Philippe Rauffet
email: philippe.rauffet@irccyn.ec-nantes.fr

Catherine Da Cunha
email: catherine.da-cunha@irccyn.ec-nantes.fr

Alain Bernard
email: alain.bernard@irccyn.ec-nantes.fr

Designing and managing Organizational Interoperability with organizational capabilities and roadmaps

This paper discusses organizational interoperability issues in through the study of two cases. Then it presents a framework which can help to design and manage this interoperability, by driving the development of "organizational capabilities".

Introduction

Companies are nowadays dynamic structures, which are rearranged on a regular basis depending on projects and alliances. Organizations become therefore multi-functional, multi-products, or geographically distributed. These reconfigurations have significant consequences in terms of performance at all levels, and they sometimes trigger such an organizational heterogeneity. Organizational interoperability is therefore a key challenge for companies so as to make their resources (sites, people, products, software…) work together, with the same practices and the same processes.

ATHENA defines interoperability as "the ability of two or more systems or components to exchange information and to use [without any misinterpretation or loss of sense] the information that has been exchanged" [1].

As this definition emphasizes, interoperability is only concerned with communication and information sharing so as to make interactions possible. By contrast enterprise integration is broader and is related to cooperation and coordination of the organizational entities and their interactions [2]. However, Vernadat shows that these two notions are linked, since interoperability is a key for supporting enterprise integration. Thus it is essential to make systems interoperable and interacting in order to control and manage complex organizations and drive their performance.

In order to reach this interoperability, enterprises must limit "heterogeneity" [3]. Heterogeneity can be differentiated in three types:

-a semantic heterogeneity, composed of syntax and language problems (signification of speech, sense of the knowledge which can be proceeded and manipulated in the enterprise). That comes from the fact that systems are defined, built and used by different people, in different places at different times, with different aims, with different vocabularies.

-an organizational (or functional) heterogeneity, caused by the differences in practices and business processes. Indeed, different enterprises or departments are growing and developing their own organization isolated from another. Consequently, same tasks may not be processed in the same way in two different organizational units and problems can occur at their cooperation border.

-a technical (or material) heterogeneity, resulting from the use of several materials to transport, transmit, and operate information which are not compatible.

This paper only focuses on organizational interoperability. It deals with some current works fulfilled in the ANR project Pilot2.0 [4] around a system managing organizational capabilities and named 5 steps ® [5]. Section 2 deals with organizational interoperability issues in two different cases. Section 3 discusses on approaches to design interoperability. Section 4 presents the key concept (organizational capability [6]) and the mechanisms of 5 steps ®, and shows how they could be used among other things to increase interoperability between heterogeneous organizational units and communication between functional and operational people. Finally, limits and perspectives are discussed in last section.

Organizational interoperability issues in a globalized industrial group and in a French public administration

The following part deals with two different case studies, Valeo group and General Council of Vaucluse. These two organizations are members of the research Pilot2.0 project, supported by the French National Research Agency (ANR).

The exchanges with these project partners enabled us to analyze their organizational context and to identify their interoperability issues. The summary of this analysis is given in Fig. 1.

These two organizations differ in their missions, their organizational structure and their environment. However, they present some common problems:

-There are some difficulties for limiting heterogeneity in functional practices and processes;

-Communication and interactions are rough to animate between different actors (functional with operational people, new sites with elder sites, functional domain with another one, internal services with suppliers and associations…); -Finally, there is a missing means to manage heterogeneous, complex (multi -products or -services, multifunctional), and extended structures.

From this analysis and these exchanges emerged some levers for overcoming these interoperability issues:

-It is necessary to gather and capitalize local innovations (good practices capture)

-These practices must be deployed broadly, whatever culture and maturity of sites (best practices modeling and deployment)

-Finally, a tool to manage the modeling and the diffusion of these practices is required.

This short view of two different cases enables to characterize organizational interoperability problems met by enterprises and administration. The next section focuses on approach to solve these problems and increase interoperability.

Organization

Interoperability design approaches

Systems, as software, process or organization, are not originally interoperable, excepted when they were originally designed as open systems (i.e. quite "universal" solutions, able to be plugged and played with any other system).

It seems therefore necessary that some approaches are required to increase systems interoperability [7]. One of these approaches is focused on the way to design interoperability [8]. As Fig. 2 outlines, it is composed of three levels of abstraction, from a specific view to a generic standpoint:

-Good practices / Good solutions: Good practices are approaches that have been implemented and used in industries while good solutions are those developed as research results and there is not yet real implementation in the companies.

-Design patterns: they can be seen as proven solutions to a problem in a given context. They are also ways to describe best practices and good solutions, and to capture experience in a way that it is possible for others to reuse this experience.

-Design principles: design principles can be defined as a fundamental truth which stands for evidence, and is used as rules to follow to orient design decision-making.

This design approach proposes therefore a "process" to transform a heterogeneous organization into an interoperable system. It generalizes and diffuses broadly the best practices and the accurate solutions, by using patterns or principles. On the particular organizational interoperability challenge, some design principles have been already established. As described by Vallespir [9], two solutions appear: the standardization and the mutual adjustment between practices and processes.

The first one aims at defining standard practices and implementing them everywhere. Thus each enterprise or each business unit has to comply with these standardized best practices (Fig. 3). The other one proposes the mediation of practices coming from different organizational units. Each entity conserves therefore its specificities. Nevertheless, it has to "translate" its working standards and process modeling, so as to be understandable by and interoperable with others (Fig. 4). Thus cases study and literature identified organizational interoperability challenges. Then interoperability design approaches were explained. The following section is now describing a framework able to design and manage this organizational interoperability in enterprises.

Organizational capability and 5 steps ® framework for interoperability design and management

Sharing good practices and processes can improve organizational interoperability, as mentioned in §3, either by standardization, either by adjustment.

However, how can a system be driven towards interoperability? How can these good practices be diffused, with which patterns?

A possible answer is organizational learning. Indeed, management of functional progresses and diffusion of good practices on all concerned entities should be able to raise enterprise interoperability.

This section presents 5Steps®, a framework developing and managing organizational capabilities. First organizational capability is defined. It is the key concept of 5Steps®, and it makes organizational learning processes actionable. Then the mechanisms of this framework are given. They enable the modeling of captured good practices and solutions around capabilities, and their diffusion throughout organization. Finally all these features are analyzed in an organizational interoperability standpoint, and their abilities for helping interoperability design and management are discussed.

The concept of organizational capability

According to Renard and Gilles Saint-Amant [6] and Ibrahima Fall [10] organizational capability can be defined as a "know how to act", a potential of action, resulting from the combination and the coordination of "action levers" (resources, knowledge and competencies) of the organization. This potential can be mobilized through the value flow of the company to perform a specific objective.

Several characteristics emerge from this concept: -like knowledge, capability is systemic [11], i.e. it is broader than the sum of its components.

-An organizational capability cannot be considered separately from the "action process" [12,13]. It constitutes a relevant unit of analysis (the action is indicative of the expression of the capability).

-Finally its behaviour is adaptive and linked to its "path dependency". This means that any capability is the product of a previous capability, and is the result of an active learning, both on individual and organizational levels [14,13].

All these elements are summed up on Fig. 5. The study will now focus on a framework enabling to model capabilities and their progress, and then to deploy and to manage them.

5Steps® Roadmapping

5Steps® [5] is a framework developed by Valeo and MNM Consulting three years ago. Supported by a formalism, the roadmaps, and a software tool, it is now implemented on the whole Valeo Group. It is used to diffuse best practices, to integrate new sites, and to ease interactions between heterogeneous entities.

Moreover, the research and development works around this framework occur in the framework of Pilot2.0, supported by the French National Agency for Research (ANR) since December 2007.

Capturing and modeling functional

domains with roadmaps. 5Steps® puts organizational capability in the core structure of its performancedriven system, as shown in Fig. 6. The roadmap structure uses the bases of organizational capabilities, by crossing in a matrix form "action levers" and the "path dependency" (figured by the maturity levels). The writing of roadmaps is based on the capture of good practices and good solutions. For instance, at Valeo, the good practices are identified and gathered by quality managers and auditors, or during exchanges between operational and functional managers. In parallel good solutions can be also proposed by functional experts to introduce new standards. Then capabilities development is modeled by roadmaps. Roadmaps' writers use the content of this repository which can be reused everywhere according to subjects of capability to develop.

For example, a roadmap for developing the capacity of managing Information Systems in plants was used and implemented in Valeo (cf. Fig. 7).

Diffusing and measuring roadmaps.

5Steps® also proposes a solution to manage these roadmaps. Thus roadmaps implementation, animation and evaluation behave the following scheme:

As the model emphasizes (Fig. 8):

-Roadmap subjects are generated by strategic managers, who identify the interoperability key issues to be solved.

-The roadmaps are then written by functional managers, who organize capabilities following the maturity path of roadmaps. For this purpose they gather best practices and propose good solutions which improve organizational interoperability.

-These roadmaps are transmitted to middle and operational managers, who discuss objectives, in terms of level to reach and delay to respect for level achievement. This step is focused on the choice of organizational units which will have to implement roadmaps. Moreover, this deployment phase is looking for homogeneity, since roadmaps are diffused broadly. For instance, Valeo group modelled 40 roadmaps of 6 different functional networks that it deployed on 210 sites (a site is a production place for on products, a plant able to produce several products). It aims at sharing and making the same standards actionable on all organizational entities.

-Roadmaps are implemented, with the support of a software application which ensures the broad diffusion and enables a self-assessment of deliverables to achieve.

