[END_REF] could have a similar influence on pK a .

Comparison of Lime and SEP for Imaging Endocytosis and Exocytosis. We next tested the performance of

Lime in live-cell imaging and compared it with SEP. The transferrin receptor (TfR) fused to a pH-sensitive GFP (e.g., TfR-SEP) has been widely used for the detection of single endocytosis and exocytosis events in cultured cells. 15,18,[START_REF] Taylor | A High Precision Survey of the Molecular Dynamics of Mammalian Clathrin-Mediated Endocytosis[END_REF][START_REF] Sposini | Imaging Endocytic Vesicle Formation at High Spatial and Temporal Resolutions with the Pulsed-PH Protocol[END_REF][START_REF] Merrifield | Coupling between Clathrin-Coated-Pit Invagination, Cortactin Recruitment, and Membrane Scission Observed in Live Cells[END_REF] We first imaged and quantified endocytic events in the BSC-1 cell line using the pulsed-pH protocol with time-lapse total internal reflection fluorescence (TIRF) imaging at 0.5 Hz. We

C

196 used a previously developed algorithm for the determination of 197 the scission events. [START_REF] Taylor | A High Precision Survey of the Molecular Dynamics of Mammalian Clathrin-Mediated Endocytosis[END_REF][START_REF] Sposini | Imaging Endocytic Vesicle Formation at High Spatial and Temporal Resolutions with the Pulsed-PH Protocol[END_REF] Cells expressing TfR-Lime had brighter 198 fluorescence than the ones expressing TfR-SEP under identical 199 imaging conditions. A Western blot analysis revealed that the 200 amounts of TfR-SEP and TfR-Lime in cells were not 201 significantly different (Figure S3). A two-fold increase in 202 exposure time was required to obtain images of TfR-SEP with f3 203 intensities that were comparable to TfR-Lime (Figure 3A,B).

204

Quantitative analysis of the average fluorescence intensity at 205 the time of scission revealed that TfR-Lime offers approx-206 imately a two-fold improvement in terms of the fluorescence 207 brightness over TfR-SEP, at both pH = 5.5 and pH = 7.4 208 (Figure 3C,D). For reasons that remain unclear, the 209 fluorescence intensities for TfR-Lime endocytic events are 210 more variable than for TfR-SEP endocytic events. Both TfR-211 SEP and TfR-Lime reported similar frequencies of events 212 detected (Figure 3E), confirming that most endocytic events 213 can be detected with both reporter proteins. [START_REF] Merrifield | Coupling between Clathrin-Coated-Pit Invagination, Cortactin Recruitment, and Membrane Scission Observed in Live Cells[END_REF] The average 214 fluorescence time course of endocytosis of TfR-SEP and TfR-215 Lime reflected similar kinetics, yet again with a substantially 216 higher fluorescence intensity for TfR-Lime (Figure 3F,G). We 217 next imaged exocytic events using continuous TIRF micros-218 copy imaging at 10 Hz (Figure 3H). An exocytotic event 219 manifests as sudden bursts of fluorescence due to the change in pH experienced by the TfR construct followed by dissipation of fluorescence as the TfR construct diffuses away from the exocytosis site, as previously described. 18,[START_REF] Xu | Dual-Mode of Insulin Action Controls GLUT4 Vesicle Exocytosis[END_REF] In agreement with our results for endocytosis, cells expressing TfR-Lime exhibited a significantly higher fluorescence amplitude than TfR-SEP (Figure 3I), yet still reported comparable event frequencies, and had similar fluorescence decay kinetics at sites of exocytosis events, which reflects the rate of protein diffusion away from the fusion site (Figure 3J,K). Overall, our results indicated that Lime has clear advantages over SEP as a useful reporter for TfR exocytosis and endocytic vesicle detection.