-Roadmap measurement by self-assessment provides some reporting on progress states and some feedbacks for improving roadmap's content. The selfassessment is also reported into scorecards. These data are audited regularly, and become inputs of a Business Intelligence tool (like for instance at Valeo). Measures aggregation and consolidation allow monitoring capabilities progress. The progress status can be compared between organizational entities, giving a clear view of maturity of organizational capabilities among several plants, products branches, functional divisions… -Finally, in a process of continuous improvement, feedbacks are collected about the content and the deployment of roadmaps. New versions of roadmaps can be proposed, to upgrade these interoperability design patterns. Thus, 5Steps® framework proposes a model to continuously increase interoperability design, by taking account of problems met by operational people and by consequently improving the patterns.

Designing and managing interoperability with organizational capability and roadmapping.

As emphasized in its definition (cf. §4.1), organizational capability results from the coordination and the interactions of "action levers", and is the product of an organizational learning process (path dependency). Capability is therefore a progressive learning way to make a set of resources, knowledge and competencies interoperable, in order to use them to achieve a specific objective.

On another hand, a roadmap aims at managing an organizational capability, i.e. several resources, as manager, applications and suppliers, in order to improve their coordination and their communication. The progressive combination of these resources therefore enables to acquire a collective capability, and to develop their interoperability. Thus roadmaps are interoperability "design pattern" (cf. §3), which transform "good practices" and "good solutions" in a model implementable everywhere.

Thus roadmapping can constitute a support for designing organizational interoperability, by following the three abstraction levels of interoperability design approaches presented in §3 (cf. Fig. 9):

-Good practices are collected on operational ground, and good solutions are proposed by functional experts of organizations.

-From this collection, roadmaps are modeled, providing "interoperability design patterns" for organizations.

-Finally, the management method and the tool supporting it give some rules to diffuse, implement, measure and improve these interoperability patterns, as a kind of "design principles". Moreover the roadmapping measurement system provides a new means to assess the interoperability degree of an organization at two levels of precision:

-the local roadmap assessment indicates for each site how resources are coordinated so as to perform the organizational objectives.

-the consolidation and the comparison of different roadmaps indicates how plants, how business units are interoperable in order to provide the same results with the same quality level.

These indicators could therefore represent how much an organization increased its interoperability, and which efforts it has to continue to improve this interoperability.

Discussion

This paper presents organizational capability and roadmapping as tools for helping the design and the management of organizational interoperability:

-practices could be standardized and harmonized, and all organizational entities follow the same way to reach the same interoperability level.

-Semantic interoperability is also improved, Indeed roadmaps can be viewed as such "ontologies" [15,16], by giving a shared and unique description of a functional domain and their development.

-Roadmap measurement can provide new indicators on local coordination of resources and global interoperability of plants, business units… These elements have seemingly positive impacts where they have been implemented. Thus a survey within the Valeo Group [17] shows that 74% of managers using this formalim consider that the roadmapping framework makes explicit and accessible the best practices repositories. Moreover, for 80% of middle management and 75% of operational people, it incites communication and information sharing between plants.

However some limits also appeared in the implementation of the roadmapping:

-The roadmap structure introduces some new heterogeneity problems: this one aims at increasing organizational interoperability by driving and coordinating several « levers », but getting resources working together can oppositely trigger off such a complexity and therefore a bigger risk of heterogeneity.

-Roadmapping is based on a standardization of practices, and can therefore be too « normative ». Even if communication and interactions are improved by the use of these actionable design patterns, each actor observes a limitation of her/his autonomy about his/her way of working and achieving his/her tasks. It is why some negotiation and collaboration loops are planned in the method, but they must be bettered up.

For this last purpose, a current work is made to better adapt these generic roadmaps to the heterogeneity of organizations. It is based on the identification and the determination of some groups of specificities. For instance, a roadmap concerns with only a functional domain. However, some disparities can appear in its deployment, according to some cultural or historical criteria (which bring about some difficulties for understand or have the means to implement the roadmap). To solve this issue, feedbacks must be of course collected and organized.

But self-assessment data can also be analyzed to provide further information, in order to identify a singularity in roadmap deployment on a site, and to find the cause of this singularity (cf. Fig. 10).

Conclusion

This part emphasized that organizations can be driven towards interoperability with the support of organizational learning processes.

Organizational capabilities and roadmaps provide solutions to design and manage this interoperability development, based on two distinct levels:

-At a first stage, organizational capability constitutes a design pattern, which allows combining different "action levers" and driving them on the road towards interoperability.

Thus organizational knowledge and synergy are improved [18].

-At a second stage, roadmaps enable to implement these set of best practices as "interoperability patterns", giving some rules for the animation and the control of interoperability management.

Nevertheless the current implementation ofpresented concepts and framework results rather in a "standardization of practices", as defined by Vallespir in §3. It is why the future works will look for turning the current system into a more flexible system taking account of some adjustment, e.g. by determining some groups of specificities for the deployment of roadmaps.

Fig. 1 :

 1 Fig.1: Analysis of interoperability issues in Valeo group and CG84

Fig. 1 :

 1 Fig.1: Abstraction levels for designing interoperability [8]

Fig. 2 :

 2 Fig.2: Standardization of practices [9]

Fig. 4 :

 4 Fig.4: Adjustment of practices [9]

Fig. 5 .

 5 Fig.5. Organizational capability definition

Fig. 6 .

 6 Fig.6. Roadmap and organizational capability

Fig. 9 .

 9 Fig.9. Model of organizational interoperability design and management with roadmaps

Fig. 10 .

 10 Fig.10. Roadmap's improvement perspectives


Gerardo Rubino

Alan Krinik

Pomona, US G Calpoly

Rubino

The power-dual and the exponential-dual matrices -Applications to the computation of matrix powers and matrix exponentials

Original path

The problem addressed here is to find the solution to a linear system of difference equations or of differential equations.

The approach here described comes from the analysis of Markovian models, mainly queuing models, in discrete time or in continuous time.

At the beginning, we developed a methodology based on the concept of duality in Markov processes proposed by Siegmund and developed in Anderson's book1 , and we used it to derive closed-form expressions of the transient distributions of some basic queueing models (M/M/1, M/M/1/H, M/M/1 and M/M/1/H with "catastrophes", etc.).

Later, we extended this duality idea to arbitrary differential systems, not necessarily coming from stochastic processes. This paper describes new results in this direction. Back to general remarks

Outline

Using the dual to analyze Markovian queues

From a given Markov process X , living on the nonnegative integers, (i) we build a new one, its Siegmund's dual X * , if it exists, which is not always the case.

The transformation is linear (the generators are linearly related), and its main property is that the solutions to both systems are also linearly related, and in the two directions: X → X * and X * → X .

Next, (ii) we find the transient distribution of X * , in general using Uniformization, to move to discrete time and thus analyzing the transient distribution of a Markov chain Y * (typically counting paths by means of lattice path combinatorics tools).

To finish, (iii) we come back at X using the inverse dual (linear) transformation.

In cases with some regularity in the structure, this can produce closed-forms solutions, and it can also work when the dimension of the initial problem is infinite (e.g. for the M/M/1 queue).

Introduction Duality concepts Back to general remarks

Outline

Comments

This procedure is useful when X * is easier to analyze, in a way or another.

As we see, there is an initial continuous time phase, then a discrete time part process, where a transient distribution is needed.

In some cases, we have a discrete time problem for which we look for transients from the beginning (instead of ODEs we have then difference equations).

In this talk we will briefly describe a general approach where transients in continuous time (algebraically speaking, matrix exponentials) or in discrete time (matrix powers) are needed. The former are built using the latter.

To simplify the discussion, we only present here the finite case.

Talk's content The power-dual of a matrix

Let M be a square matrix (of reals, of complex numbers, ...) with dimension n, indexed on {0, 1, . . . , n -1}. The power-dual of M is a new matrix M, with dimension n + 1, indexed on {0, 1, . . . , n -1, n}, defined as follows: for i = 0, 1, . . . , n -1,

M i,j =            n-1 k=i M j,k - n-1 k=i M j-1,k if j ≤ n -1, 1 - n-1 k=0 M i,k when j = n,
and the last row

(when i = n) is ( 0 0 . . . 0 1 ). Example in dimension 2: M = a b c d ; M =   a + b c + d -(a + b) 1 -(c + d) b d -b 1 -d 0 0 1   Back to general remarks Power-duals
Example in dimension 3:

M =   1 -1 2 2 0 -2 1 -3 3   ; M =     2 -2 1 0 1 -3 2 1 2 -4 5 -2 0 0 0 1    
A stochastic matrix:

P =   2/5 0 3/5 1/6 1/2 1/3 0 1/4 3/4   ; P =     1 0 0 0 3/5 7/30 1/6 0 3/5 -4/15 5/12 1/4 0 0 0 1     ;
as we see, in this case the power-dual is not a stochastic matrix; Another stochastic matrix:

P = 2/5 3/5 1/6 5/6 ; P =   1 0 0 3/5 1/15 1/3 0 0 1   ;
here, the power-dual is also a stochastic matrix.

Introduction Duality concepts Back to general remarks

Power-duals

Some properties

Result: if M is the power-dual of matrix M, then the sum of the elements of any row of M is 1, and the last row of M is composed only of 0s except for the last element which is 1.

Inversion Lemma: for 0 ≤ i, j ≤ n -1, we have

M i,j = i k=0 M j,k - i k=0 M j+1,k .
Central result: For any two square matrices A and B with the same dimension, we have AB = B A.