Imaging Neuronal Activities with Lime and Its

Variants. To evaluate the performance of Lime for imaging neuronal activity, we first attempted to image synaptic vesicle release using synaptophysin-Lime. Lime was incorporated into the intravesicular loop region of vesicle protein synaptophysin. [START_REF] Zhu | Two Pathways of Synaptic Vesicle Retrieval Revealed by Single-Vesicle Imaging[END_REF] Synaptophysin-Lime expressed in dissociated neurons exhibited well-localized fluorescence at the synapse with low f4 background fluorescence (Figure 4A). Under electric stimulation to evoke action potentials (APs), synaptophysin-Lime responded with a fluorescence change ΔF/F 0 of 19.0 ± 2.9% increase upon 10 stimuli by a field electrode at 10 Hz (Figure 4B,C). Synaptophysin-SEP exhibited a change of 22.6 ± 2.4% under the same conditions. Synaptophysin-Lime also exhibited TIRF imaging was performed on an inverted microscope (IX71; 369 Olympus) equipped with an Apochromat N oil 60× objective (NA 370 1.49), a 1.6× magnifying lens, and an EMCCD camera (Quan-371 tEM:512SC; Roper Scientific). Fluorescence emission was collected 372 via a 525/50 filter (Chroma). Samples were illuminated by a 473 nm 373 laser (Cobolt) for SEP and Lime imaging with identical laser intensity 374 and camera settings. The ppH protocol for imaging endocytosis was 375 performed as described previously [START_REF] Sposini | Imaging Endocytic Vesicle Formation at High Spatial and Temporal Resolutions with the Pulsed-PH Protocol[END_REF] with a frame rate of 0.5 Hz (100 376 ms exposure time). Cycles of one image at pH 7.4, followed 2 s later 377 by one image at pH 5.5, were repeated for at least 5 min. Imaging of 378 exocytosis events was performed with continuous imaging at 10 Hz 379 for 1 min.

380

To perform a Western blot, BSC-1 cells in 6-well plates were 381 transfected with 0.5 μg plasmid/well encoding either TfR-SEP or 382 TfR-Lime. Cells were lysed 30 h post-transfection in RIPA buffer supplemented with antiproteases, and 80 μg of each lysate was loaded and run on a 4-20% polyacrylamide gel. Proteins were transferred to a nitrocellulose membrane and blotted using anti-GFP (Roche ref.

11814460001) and antiactin (Sigma-Aldrich ref. A2066) primary antibodies.

Analysis of Endocytosis and Exocytosis Events with TfR-

Lime. Detection and quantification of endocytosis events were conducted using custom-made Matlab (R2018b) scripts that have previously been described. [START_REF] Sposini | Imaging Endocytic Vesicle Formation at High Spatial and Temporal Resolutions with the Pulsed-PH Protocol[END_REF] In short, a sudden, punctate, fluorescence increase appearing in pH 5.5 images was detected as being an endocytic event if (1) it was visible for more than 3 frames (i.e., 8 s) and (2) it appeared at the same location as a pre-existing fluorescence cluster detectable in pH 7.4 images. Candidate events from each cell were validated using a trained support vector machine as described previously. [START_REF] Sposini | Imaging Endocytic Vesicle Formation at High Spatial and Temporal Resolutions with the Pulsed-PH Protocol[END_REF] The event frequency was expressed per cell surface area measured on the cell masks. Imaging Neuronal Exocytosis with Synaptophysin-Lime. All procedures followed the European legislation for animal experimentation (directive 2010/63/EU), and animal manipulations were approved by the institutional Ethical Committee. Wild-type rats were of the Sprague Dawley strain Crl:CD (SD), which are bred by Charles River Laboratories worldwide following the international genetic standard protocol (IGS). Hippocampal CA3-CA1 regions without dentate gyrus were isolated from 0-to 2-days-old rats of mixed gender, plated on poly-ornithine-coated coverslips, transfected 7 days after plating using calcium phosphate-mediated gene transfer, and imaged 14-21 days after plating as previously described. [START_REF] De Juan-Sanz | Axonal Endoplasmic Reticulum Ca2+ Content Controls Release Probability in CNS Nerve Terminals[END_REF] These cultures resulted in mixed neuron-astrocyte cultures, which were maintained in culture media composed of MEM (Thermo Scientific), 0.5% glucose, 0.1 g/L bovine transferrin (Millipore Sigma), 0.25 g/L insulin (Millipore Sigma), 1% Glutamax (Thermo Scientific), 5-10% fetal bovine serum (Thermo Scientific), and 2% N-21 (Bio-Techne).

Culture media was replaced 2-3 days later with culture media containing 2 μM cytosine β-D-arabinofuranoside (Millipore Sigma) to limit astrocyte overgrowth. Cultures were incubated at 37 °C in a 95% air/5% CO 2 humidified incubator for 14-21 days prior to use.

Neuronal live-cell imaging was performed using a custom-built laserilluminated epifluorescence microscope with an Andor iXon Ultra (model no. DU-897U-CSO-#BV) back-illuminated electron-multiplying charge-coupled device camera cooled to -90 °C. Experiments were performed at a clamped temperature of 36.5 °C using both an in-line solution heater (640104: SHM-6, multichannel systems) and a chamber heater platform (64-0285 PH-2, multichannel systems).