Corollary: for any square matrix M and any nonnegative integer k,

M k = M k .
In words, the power-dual of the kth power of M is the kth power of M's power-dual. The exponential-dual of a matrix

Let M be a square matrix (of reals, of complex numbers, ...) with dimension n, indexed on {0, 1, . . . , n -1}. The exponential-dual of M is a new matrix M, with dimension n + 1 indexed on {0, 1, . . . , n -1, n}, defined as follows: for i < n,

M i,j = n-1 k=i M j,k - n-1 k=i M j-1,k and the last row (i = n) is ( 0 0 . . . 0 ). Example in dimension 2: M = a b c d ; M =   a + b c + d -(a + b) -(c + d) b d -b -d 0 0 0  
We can observe that if we denote by W a square matrix with the same dimension as M, composed of 0s except the last column composed of 1s, then we have M = M -W .

Duality concepts Back to general remarks

Exponential-duals

Example, dimension 3:

M =   1 2 3 -1 2 -3 0 2 -2   ; M =     6 -8 2 0 5 -6 1 0 3 -6 1 2 0 0 0 0    
An infinitesimal generator:

M = -λ λ µ -µ ; M =   0 0 0 λ -(λ + µ) µ 0 0 0   ;
as we see, the exponential-dual is also an infinitesimal generator; Another infinitesimal generator:

M =   -3 1 2 6 -7 1 1 2 -3   ; M =     0 0 0 0 3 -9 5 1 2 -1 -4 3 0 0 0 0     ;
here, the exponential-dual is not an infinitesimal generator.

Talk's content

Introduction Outline

Duality concepts

Power-duals Exponential-duals Back to general remarks Take H = 4, to shorten the slides. Denote Λ = λ + µ + γ. We have

A =       -λ λ 0 0 0 µ + γ -Λ λ 0 0 γ µ -Λ λ 0 γ 0 µ -Λ λ γ 0 0 µ -(µ + γ)      
If we move to discrete time using Uniformization, the computation of e At translates into that of evaluating U k where using the notation p = λ/Λ, q = µ/Λ, r = γ/Λ,

U = I -A/Λ is U =       q + r p 0 0 0 q + r 0 p 0 0 r q 0 p 0 r 0 q 0 p r 0 0 q p      
The powers of U are hard to obtain in closed-form here.

If we use the exponential-dual (plus Uniformization), we must compute the powers of

P =        
1 0 0 0 0 0 p 0 q 0 0 r 0 p 0 q 0 r 0 0 p 0 q r 0 0 0 p q + r r 0 0 0 0 0 1

       
Due to those '1' at the corners, the use of lattice path combinatorics is much easier here than with matrix U of previous slide. That's an example of situation where we can develop the process until its end.

The infinite case can also be solved in the same way.

The formal solution of the ODEs (Chapman-Kolmogorov equations) is given below, using the notation ρ = λ/µ, for the case of the queue starting empty at time 0 (the general case is similar while more complex): In the formula, π j denotes p j (∞) (steady-state), easy to evaluate symbolically.

p j (t) = π j - 2µρ

Continuous-Time Markov Chains, W. J. Anderson, published by Springer in 1991 (out of press).

Duality concepts

Back to general remarks Exponential-duals

Some properties

Observe first that if M is the exponential-dual of matrix M, then the sum of the elements of any row of M is 0, and the last row of M is composed only of 0s. Inversion Lemma: 0 ≤ i, j ≤ n -1, we have

Important result: for any two square matrices A and B with the same dimension, we have AB = B A.

Corollary: for any square matrix M and any nonnegative integer k,

In words, the exponential-dual of the kth power of M is the kth power of M's exponential-dual.

More results:

Let A and B be two square matrices with the same dimension, and α be a scalar. We have

Duality concepts Back to general remarks Exponential-duals

Main result

Let M be an arbitrary square matrix. Using previous properties,

In words, the exponential-dual of the exponential of M is the exponential of the power-dual of M.

This result explains how the evaluation of the power-dual appears when we work on matrix exponentials.

In this short presentation with skipped some other technical results that are needed to prove previous theorem.

Observe the difference: Other general properties:

If P is a stochastic matrix, then the first row of P is (1 0 . . . 0 ). But recall that P needs not be stochastic.

If A is an infinitesimal generator, then the first row of A is (0 0 . . . 0 ). Recall that A needs not be an infinitesimal generator.

Connection with the dual of a Markov process as defined by Siegmund in 1976 and developed in Anderson's book: if A is the infinitesimal generator of Markov process X and if A is also an infinitesimal generator, the associated Markov process X * is the dual of X .

Duality concepts Back to general remarks

Overview

This work generalizes the use of duality in the theory of Markov processes as proposed by Siegmund (see Anderson's book). The dual doesn't always exist.

In previous work we proposed to define what we call here exponential-dual, adopting a purely algebraic approach, when we realized that the stochastic framework was not necessary for the interest we had in the concept. Many proofs needed to be rewritten.

Here, we decompose explicitly the cases of discrete time and continuous time, and exhibit their connections, making the construction more complete and consistent.

On dual's existence (Markov processes)

This concerns the historical origin of this work, in the stochastic processes area.

For instance, consider an homogeneous "ring" illustrated here for 3 states:

Whatever the size of the ring and the value of λ, the associated Markov process has no dual (for those familiar with duality in Markov processes, the problem is the monotonicity condition impossible to satisfy here).

Consider now the general Markov process with 3 states:

Here, the dual exists iff

Duality concepts Back to general remarks

Zoomming on an example

Consider here the M/M/1/H queue plus "catastrophes" (or failures) as shown in the following Markovian graph: Thanks for your time.


G Rubino

The talk will consist of the following points:

1 -a brief refresher on the Splitting method, 2 -the flow model,

-the Creation and Destruction Processes (CP and DP)

and our multilevel extensions, 4 -our Splitting method, designed to work also with dependent components.

• This work is about analyzing dependability properties of complex systems, by means of Monte Carlo techniques. • More specifically, we consider transportation systems where some fluid is sent through a network from a source to a destination, going from node to node through directional links having some capacity. • The flow can make a fork at a given node, or a join.

• The links capacities are random variables, and we call failures the events "moving from the standard (nominal) maximum capacity to a smaller value". They are usually supposed to be independent of each other. They are also called the model's components. • The problem is a static one (no time variable). At a given time, where the system is considered, the links have some capacities, sampled from the capacities' distributions.

• Because of the links capacities, there is a maximal amount of flow that can be transported by the network, MF, from source to destination, which is a random variable.

• We are given a demand value, a minimal amount of fluid we need to be able to transport. We want to evaluate the number ζ = P{MF < d}, an important dependability metric in this context, an unavailability one. • In many situations, and the one considered here, the event {MF < d} is rare, that is, ζ ≪ 1. How to estimate it is the topic of this talk. • The main families of techniques to deal with rare events are Importance Sampling and Splitting (also appearing under other different names), plus some other special ones such as Recursive Variance Reductions. • In this paper we deal with Splitting, which is specifically designed for problems defined on stochastic processes (here, the setting is static). • Because of that, we must first transform the static model into a dynamic one.

• After presenting the transformation, we will describe a particular implementation of the Splitting approach that gives very good results. • Then, we will relax the independence assumption (between capacities), and show how the proposed method can also deal with this extension to the original model.

The method was actually designed to deal with such an extension. • To deal with the dependent components case, we use Marshall-Olkin copulas.

• We have a stochastic process X = {X(t)} t≥0 living in some space S, and two subsets of states S A and S B . Assume them disjoint, with X(0) ∈ S A , S B . • Let us denote τ A (resp. τ B ) the hitting times of X in S A (resp. in S B ).

• We are interesting in evaluating ζ = P{τ B < τ A }, and we consider the case where τ B ≪ τ A with high probability.

X(0) S S A S B G. Rubino 1-Splitting Starting idea X(0) S S A S B L * S *
• First idea: simulate several independent copies of X. When one of them gets close to S B at some time τ, make several copies of it, starting at τ. • For that purpose, define an intermediate set S * ⊃ S B , with S * ∩ S A = / 0, with border L * , and split X when it touches L * .

1-Splitting

Starting idea

X(0) S S A S B L 1 L 2 L 3 S 1 S 2 S 3
• The procedure can be applied recursively, cloning trajectories that cross the borders L 1 , L 2 , L 3 of a sequence of subspaces S 1 ⊃ S 2 ⊃ S 3 ⊃ S B . • Second (possible) idea: when a trajectory that crossed border L i comes back at L i before reaching L i+1 , kill it (the RESTART variation).

• This is widely used in queuing models for performance evaluation of systems, and in dependability models. • In some cases we want to evaluate P(τ B < T) where T is an a.s. finite stopping time.

• In other cases, this type of probabilities help in analyzing other targets.

Example:

• suppose that the states in S B are "bad" states, and that τ A = τ 0 , the return time to the initial state. • τ B is the system's life-time.

• We are interested in evaluating the MMTF of the system, the Mean Time To Failure, = E(τ B ). In the highly reliable case, MTTF≫ 1. • It can be shown that

E(τ B ) = E min(τ 0 , τ B ) P(τ B < τ 0 ) .
The numerator is easy to estimate; the estimation of the denominator is a typical rare event problem. • In the models we are dealing with, h(X(t)) will tend to stay low, to come back quickly to 0 (or to negative values, depending on how we defined h), and it will be rare to observe it getting close to ℓ.

1-Splitting

• In a queueing system modeled by some process, a typical example is h(x) = # of customers in some queue, when system's state is x. • In a dependability model, a typical example is h(x) = # of failed components, when system's state is x. • In both cases, h(x) measures somehow how close we are to the bad states (too many customers in the queue, or too many components down in the dependability case).

1-Splitting

Crude Monte Carlo t h(X(t))

0 ℓ ζ = 1 7
We estimate ζ by dividing the number of trajectories reaching ℓ by the total number of copies simulated.