Action potentials (AP) were evoked by passing 1 ms current pulses, yielding fields of approximately 7 V cm -1 via platinum-iridium electrodes using an imaging chamber that allows field stimulation (64-0226 RC-21BRFS, multichannel systems). Cells were continuously perfused at 0.1 mL/min with a Tyrode's solution containing (in mM) 119 NaCl, 2.5 KCl, 2 CaCl 2 , 2 MgCl 2 , and 30 glucose (Sigma-Aldrich G8270), 10 μM 6-cyano-7-nitroquinoxaline-2,3-dione (CNQX), and 50 μM D,L-2-amino-5-phosphonovaleric acid (AP5) to block postsynaptic AMPA and NMDA receptor responses, buffered to pH 7.4 using 25 mM HEPES. Synaptophysin-Lime was generated by

  80 to provide molecular insight into its pH sensitivity. We 81 demonstrated the utility of Lime by imaging endocytosis and 82 exocytosis in cultured cell lines and neuronal activities in 83 dissociated neurons. 84 ■ RESULTS AND DISCUSSION 85 Rational Engineering and In Vitro Characterization of 86 Lime. Our approach to rationally improving SEP was to 87 incorporate mutations from previously reported GFP variants

Figure 1 .

 1 Figure 1. Rational engineering and in vitro characterization of Lime. (A) Design and genealogy of Lime and its variants. (B) Fluorescence spectrum of Lime at pH 5.5 (dotted lines) and pH 7.5 (solid lines). (C) pH titration curves of Lime.

Figure 2 .

 2 Figure 2. Structural analysis of Lime. (A) SEP mutation positions (from EGFP) are mapped on the structure (left: side view; right: top view) of GFP (PDB 1EMA). The chromophore is shown in green sticks, and mutated residues are shown in gray sticks. (B) In GFP, the side chain of residue S147 points outward from the β-barrel structure. (C) In Lime, the side chain of residue S147D points inward and directly interacts with the chromophore. (D) In the pH-sensitive GFP S65T/H148D variant (PDB 2DUF), H148D is pointing inward and directly interacts with the chromophore. Conformations of residue S147 and H148G in deGFP1 at pH 5.6 (E) (PDB 1JBY) and pH 9.0 (F) (PDB 1JBZ).

  adopts a more planar conformation compared to the lower pH structure. The two dihedral angles indicating the chromophore planarity are 8.7 and -5.6°in the pH 10 state, compared with 19.9 and -15.0°in the pH 6 state (FigureS2C,D). Second, the carboxylate group on the S147D side chain in the high pH structure sits closer to the chromophore (distance of 2.7 Å) than in the low pH structure (distance of 3.2 Å) (FigureS2C,D). These conformational changes in the chromophore and its proximity to the S147D side chain are likely associated with the large fluorescence change between high and low pH states.

Figure 3 .

 3 Figure 3. Imaging endocytosis and exocytosis with TfR-Lime. (A,B) Fluorescence images of TfR-SEP (A) and TfR-Lime (B) expressing cells at pH 7.4 and 2 s later at pH 5.5. Scale bar, 5 μm. Fluorescence images of TfR-Lime expressing cells at pH 7.4 and 2 s later at pH 5.5. Scale bar, 5 μm. (C,D) Average fluorescence at the time of scission with TfR-SEP or TfR-Lime at pH 7.4 (C) or pH 5.5 (D) (unpaired t-test, p = 0.027 and 0.029, respectively). (E) Frequency of detected endocytic events (unpaired t-test, p = 0.404). (F,G) Average fluorescence time course of endocytosis events reported by TfR-SEP ((F) 3712 events, 8 cells) and TfR-Lime ((G) 4269 events, 7 cells). Black lines represent 95% confidence intervals for randomized events to determine the level of enrichment at sites of scission. (H) Examples of exocytosis events in consecutive frames for cells transfected with TfR-SEP (top) and TfR-Lime (bottom). Scale bar, 1 μm. (I) Average fluorescence at the time of exocytosis for events with TfR-SEP and TfR-Lime (unpaired t-test, p = 0.027). (J) Frequency of detected exocytic events (unpaired t-test, p = 0.297). (K) Average fluorescence time course of exocytosis events reported by TfR-SEP (528 events, 8 cells) and TfR-Lime (721 events, 6 cells), aligned to the time of detection (time 0).

Figure 4 .

 4 Figure 4. Imaging neuronal activities with synaptophysin-Lime and ArcLimeade. (A) Representative image of neurons expressing synaptophysin-Lime. (B) Synaptophysin-Lime response time course under stimulation of 10 action potentials (AP) at 10 Hz (n = 8). Comparison between synaptophysin-Lime and synaptophysin-SEP in terms of fluorescence change upon 10 stimuli by the field electrode at 10 Hz (C), baseline fluorescence (D), and surface fractions (E). (F) Representative image of neurons expressing ArcLimeade. (G) Simultaneous fluorescence imaging (top) and whole-cell patch-clamp membrane voltage measurements (bottom) from cultured neurons expressing ArcLimeade. (H) Representative fluorescence changes of ArcLimeade to voltage steps from -110 to 50 mV, normalized to the fluorescence at -70 mV. (I) Fluorescence change vs membrane voltage (F-V) curve for ArcLimeade and ArcLight-Q239.

  368

  The average fluorescence intensity was measured in a 4-pixel diameter circle (430 nm) minus a local background (annulus of 4-pixel inner diameter, 10-pixel outer diameter) centered on the detected vesicle (images at pH 5.5) until the vesicle disappears. Causes of vesicle disappearance include movement away from the cell surface and vesicle acidification. Figure 3F,G represents averages ± SEM of the fluorescence (at pH 7.4 and 5.5) aligned at the time of vesicle detection (time 0) of all detected events. Detection and quantification of exocytosis events were performed with custom-made Matlab (R2018b) scripts as described previously. 46 Exocytosis events were detected with an ad hoc threshold in a differential movie and further validated with automated and user-defined criteria. The event frequency was expressed per cell surface area measured on the cell masks. The average fluorescence in a 5-pixel diameter area (minus a local background) centered on the appearing vesicle was measured for each event. The decrease in fluorescence reflects diffusion of TfR in the plasma membrane after exocytosis.

Table 1 .

 1 In Vitro Characterization of GFP-Based pH-Sensitive FPs

		extinction coefficient (EC, M -1 cm -1 )	quantum yield (QY)	molecular brightness (EC × QY/1000)	absorption maxima (nm)	excitation maxima (nm)	emission maxima (nm) pK a	n H	fold change (pH 5.6 to 7.2)
	EGFP	57,000	0.60	34	488	488	509	5.5	1.0	N.D. a
	SEP	37,000	0.80	30	490	488	511	7.0	1.7	66 ± 6
	SEP-D	31,000	0.80	25	490	488	511	7.2	1.3	70 ± 1
	Lime	45,000	0.77	35	483	488	510	7.1	1.6	82 ± 9
	mLime	36,000	0.72	26	483	488	509	7.1	1.6	86 ± 2
	Limeade	31,000	0.72	22	483	488	509	7.1	1.5	99 ± 1
	mLimeade	29,000	0.77	22	483	488	509	7.1	1.5	98 ± 1

a N.D., not determined.

Table 2 .

 2 In Vitro Characterization of EGFP-and mClover3-S147D Mutants

							emission
		extinction coefficient (EC, M -1 cm -1 )	quantum yield (QY)	molecular brightness (EC × QY, ×1000)	absorption maxima (nm)	excitation maxima (nm)	maxima (nm)	pK a	n H	fold change (pH 5.6 to 7.2)
	EGFP-	32,000	0.53	17	488	488	509	7.0	1.4	43 ± 2
	S147D
	mClover3-	45,000	0.61	27	507	508	518	7.3	1.0	67 ± 2
	S147D