G. Rubino τ A τ B τ 1 τ 2 a trajectory of process h(X(t)) t h(X(t)) 0 ℓ ℓ 1 ℓ 2
• Now, suppose that the set of values of h is partitioned using thresholds ℓ 1 , ℓ 2 , . . . , defining an associated sequence of embedded subspaces of S.

• In the figure, x ∈ S 1 ⇐⇒ h(x) ≥ ℓ 1 , x ∈ S 2 ⇐⇒ h(x) ≥ ℓ 2 , plus, for instance, x ∈ S B ⇐⇒ h(x) ≥ ℓ and x ∈ S A ⇐⇒ h(x) = 0. 1-Splitting Thresholds τ 1 t h(X(t)) 0 ℓ 1 ℓ 2 ℓ
• Crossing the levels ℓ 1 , ℓ 2 , . . . is equivalent to crossing the borders L 1 , L 2 , . . . defined on S (in the previous description). • Now, each time a trajectory reaches level or threshold ℓ i (at time τ i ), we clone it.

τ 1 τ 2 t h(X(t)) 0 ℓ 1 ℓ 2 ℓ
• Crossing the levels ℓ 1 , ℓ 2 , . . . is equivalent to crossing the borders L 1 , L 2 , . . . defined on S (in the previous description). • Now, each time a trajectory reaches level or threshold ℓ i (at time τ i ), we clone it.

τ B τ 1 τ 2 t h(X(t)) 0 ℓ 1 ℓ 2 ℓ
• Crossing the levels ℓ 1 , ℓ 2 , . . . is equivalent to crossing the borders L 1 , L 2 , . . . defined on S (in the previous description). • Now, each time a trajectory reaches level or threshold ℓ i (at time τ i ), we clone it.

τ B τ 1 τ 2 t h(X(t)) 0 ℓ 1 ℓ 2 ℓ • D i = event {τ i < τ A }. • If we have k thresholds, D k ⊂ D k-1 ⊂ • • • ⊂ D 2 ⊂ D 1 G. Rubino 18 / 63
1-Splitting Probabilitiy of embedded events

• Assume we have k thresholds ℓ 1 , ℓ 2 , . . . , ℓ k = ℓ. The event of interest is D k .

• Easy to check that we have

P{D k } = P{D k | D k-1 } p k P{D k-1 | D k-2 } p k-1 • • • P{D 2 | D 1 } p 2 P{D 1 } p 1 = ζ = k ∏ h=1 p h
• At the heart of Splitting, we have the following result:

ζ = k ∏ h=1 p h → E{ ζ } = ζ ,
where p h is the standard estimator of p h . • That is, we have an unbiased estimator of the target built from unbiased (crude) estimators associated with a single threshold case (which is built such that reaching the threshold is not rare).

τ B τ 1 τ 2 t h(X(t)) 0 ℓ 1 ℓ 2 ℓ 3 p p p ζ ζ = p 1 p 2 p 3 G. Rubino τ 1 t h(X(t)) 0 ℓ 1 ℓ 2 ℓ 3 # of successful trajectories = 1 total # of trajectories = 3 p 1 p 1 = 1 3 G. Rubino τ B τ 1 τ 2 t h(X(t)) 0 ℓ 1 ℓ 2 ℓ 3 p p p ζ ζ = 1 3 1 3 1 3 = 1 9 G. Rubino 22 / 1-Splitting Estimation t h(X(t)) 0 ℓ 1 ℓ 2 ℓ 3 N N 1 N N R 1 R R ζ = R 1 N 0 R 2 N 1 R 3 N 2 • • • R k N k-1 ,
where k is the # of thresholds, N i is the # of clones built after reaching level ℓ i , and R j is the # of clones having reached level ℓ j .

• We want to have (to observe) R k > 0 (enough times). Otherwise, the procedure will return 0, or a too small fraction, typical rare event issue. • Choosing how many levels to use (parameter k), positioning the levels, selecting the control parameters N 0 , . . . ,

N k-1 , R 1 , . . . , R k is complicated.
There is a large # of possible configurations. • Computing the variance of the Splitting estimator is too hard. So, there is no theorem allowing to have a theoretical comparison of the different possible configurations. • There is also no formal result leading to optimal configurations of the procedure (sufficient conditions), except in simple particular cases. • What people have is more or less simple models where it is possible to find optimal values. Then, they are used as heuristic guidelines to choose all those hyper-parameters in the variant we are running.

2-Flows

G. Rubino

2-Flows

Classical deterministic model

X 1 X 2 X 3 X 4 X 5 . . . . . . . . . . . . s t G = (V , E , X)   
V : set of n nodes, with a source (blue) and a terminal (green); E : set of m arcs; X : vector of arc capacities, (X 1 , . . . , X m ) ≥ (0, . . . , 0).

• Flow: a function on the arcs, satisfying local balance at the interior (red) nodes.

• Result: the sum of the flows on the arcs leaving s is equal to the sum of the flows of the arcs arriving at t, and it is called the flow value. • V(X): the maximum possible flow value, underlying the dependence on the capacities. Algorithms to compute it: Ford-Fulkerson, Edmonds-Karp, etc.

X 1 X 2 X 3 X 4 X 5 . . . . . . . . . . . . s t
• The capacity X i of arc i is now a random variable (links can fail).

• X = (X 1 , . . . , X m ) is a random vector with values in some S = (S 1 , . . . , S m ).

• We are given a demand d, and the question is to know if the transportation network can transport at least that quantity of fluid. • The target of the analysis is the estimation of ζ = P{V(X) < d} .

• If the network is large, ζ will be difficult or impossible to compute (NP hard territory). If the network is highly reliable (ζ ≪ 1), Standard (Crude) Monte Carlo will suffer if applied naively.

2-Flows

Stochastic extension

Our goals:

• To focus on the case of ζ ≪ 1 (the rare event case).

• To convert this static model into a dynamic one (a stochastic process) and apply Splitting to it, for the rareness issue. • To find a Splitting procedure also able to deal with the case of dependent arc failures (this led to use Marshall-Olkin copulas to model the dependencies).

3-From static to dynamic

The Multilevel Creation Process

We build now a dynamics on the same structure. Basic principles:

• At time t = 0, all links are failed (all the capacities are 0).

• At t = 0, we start m artificial "repairing processes" in parallel, for all links, where link i is repaired after an Exponentially distributed random delay τ i . • The m r.v.s τ 1 , . . . , τ m are independent.

• Observe that, now, X = {X(t), t ≥ 0}.

Our goal:

We want that, in the new dynamic system, P(V(X(1)) < d) = ζ , where ζ is the target defined in the initial static context.

t 0 = 0 t n = 1 t 1 t 2 t n-1 • • • • • • p 1 p 2 p n p 0 t X i (t) λ
We will choose the rate λ , and the points t 1 , t 2 , . . . , t n-1 such that n, where t 0 = 0 and t n = 1, and P(τ i > 1) = p 0 .

P(t k-1 < τ i ≤ t k ) = p k , k = 1, . . . ,
t 0 = 0 t n = 1 t 1 t 2 t n-1 • • • • • • p 1 p 2 p n p 0 t τ i X i (t) . . . M 1 M 2 M n ◮ If 0 < τ i ≤ t 1 X i (t) : 0 → M 1 and keep this value forever, If t 1 < τ i ≤ t 2 X i (t) : 0 → M 2 and keep this value forever, . . . If t n-1 < τ i ≤ 1 X i (t) : 0 → M n and keep this value forever, If 1 < τ i keep X i (t) = 0 forever. t 0 = 0 t n = 1 t 1 t 2 t n-1 • • • • • • p 1 p 2 p n p 0 t τ i X i (t) . . . M 1 M 2 M n If 0 < τ i ≤ t 1 X i (t) : 0 → M 1 and keep this value forever, ◮ If t 1 < τ i ≤ t 2 X i (t) : 0 → M 2 and keep this value forever, . . . If t n-1 < τ i ≤ 1 X i (t) : 0 → M n and keep this value forever, If 1 < τ i keep X i (t) = 0 forever. t 0 = 0 t n = 1 t 1 t 2 t n-1 • • • • • • p 1 p 2 p n p 0 t τ i X i (t) . . . M 1 M 2 M n If 0 < τ i ≤ t 1 X i (t) : 0 → M 1 and keep this value forever, If t 1 < τ i ≤ t 2 X i (t) : 0 → M 2
and keep this value forever, . . .

◮ If t n-1 < τ i ≤ 1 X i (t) : 0 → M n and keep this value forever, If 1 < τ i keep X i (t) = 0 forever. t 0 = 0 t n = 1 t 1 t 2 t n-1 • • • • • • p 1 p 2 p n p 0 t τ i → X i (t) . . . M 1 M 2 M n If 0 < τ i ≤ t 1 X i (t) : 0 → M 1 and keep this value forever, If t 1 < τ i ≤ t 2 X i (t) : 0 → M 2 and keep this value forever, . . . If t n-1 < τ i ≤ 1 X i (t) : 0 → M n and keep this value forever, ◮ If 1 < τ i keep X i (t) = 0 forever.

3-From static to dynamic

Values to use

t 0 = 0 t n = 1 t 1 t 2 t n-1 • • • • • • p 1 p 2 p n p 0 t X i (t) λ
After some algebra, we can take λ = -ln(p 0 )

t k = ln(1 -p 1 -p 2 -. . . -p k ) ln(p 0 ) , k = 1, . . . , n -1. G. Rubino

3-From static to dynamic

On the implementation

• When building trajectories, we need to take the τ i s from smallest to largest.

• In other words, we must work with the order statistic of the sequence of repair times, τ (1) , . . . , τ (m) . • For this task, it's more efficient to use the properties of the Exponential, in particular of the min of independent Exponential r.v.s.

• We know that • τ (1) is Exponential with rate Λ = λ 1 + • • • + λ m , • argmin{τ i } = λ i /Λ, • knowing that τ (1) = τ h , τ (2) is Exponential with rate Λ -λ h , • etc. t t = 1 t = 0 d V max V(X(t))
• Following the "repairs", V(X) grows by jumps.

• V max is the max possible value (for instance, the value corresponding to putting all the links at their max capacities).

• The point is to see if V(X(1)) is above or below d.

4-Splitting on the artificial dynamic process

Splitting on the Multilevel Creation Process

t 2 2 3 3 4 6 ℓ 1 ℓ 2 t = 1 t = 0 d V max V(X(t))
• An appropriate application of Splitting requires, once configured,

• to define the thresholds on the time axis,

• then to clone the paths that reach next level before crossing the horizontal line "y = d".

• The resulting estimation (in the picture

) is ζ = 2 2 3 4 3 6 = 0.375
Recall that our ultimate goal is to be able to deal with dependent components.

• We must choose how to model dependencies. One of the easiest ways is to use Marshall-Olkin copulas, based on the idea that failures happen in shocks. • A shock is the simultaneous failure of a subset of components; it's just a way to use data reporting, for instance, correlations between individual failure events. • The process of going from correlation data to shocks and their associated failure rates is out of the scope of the talk (see the references). • Given these remarks and after observing that the Creation Process isn't well adapted to this setting, we made the same kind of multi-level extension to the dual Destruction Process (DP). • In a DP, the m components start operational, and they fail one after the other according to a similar dynamic process as before. • But moving Splitting to the DP context is not straightforward and we had to make some important changes.

t 0 = 0 t n = 1 t 1 t 2 t n-1 • • • • • • p 1 p n p n-1 p 0 t X i (t) λ
We will choose the rate λ , and the points t 1 , t 2 , . . . , t n-1 such that P(t k < τ i ≤ t k+1 ) = p k , k = 0, . . . , n -1, where t 0 = 0 and t n = 1, and P(τ i > 1) = p n .

t 0 = 0 t n = 1 t 1 t 2 t n-1 • • • • • • p 1 p n p n-1 p 0 t τ i X i (t) . . . M 1 M n M n-1 ◮ If 0 < τ i ≤ t 1 X i (t) : M n → 0 and keep this value forever, If t 1 < τ i ≤ t 2 X i (t) : M n → M 1 and keep this value forever, . . . If t n-1 < τ i ≤ 1 X i (t) : M n → M n-1 and keep this value forever, If 1 < τ i keep X i (t) = M n forever. t 0 = 0 t n = 1 t 1 t 2 t n-1 • • • • • • p 1 p n p n-1 p 0 t X i (t) λ Values: λ = -ln(p n ), t k = ln(1 -p 0 -p 1 -. . . -p k-1 ) ln(p n ) , k = 1, . . . , n -1.
• In the talk, I omitted details about the implementation of the Splitting algorithms. • In particular, the hyper-parameters issues are not described, and we don't discuss the different versions of Splitting that can be used. In the paper, we followed some guidelines of this type, and used pilot runs for fine tuning them, before simulating. • Another important issue in this area is variance control, because we lack results concerning the variance of the different versions of the method. As said before, we only have optimality results on pretty simple models providing guidelines on the design of the procedures. • Another important practical issue is data leading to the shock-based model for handling dependencies. There is a reference at the end where we explored this issue with some mathematical details about the technical side.

• From our viewpoint, the most interesting aspect of this proposal is the fact that it fits with the problem of relaxing the usual independent components assumption in the models. • The other point to underline is the good behavior of the method when the failure of the system becomes very rare. • It will be useful to try Importance Sampling on this problem, perhaps the so-called Zero Variance sub-family, for comparison purposes.

• M. Lomonosov. "On Monte Carlo estimates in network reliability". In: Prob. in Eng. and Informational Sciences 8 (1994), pp. 245-264.

The Creation Process is defined in this paper.

  • P. Glasserman et al. "Splitting for Rare Event Simulation: Analysis of Simple Cases". In: Proceedings of the 1996 Winter Simulation Conference. San Diego, California: IEEE Computer Society Press, 1996, pp. 302-308. • M. J. J. Garvels. "The Splitting Method in Rare Event Simulation". PhD thesis. Faculty of mathematical Science, University of Twente, The Netherlands, 2000. • M. J. J. Garvels, D. P. Kroese, and J.-K. C. W. Van Ommeren. "On the Importance Function in Splitting Simulation". In: European Transactions on Telecommunications 13.4 (2002), pp. 363-371. • M. Villén-Altamirano and J. Villén-Altamirano. "On the efficiency of RESTART for multidimensional state systems". In: ACM Transactions on Modeling and Computer Simulation 16 (3 July 2006), pp. 251-279. • G. Rubino and B. Tuffin (editors and co-authors). Rare Event Simulation Using Monte Carlo Methods. Wiley, 2009.

  • Héctor Cancela, Leslie Murray, and Gerardo Rubino. "Efficient Estimation of Stochastic Flow Network Reliability". In: IEEE Transactions on Reliability 68, 3 (Sep. 2019), pp. 954-970. This paper proposes the Multilevel extension of the initial Creation Process and the associated Splitting approach to the flow problem described here. • O. Matus, E. Moreno, J. Barrera and G. Rubino. "On the Marshall-Olkin Copula Model for Network Reliability Under Dependent Failures". In: IEEE Transactions on Reliability 68, 2 (June 2019), pp. 451-461. This paper illustrates the use of the Marshall-Olkin copulas in the area and the benefits in accuracy this tool offers. • Héctor Cancela, Leslie Murray, and Gerardo Rubino. "Reliability Estimation for Stochastic Flow Networks with Dependent Arcs". In: IEEE Transactions on Reliability (2022), to appear (the revised version was accepted this week). paper corresponds to the talk. It has some numerical results on a couple of models. • Zdravko I. Botev, Pierre L'Ecuyer, Gerardo Rubino, Richard Simard, Bruno Tuffin. "Static Network Reliability Estimation via Generalized Splitting". In: INFORMS J. of Computing (2013), 25, 1.

	This

A related work on applying Splitting to a more classical network reliability problem.

G. Rubino

G. Rubino 27 / 63

3-From static to dynamicG. Rubino

2-Flows

Capacities Assume (only for the presentation) that

• the capacities X 1 , . . . , X n are discrete i.i.d.

• with P(X i = M j ) = p j , j = 1, . . . , n (homogeneous situation),

• M 1 > M 2 > . . . > M n > 0,

• plus P(X i = 0) = p 0 .

We say that • if X i = 0, the link is failed,

• if M 2 ≥ X i ≥ M n , the link is partially working,

• if X i = M 1 , the link is fully operational.

• The homogeneous assumption is just for simplicity in the presentation.

• Everything here also works when each X i has a different distribution. The notation will need to be (M i,j ), (p i,j ), where i = 1, . . . , m and for component i, j = 0, 1, 2, . . . , n i , with 1

• The trajectories crossing level d before t = 1 will be above d at t = 1 (this is the typical case). • After points such as (1) and (2), it is then useless to continue the paths.

• Crude Monte Carlo estimation of ζ is the ratio between the # of red points and the total # of simulated trajectories.

• In the highly reliable case, the trajectories will strongly tend to go up (repairs are fast). • Only very few of them will cross the t = 1 border below d (building red points).

• A way of making more efficient the process would be to "stimulate" in some way the trajectories to go to the right (for instance, slowing down the repairs).

G. Rubino

Notation on capacities for the Destruction Process Now,

• If X i = 0, arc i is totally failed.

All links start fully up (value M n in the homogeneous case), and link i will fail at time τ i , assumed to be Exponentially distributed.

We call failure the move from M n to some M k , k ≤ n -1, or to 0.

G. Rubino

4-Splitting on the artificial dynamic process

Implementing the Multilevel Destruction Process

M n → M 1 and keep this value forever, . . .

• Following the "failures", V(X) decreases by jumps.

• V max is the max possible value (for instance, the value corresponding to putting all the links at their max capacities).

• The trajectories crossing level d before t = 1 will be below d at t = 1 (this is the typical case). • After red points it is then useless to continue the paths.

• Crude Monte Carlo estimation of ζ is the ratio between the # of red points and the total # of simulated trajectories. Here,

• In the highly reliable case, the trajectories will rarely go down.

• Only very few of them will cross the "y = d" line before t = 1 (building red points).

• A way of making more efficient the process would be to "stimulate" in some way the trajectories to go down.

G. Rubino

Bad application of the Splitting idea

• All this means that a similar use of the Splitting idea shown before doesn't work.

• Using thresholds on the time axis will push the clones to the right, and we want them to go down.

G. Rubino

• The idea is to put the thresholds on the vertical axis, on the V(X(t)) values,

• and to clone the trajectories from the crossing points when they happen before t = 1.


Jean-Pierre Lefort
email: jeanpierre970@yahoo.fr

John Renouf
email: johnrenouf@orpheusinternet.co.uk

Guzel Danukalova
email: danukalova@ufaras.ru

Jean-Pierre Lefort

J-P Lefort

Transit boosting of trans-Channel katabatic winds by depressions associated with west European storms tracks: impact on Pleistocene loess deposits

Keywords: Westernmost Europe, British-Irish Ice Sheet, katabatic wind, Channel storms track, loess transfer

Onshore and offshore sedimentological, geochemical, geomorphological, paleontological and geochronological studies of loess deposits located under and around the English Channel revealed that they were transported by katabatic winds generated by the British-Irish Ice Sheet. Katabatic winds, which are low-altitude wind flows, were able to jump over the low southern British hills but were stopped by the higher Brittany and Normandy hills. This regional topography is interrupted by a north-south corridor linking the northern and southern shores of Brittany where loess propagated down to the mouth of Loire River. This long transit shows that the total distance travelled by the katabatic wind was around 750 kilometres, which represents an unusual distance for the propagation of this wind under continental conditions.