https://doi.org/10.1021/acssensors.3c00484 ACS Sens. XXXX, XXX, XXX-XXX

https://doi.org/10.1021/acssensors.3c00484 ACS Sens. XXXX, XXX, XXX-XXX

https://doi.org/10.1021/acssensors.3c00484 ACS Sens. XXXX, XXX, XXX-XXX I

This work was supported by grants from the Canadian 558 Institutes of Health Research (CIHR, FS-154310 to R.E.C.) 559 and the Natural Sciences and Engineering Research Council of 560 Canada (NSERC, RGPIN 2018-04364 to R.E.C. and RGPIN-561 2016-06478 to M.J.L.) and the Agence Nationale de la 562 Recherche (LocalEndoProbes, ANR-17-CE16-0012, to D.P.). 563 Y.W. was supported by the Alberta Parkinson Society 564 Fellowship and the National Natural Science Foundation of 565 China (nos. 31870132 and 82072237). S.S. was supported by a 566 Sir Henry Wellcome Postdoctoral Fellowship (Wellcome 567 Trust, 218650/Z/19/Z). Data were collected at beamline 568 CMCF-ID at the Canadian Light Source, a national research 569 facility of the University of Saskatchewan, which is supported 570 by the Canada Foundation for Innovation (CFI), the NSERC, 571 the National Research Council (NRC), the CIHR, the 572 Government of Saskatchewan, and the University of 573 Saskatchewan. A.S.A. and E.R.S. were supported by Howard 574 Hughes Medical Institute. J.d.J.-S. and A.A.V. were supported

■ AUTHOR INFORMATION

Authors Yi Shen -Department of Chemistry, University of Alberta, Edmonton, Alberta T6G 2G2, Canada; orcid.org/0000-0003-4117-0279; Email: yi.shen@ualberta.ca Robert E. Campbell -Department of Chemistry, University of Alberta, Edmonton, Alberta T6G 2G2, Canada; Department of Chemistry, Graduate School of Science, The University of Tokyo, Tokyo 113-0033, Japan; orcid.org/0000-0003-0604-092X; Email: campbell@chem.s.u-tokyo.ac.jp Authors Yurong Wen -Department of Biochemistry, Faculty of Medicine & Dentistry, University of Alberta, Edmonton, Alberta T6G 2H7, Canada; Center for Microbiome Research of Med-X Institute, The First Affiliated Hospital, Xi'an Jiaotong University, Xi'an, Shaanxi 710061, China Silvia Sposini -CNRS, Interdisciplinary Institute for Neuroscience, UMR 5297, University of Bordeaux, Bordeaux 33076, France; Department of Metabolism, Digestion and Reproduction, Institute of Reproductive and Developmental Biology, Imperial College London, London SW7 2BX, United

F

replacing SEP with Lime in synaptophysin-SEP. Synaptophysin-Limeexpressing neurons were imaged at 10 Hz, and responses to 10 AP at 10 Hz were recorded. Images were analyzed using the ImageJ plugin Time Series Analyzer V3, where typically 30-40 regions of interest (ROIs) corresponding to synaptic boutons were selected, and fluorescence was measured over time.

Imagining Neuronal Membrane Voltage with ArcLimeade.

All procedures involving animals were conducted in accordance with protocols approved by the Howard Hughes Medical Institute Janelia Research Campus Institutional Animal Care and Use Committee and Institutional Biosafety Committee. Hippocampal neurons extracted from P0 to 1 Sprague Dawley rat pups were transfected with pcDNA3.1-CAG-ArcLimeade by electroporation (Lonza, P3 Primary Cell 4D-Nucleofector X kit). Neurons were plated onto 24-well glassbottom plates (MaTek) coated with poly-D-lysine (Sigma). Neurons were cultured for 8-12 days in NbActiv4 medium (BrainBits). Imaging buffer (containing the following (in mM): 145 NaCl, 2.5 KCl, 10 glucose, 10 HEPES pH 7.4, 2 CaCl 2 , and 1 MgCl 2 ) was used. Cultures were illuminated with a SPECTRA X light engine (Lumencor) and observed through a 40× oil objective (NA = 1.3, Nikon) on an inverted Nikon Eclipse Ti2 microscope. Excitation and emission light was passed through a FITC filter set (475/50 nm (excitation), 540/50 nm (emission), and a 506LP dichroic mirror (FITC-5050A-000; Semrock)) to image ArcLimeade. Fluorescence was collected by a CMOS camera (ORCA-Flash 4.0, Hamamatsu). A stimulus isolator (A385, World Precision Instruments) with platinum wires was used to deliver field stimuli to elicit action potentials in cultured neurons. For voltage-clamp experiments used to generate fluorescence-voltage curves, voltage steps (from -110 to +50 mV in 20 mV increments for 1 s) were applied to cells held at -70 mV. All recordings were done at room temperature.

■ ASSOCIATED CONTENT

* sı Supporting Information

The Supporting Information is available free of charge at https://pubs.acs.org/doi/10.1021/acssensors.3c00484.

(Table S1) X-ray data collection and refinement statistics, (Figure S1) sequence alignment, (Figure S2) comparison of Lime structures at pH 10 and pH 6, and (Figure S3) comparison of TfR-SEP and TfR-Lime expression levels in BSC-1 cells (PDF)