Introduction

There are currently three major solutions to explain the collection, transfer and deposit of the loess particles during the Upper Pleistocene around the English Channel. All three models involve different modes of transport by the wind.

-The first one, based on the "General circulation Model" involves the long-standing eastward atmospheric circulation above Western Europe [START_REF] Rousseau | Link between European and North-Atlantic abrupt climate changes over the last glaciation[END_REF][START_REF] Sima | Imprint of North-Atlantic abrupt climate changes on Western European loess deposits as viewed in a dust emission model[END_REF][START_REF] Rousseau | North Atlantic abrupt climatic events of the last glacial period recorded in Ukrainian loess deposits[END_REF] and use numerical modelling to investigate the relationship between the North-Atlantic abrupt climate changes and the environmental variations revealed by loess profiles. Unfortunately, this model does not incorporate the most recent onshore or offshore data from Brittany.

-The second one or "Storm Model" suggests that the transport of loess particles was associated with an eastward drift of periodic storms passing above the Western English Channel [START_REF] Antoine | Rapid and cyclic aeolian deposition during the Last Glacial in European loess: a high-resolution record from Nussloch Germany[END_REF]. It partly used some onshore data [START_REF] Lautridou | Les minéraux lourds du loess du pays de Caux: un nouveau critère stratigraphique[END_REF][START_REF] Lautridou | Le cycle glaciaire Pleistocène en Europe du Nord-Ouest et plus particulièrement en Normandie[END_REF][START_REF] Antoine | The loess and coversands of northern France and southern England[END_REF] but does not explain why the transfer of the particles transported by the anticlockwise vortex generated by the Downloaded from http://pubs.geoscienceworld.org/jgs/article-pdf/doi/10.1144/jgs2021-078/5433607/jgs2021-078.pdf by CNRS_INSU user low-pressure centre was effective in the north-south direction but not in the south-north direction.

Neither of these two models takes into account of the close proximity of the British-Irish Ice Sheet.

-Differing from these two models, the "katabatic Model" [START_REF] Lefort | Origin and emplacement of the loess deposited in Northern Brittany and under the English Channel[END_REF]2019) is based on field evidences alone and considers, for the first time, the past wind field not just in two dimensions (2D) but also in three (3D). This model incorporates different onshore and offshore studies such as: 1/the contribution of topography, 2/the rare earth elements, 3/ the loess fauna, 4/ the orientation of loess cliffs, 5/ the contribution of Icelandic basalts, 6/ the orientation of Neanderthals dwelling, 7/ the granulometry of loess, 8/ the distribution of heavy and flaky minerals and 9/ the thickness of the initial wind field; it is probably very close to the past reality. However, the problem of the distance of propagation of the katabatic winds remains to be discussed to firmly establish this model; this is the main purpose of this paper. Before discussing the problems of interference between the katabatic winds and the depression centres, it is necessary to summarize a certain number of results obtained at a regional scale for clarity.

Palaeometeorological backgrounds

The katabatic winds in the peri-Channel context Katabatic winds, which are also known as "gravity" or "down slope" winds, originate over a glacier or an ice sheet. In Antarctica [START_REF] Parish | The role of katabatic winds on the Antarctic surface wind regime[END_REF], wind speeds above 10 m/s are commonly recorded; they result from the drainage of the chilled air seaward from the interior ice cap [START_REF] Rubin | Antarctic meteorology[END_REF]. Two parameters are of particular interest and need to be commented in greater detail.

-All the available data collected in Brittany and Normandy indicate a British or Channel origin for the winds blowing from the North and Northwest [START_REF] Lautridou | Les minéraux lourds du loess du pays de Caux: un nouveau critère stratigraphique[END_REF][START_REF] Lautridou | Le cycle glaciaire Pleistocène en Europe du Nord-Ouest et plus particulièrement en Normandie[END_REF][START_REF] Antoine | The loess and coversands of northern France and southern England[END_REF][START_REF] Lefort | CNRS_INSU user ACCEPTED MANUSCRIPT / CLEAN COPY stands of the English Channel[END_REF][START_REF] Lefort | Impact of katabatic winds on the on the environment of Neanderthals during the middle Palaeolithic in westernmost Europe[END_REF] (Figure 1). It is certain that the wind directions based on the heavy minerals are not very accurate since the sources of these particles are often spread over a large longitudinal distance. By contrast, the orientation of the cliffs which collected the aeolian dust provides [START_REF] Lefort | CNRS_INSU user ACCEPTED MANUSCRIPT / CLEAN COPY stands of the English Channel[END_REF] more accurate wind directions since the south, west and east facing cliffs are generally free of important loess thicknesses [START_REF] Lefort | Impact of katabatic winds on the on the environment of Neanderthals during the middle Palaeolithic in westernmost Europe[END_REF]). The more useful criterion for the determination of the wind direction is certainly that of granulometric study [START_REF] Bigot | Essai de modélisation de l'apport loessique en Bretagne (France)[END_REF]. The general direction given by the drift of the Icelandic ice-rafts is also useful but a little bit more speculative [START_REF] Lefort | Les basalts 'islandais' de l'ile d'Ouessant: comment sont-ils parvenus sur l'ile? Une histoire compliquée d'age Pleistocène[END_REF].

Figure 1 -The vertical thickness of a katabatic flow over a moderate slope is usually limited between 200 to 220 metres and the maximum velocity concentrates at about 25 m above the ice sheet [START_REF] Renfrew | The dynamics of idealized katabatic flow over a moderate slope and ice shelf[END_REF]. In Brittany, the difference in altitude between the upper and lower limits of the initial loess deposits recorded offshore and onshore is always slightly lower or close to these thicknesses [START_REF] Lefort | CNRS_INSU user ACCEPTED MANUSCRIPT / CLEAN COPY stands of the English Channel[END_REF] (see the discussion showing that loess erosion was very limited in this paper) and demonstrates that loess of northern Brittany was transported by low and not by high altitude winds. Another consequence of the limited thickness of the "heavy" katabatic flow is that such winds were not able to override high grounds higher than 220 metres. This is in accordance with the observation that there is no loess south of the two hilly areas of Downloaded from http://pubs.geoscienceworld.org/jgs/article-pdf/doi/10.1144/jgs2021-078/5433607/jgs2021-078.pdf by CNRS_INSU user Brittany (mean altitude 240 m) and Normandy hills (mean altitude 254 m). Furthermore, between these two hilly areas an onshore corridor oriented in a North-South direction (Figure 2) is linking the English Channel to the Bay of Biscay. Because the altitudes within this corridor are always less than 150 m, katabatic winds were able to propagate southwards down to the mouth of the Loire River. In the south of England (except in the SW Peninsula), katabatic winds were able to override the southern English Hills characterized by an approximate difference of less than hundred metres respect with the outwash plain. Because the strength of the katabatic winds was decreasing in a southward direction, loess was deposited on the lee side of the hills in southern England and on the windward side in northern Brittany [START_REF] Lefort | Impact of katabatic winds on the on the environment of Neanderthals during the middle Palaeolithic in westernmost Europe[END_REF].

Previous discussions on the distances reached by katabatic winds

-Onshore, it is often considered that the propagation of katabatic winds is very limited. Long distance transport of dust particles by katabatic winds was first proposed for the Ukraine [START_REF] Tutkovsky | Fossil deserts of the Northern hemisphere[END_REF] in an area located at 400 kilometres from the Fennoscandian Ice sheet, but this assumption has been challenged. In Alaska, located close to an ice sheet but in a mountainous area, loess has been transported over a distance of 80 kilometres [START_REF] Thorson | Eolian deflation by ancient katabatic winds: A late Quaternary example from the north Alaska Ranges[END_REF]. The assumption that katabatic winds may not travel over long distances and are limited to a displacement of 20 to 100 kilometers (Adolph and Wendler, 1995) is often accepted but is contradicted by satellite observations [START_REF] Bromwich | Satellite observations of katabatic-wind Propagation for Great Distances across the Ross Ice shelf[END_REF] since NASA infrared satellite photos demonstrate that katabatic winds may propagate over distances of more than 1000 km above the sea, during 14% of the time.

Downloaded from http://pubs.geoscienceworld.org/jgs/article-pdf/doi/10.1144/jgs2021-078/5433607/jgs2021-078.pdf by CNRS_INSU user ACCEPTED MANUSCRIPT / CLEAN COPY -If we incorporate the results of the different studies dedicated to the heavy and flaky minerals in England and Brittany [START_REF] Lefort | CNRS_INSU user ACCEPTED MANUSCRIPT / CLEAN COPY stands of the English Channel[END_REF] we can estimate that the maximum distance reached by the katabatic winds is close to 750 km, which is in the range of the distances observed by [START_REF] Bromwich | Satellite observations of katabatic-wind Propagation for Great Distances across the Ross Ice shelf[END_REF]. However, the distance reached by the katabatic winds above the sea cannot be fully compared with the distance of propagation measured between the top of the English-Irish ice sheet [START_REF] Boulton | Glaciology of the British Isles Ice Sheet during the last glacial cycle: form, flow and lobes[END_REF] and the mouth of the Loire river where a very limited migration of augite (the southernmost heavy mineral marker) can be still observed [START_REF] Lefort | CNRS_INSU user ACCEPTED MANUSCRIPT / CLEAN COPY stands of the English Channel[END_REF][START_REF] Lefort | Impact of katabatic winds on the on the environment of Neanderthals during the middle Palaeolithic in westernmost Europe[END_REF], since the north-south corridor funnelling the katabatic winds is not completely flat like the sea but affected by small topographic rugosities (Figure 2).

Figure 2

Effects of low pressure centres on katabatic winds.

-The classical descriptions of the genesis of katabatic winds always insist on the opposition between two different meteorological environments. On top of the ice sheet, the air is cold, dense and dry; by contrast above the sea (where all the katabatic winds finish in the Antarctic) the air is warmer, less dense and more humid. These differences are at the origin of two different air pressures. This pressure gradient shows that the difference in altitude between the top of the ice sheet and the sea level is not the only parameter responsible for the down-slope katabatic flow.

- [START_REF] Thorson | Eolian deflation by ancient katabatic winds: A late Quaternary example from the north Alaska Ranges[END_REF] also noted that the approach of a lowpressure centre towards a glacial plateau accelerates the velocity of the katabatic winds and generates particularly strong and gusty Downloaded from http://pubs.geoscienceworld.org/jgs/article-pdf/doi/10.1144/jgs2021-078/5433607/jgs2021-078.pdf by CNRS_INSU user ACCEPTED MANUSCRIPT / CLEAN COPY winds. The relationships between these two types of meteorological phenomena have been recently studied in greater detail in Greenland and Antarctica. In Southeast Greenland, the results of a three-dimensional model associated with a realistic case depict the basic mechanisms for the interaction between katabatic winds and an offshore cyclone. This study also suggests a close relationship between the occurrence of katabatic winds and the generation of mesoscale vortices [START_REF] Klein | Interaction of katabatic winds and mesocyclones near the eastern coast of Greenland[END_REF]. In Antarctica, the strong winds are a feature of the extended winter season and involve a complex interaction between the downslope buoyancy forcing and the pressure gradient force of cyclones. Around the coast of East Antarctica a significant majority of strong winds are associated with the enhancement of the downslope katabatic flow by the broadscale circulation, involving a deepening of pressure off the coast and an increase of pressure inland. Comparison with high-resolution numerical simulations shows that the wind field analysis is able to capture the large-scale meteorological features and the associated enhancement of the katabatic flow but also underestimates the observed wind speed if it is strongly influenced by local topographical conditions [START_REF] Turner | Strong wind events in the Antarctic[END_REF]). This study is in agreement with observations made by [START_REF] Parish | The role of katabatic winds on the Antarctic surface wind regime[END_REF] and was confirmed by [START_REF] Wenta | The Atmospheric Boundary Layer and Surface Conditions during Katabatic Wind Events over the Terra Nova Bay Polynya[END_REF].

Discussion

-There are so many similarities with the series of data collected in and around the English Channel that an application of this mechanism to Westernmost Europe during the Upper Pleistocene is more than likely. As shown for the Upper Saalian glacial episode, some of the loess particles were transported by north-south Downloaded from http://pubs.geoscienceworld.org/jgs/article-pdf/doi/10.1144/jgs2021-078/5433607/jgs2021-078.pdf by CNRS_INSU user ACCEPTED MANUSCRIPT / CLEAN COPY katabatic flows coming from the British-Irish ice sheet at a time where there was still a remnant left of the English Channel Sea [START_REF] Lefort | CNRS_INSU user ACCEPTED MANUSCRIPT / CLEAN COPY stands of the English Channel[END_REF]. Furthermore, as in Antarctica, low-pressure centres sometimes approached and even crosscut the katabatic flow. In Western Europe the migration of low-pressure centres is well documented and described in marine forecast studies [START_REF] Mayençon | Meteorologie marine[END_REF][START_REF] Bernot | Météo et stratégie[END_REF]. Now and probably also during the Upper Pleistocene the low pressure centres followed more or less (Figure 3) the axis of the Western English Channel. In terms of meteorology this axis is called « the West European Storms track » [START_REF] Antoine | Rapid and cyclic aeolian deposition during the Last Glacial in European loess: a high-resolution record from Nussloch Germany[END_REF].

Figure 3 -Katabatic winds only developed during the different glacial episodes but the Upper Saalian and Weichselian are the bestdocumented episodes. Their duration is very variable since they may not blow just for a few hours but can also last a few months (a duration of 203 days has been recorded in Antarctica) with a maximum velocity of 327 km/h. Outside the occurrence of the katabatic winds, regular storms occur about 8 times a year at our latitude and can statistically last during one week but these values can change a lot from one area to another. Under these conditions it is very difficult to know when these two meteorological phenomena were interfering with each other. It is only clear that they were both more active during the winter, even if it is impossible to detect any regular cycle.

-The excellent correspondence between the variations of the aeolian deposits observed in the site of Nussloch (Rhine Valley, Germany) [START_REF] Antoine | Rapid and cyclic aeolian deposition during the Last Glacial in European loess: a high-resolution record from Nussloch Germany[END_REF] and the atmospheric circulation over Downloaded from http://pubs.geoscienceworld.org/jgs/article-pdf/doi/10.1144/jgs2021-078/5433607/jgs2021-078.pdf by CNRS_INSU user the North Atlantic [START_REF] Rousseau | Link between European and North-Atlantic abrupt climate changes over the last glaciation[END_REF] suggests a possible link between the sedimentation rate and the change in strength of the wind. This correlation is difficult to observe in Southern England [START_REF] Catt | The contribution of loess to soils in lowland Britain[END_REF][START_REF] Barrie | Heavy mineral distribution in bottom sediments of the Bristol Channel UK[END_REF][START_REF] Catt | Loess in Cornwall[END_REF][START_REF] Parks | Thermoluminescence dating and geochemistry of loessic deposits in southeast England[END_REF] Brittany [START_REF] Lefort | CNRS_INSU user ACCEPTED MANUSCRIPT / CLEAN COPY stands of the English Channel[END_REF] and Normandy [START_REF] Lautridou | Le cycle glaciaire Pleistocène en Europe du Nord-Ouest et plus particulièrement en Normandie[END_REF], first because this kind of detail cannot be deciphered in the much thinner westernmost European stratigraphic sections but also because many of these sections are relatively close to the limits of the British-Irish Ice Sheet. Because of this proximity an increasing strength of the wind flow was more likely responsible for a stratigraphic gap than for a change in granulometry as in Germany.

-Epidotes, amphiboles, tourmalines and andalusites which are ubiquitous minerals do not allow to comment on the amount of their transit. On the contrary, augites, olivines, anatases and garnets that are unusual or very rare in the study area can be used as mineralogical markers when it is possible to locate their source. Study of these heavy minerals and of the flaky particles show a clear distinction between their transit distances: it is of 150 km for the silt minerals, 110 km for anatases, 90 km for garnets and 6 km for augites between the southern limit of the British-Irish Ice Sheet and the Mouth of the Loire River [START_REF] Lefort | CNRS_INSU user ACCEPTED MANUSCRIPT / CLEAN COPY stands of the English Channel[END_REF]. This result has been interpreted as a decreasing competency of the katabatic winds toward the south [START_REF] Lefort | CNRS_INSU user ACCEPTED MANUSCRIPT / CLEAN COPY stands of the English Channel[END_REF]). However, when looking at the details of these transfers it is possible to show that there was no mixing of the mineral markers during the katabatic transfer. Taken as a whole, it seems that there existed a succession of deflation zones (such as the British glacial outwash plain (1 figure 1), the Western English Channel (2 Figure 1) or Central Brittany (3 figure 1) alternating with areas of small uplands (such as the British hills (4 figure 1), the Armorican and Normandy hills (5 Figure 1) or the "Sillon de Bretagne" (6 Figure 1). These behaved like loess collectors or slowed down the wind flow. This process is very similar in its principle to the gold prospectors sluice boxes that were used during the gold rush times.

-During the southward transfer of the loess particles in the northsouth corridor cutting across Brittany, the Vilaine gorges (G Figure 2) located south of the Rennes Basin (R Figure 2), acted like a venturi structure. This impressive narrowing of the Vilaine river, bounded on both sides by steep to vertical cliffs reaching 50 to 60 meters (and even 80 to 100 metres with respect to the local plateau) reinforced the velocity of the katabatic winds. The result of this local acceleration is that no loess was deposited in Central Brittany (C Figure 2). Loess was only able to be deposited (D Figure 2) south of a little morphological step called the "Sillon de Bretagne" (S, Figure 2).

-If we take account of the approximate position of the low pressure centres when they were drifting along the storms track of the Channel (Figure 3) we can speculate that loess sediments deposited between the southern rim of the British-Irish Ice Sheet and Northern Brittany were ruled by the activity of the katabatic winds alone and that the extension of the sedimentation between the Rennes Basin and southern Brittany resulted from the deepening of the low pressure centre when the eastward drifting storms were crossing the katabatic flow.

-Even if these data are not directly connected to Quaternary geology, it is intriguing to learn that at present birds migrating between Northern and Southern Europe are using exactly the same route as the Upper Pleistocene katabatic flow recognized between England and the mouth of Loire River; this may be an atavism that has lasted since the Pleistocene.

Conclusions

-The easy transit of the katabatic flow recognized between the top of the British-Irish ice sheet and the mouth of Loire River (Figure 1) is well explained by the orientation of the corridor linking the northern and southern shores of Brittany (Figure 2) since it shows an almost perfect continuity with the katabatic winds blowing from England.

-Because the polar front was moving toward during the glacial episodes, the location of the storms track shown on figure 3 was not always exactly located at the same place, but the principle of a deepening of the katabatic low-pressure centre by the depression associated with the east drifting storms stays the same.

-Many authors working in Greenland and in Antarctic have insisted on the importance of topography to understand the different mechanisms associated with katabatic winds and, significantly, the presence of valleys. The onshore topography of Brittany illustrates this conclusion.

-Comparison between the limited transit of the different heavy minerals transported with loess and the long distance of propagation of the katabatic winds between England and southern Brittany shows that loess sediment was not transported as an undifferentiated sedimentary mass showing everywhere the same mineral composition but rather like the concatenation of a series of successive deflation-accumulation ensembles since the heavy minerals taken as markers are never mixed... 6). The directions of katabatic winds are shown. Onshore loess deposits and offshore conglomerates (which correspond to remnants of the offshore loess deposits) are shown with the same pattern. B: Brittany; CO: Cotentin; CS: Celtic Sea; EA: East Anglia; EC: English Channel; G: Guernsey; IS: Irish Sea; J: Jersey; LR: Loire River; N: Normandy; NS: North Sea; P: Picardie; SR: Seine River; V: Vendée; Red numbers 1 to 3 correspond with deflation zones and 4 to 6 to Downloaded from http://pubs.geoscienceworld.org/jgs/article-pdf/doi/10.1144/jgs2021-078/5433607/jgs2021-078.pdf by CNRS_INSU user ACCEPTED MANUSCRIPT / CLEAN COPY loess deposits. The British-Irish outwash plain corresponded to the deflation zone number 1.The contours of the LGM British-Irish Ice sheet are taken from Emery (2020), slightly modified.

Figure 2 .

 2 Figure 2. Distribution of the loess outcrops in Brittany and Normandy. Dash lines: Limits of the corridor funnelling the loess particles down to the mouth of Loire River. Black: loess deposits; Grey: Brittany, Vendée and Normandy hills. C: no loess deposits; D: mouth of Loire River loess deposits; G: Vilaine Gorges; R: Rennes Basin; S: Sillon de Bretagne. Arrows are katabatic wind. Note the absence of loess south of the Brittany and Normandy hills.

Figure 3 .

 3 Figure3. Interference between the katabatic wind coming from the British-Irish Ice Sheet (yellow arrows) and a storm drifting along the West European storms track. L: Low-pressure centre associated with a storm; H: high-pressure centre associated with the British-Irish Ice Sheet.

  Contours of the British Ice sheet and simplified distribution of loess deposits recognized in England, Brittany, and Normandy and under the English Channel during the Late Pleistocene (around MIS

Downloaded from http://pubs.geoscienceworld.org/jgs/article-pdf/doi/10.1144/jgs2021-078/5433607/jgs2021-078.pdf by CNRS_INSU user ACCEPTED MANUSCRIPT / CLEAN COPY Captions Figure 1.

Downloaded from http://pubs.geoscienceworld.org/jgs/article-pdf/doi/10.1144/jgs2021-078/5433607/jgs2021-078.pdf by CNRS_INSU user

Downloaded from http://pubs.geoscienceworld.org/jgs/article-pdf/doi/10.1144/jgs2021-078/5433607/jgs2021-078.pdf by CNRS_INSU user ACCEPTED MANUSCRIPT / CLEAN COPY

ACCEPTED MANUSCRIPT / CLEAN COPY


Andrea Mascherpa

Nozomii Ishii

Ayelen Tayagui

Jiang Liu

Matthieu Sollogoub
email: matthieu.sollogoub@sorbonne-universite.fr

Antony J Fairbanks
email: antony.fairbanks@canterbury.ac.nz

Lysosomal targeting of b-cyclodextrin

Keywords: carbohydrates, cyclodextrins, mannose-6-phosphate, lysosome, fluorescent probes

come

The cyclodextrins 1 (CDs) are a family of cyclic oligosaccharides, comprising 6 (a-CD), 7 (b-CD), 8 (g-CD), or more, a-(1→4)linked D-glucopyranose residues. Besides numerous other uses, they have found applications in >30 clinical settings 2 as drug delivery systems, 3 taking advantage of their lipophilic central cavity to bind to hydrophobic drugs, 4 increasing their solubility, stability, and bioavailability. Chemical modification of CDs, may improve their efficacy. For example, 2-hydroxypropyl-b-CD and sulfobutylether-β-CD 5 are both clinically used as pharmaceutical excipients for poorly water-soluble drugs. The lysosomal storage disorders (LSDs) 6 are a series of >70 rare, but incurable and often fatal, genetically inherited diseases, the clinical manifestations of which arise from the build-up of undegraded lysosomal metabolites. In general, each disease is caused by the lack of a functional enzyme in the lysosome to catalyse a required degradation pathway, resulting in the gradual occupation of undegraded metabolites (also termed accumulating substances). Amongst several treatment strategies 7 that have been investigated against the LSDs, such as enzyme replacement therapy (ERT), 8 gene therapy, chaperone therapy and substrate reduction therapy, a novel emerging and complimentary strategy is reduction of the accumulated substance. In the case of Niemann-Pick disease type C, 9 the major accumulated metabolites are glycosphingolipids and unesterified cholesterol; hydrophobic guests than can be accommodated inside the cavity of b-CD, as demonstrated by the use of b-CDs to modulate the cholesterol content of plasma membranes. 10 Testing of 2-hydroxypropyl-b-CD in both murine 11 and feline disease models for Niemann-Pick type C revealed beneficial effects, 12 so that a compassionate FDA-approved human trial of 2hydroxypropyl-b-CD therapy was undertaken. [START_REF] Ottinger | [END_REF] Although objective improvement in clinical outcomes was observed in the majority of patients, the doses required for beneficial effect in the murine model caused significant adverse effects in humans. 14 b-CDs and related compounds also have potential for therapeutic intervention in other disease states which result in the build-up of hydrophobic metabolites. For example, the accumulation of lipofuscin bisretinoids (LBs) in the lysosomal compartment of cells in the retinal pigment epithelium is thought to be the cause of retinal degradation in Stargardt's disease, 15 and is also a putative etiological agent for age-related macular degeneration. Administration of b-CDs resulted in a significant reduction in the build-up of LB's in the retinal pigment epithelium in an animal model, [START_REF] Nociari | Proc. Natl. Acad. Sci[END_REF] indicating the potential for b-CD-based therapy development. Improvement of the efficacy of administered functionalised b-CD, so that beneficial effects may be achieved at much lower dose levels so obviating adverse effects, may plausibly be achieved by specifically targeting the administered b-CD to the lysosome. Opportunely a natural 'lysosomal targeting system' exists, as, following their assembly in the endoplasmic reticulum (ER), the majority of lysosomal enzymes are trafficked to the lysosome by the phosphorylation of some the terminal mannose residues of their attached N-glycans. Installation of these mannose-6-phosphate (M6P) residues at the non-reducing termini of high mannose oligosaccharides during glycan processing in the Golgi then effects trafficking of proteins carrying such M6P-glycans to the lysosome via interactions with the two mannose-6-phosphate receptors (M6PRs); 17 namely the ~46-kDa cation dependent mannose-6-phosphate receptor (CD-M6PR), and the ~300-kDa cation independent mannose-6-phosphate receptor (CI-M6PR). 18 The M6PR system can therefore be regarded as an endogenous targeting mechanism, which may be hijacked by externally administered molecules in order to deliver them to the lysosomes. In particular the CI-M6PR migrates to the cell surface and, amongst its other functions, mediates endocytosis of externally circulating species bearing M6P-glycans 19 before then delivering them to the lysosome via intermediate endosomes. Thus, the attachment, either enzymatic 20 or chemical, 21 of M6P-terminated glycans to recombinant human lysosomal enzymes has been an area of significant interest in attempts to increase the efficacy of ERT's for the LSDs. Furthermore, CDs have been conjugated to carbohydrates, often in a multivalent fashion, to interact with proteins, often lectins. 22 Logical extension of these ideas engenders the hypothesis that chemical modification of b-CD by the attachment of M6P glycans may significantly increase the efficacy of lysosomal targeting, by CI-M6P mediated interactions. In this study we therefore undertook the chemical synthesis of a modified b-CD to which a biantennary glycan containing two M6P-residues was then covalently attached. In vitro fluorescence microscopy studies using a mammalian cell line, as a precursor to more detailed clinical investigations, then demonstrated the effect of M6P-glycan attachment, both on the rate of endocytosis and lysosomal transport. Based on a report in which the binding affinity of a series of phosphorylated oligomannosides to the CI-M6PR was probed, 23 a biantennary hexasaccharide structure 1, in which the two M6P-residues were a(1→2)-linked to the adjacent mannose units, was identified as a synthetic target. The synthetic strategy used to access 1 was based on previous approaches with some minor modifications (Scheme 1). Thus, glycosylation of the known alcohol 3 24 with the known thioglycoside donor 2 25 gave the desired b(1→4)-linked disaccharide 4. Selective removal of the Lev protection gave alcohol 5, which was triflated and reacted with tetrabutylammonium acetate with sonication, 26 to give the manno-configured disaccharide 6. Removal of the allyl protecting group gave the alcohol 7, which was used as the acceptor for glycosylation with the known selectively protected a(1→2)-linked disaccharide donor 8 27 to give tetrasaccharide 9. Regioselective reductive opening of the 4,6-benzylidene on 9 gave alcohol 10, allowing access to the 6-hydroxyl group of the central branching mannose residue. A second glycosylation using a(1→2)-linked donor 8 then gave the hexasaccharide 11. Following removal of the phthalimide protecting group and immediate acetylation to give acetamide 12, the two TIPS protecting groups were removed by treatment with BF3 etherate, to give hexasaccharide diol 13. Phosphitylation of 13 with excess dibenzyl diisopropylphosphoramidate and immediate oxidation by the addition of hydrogen peroxide then gave the bis phosphate ester 14. Finally, global deprotection by Birch reduction gave the bis-phosphorylated hexasaccharide 1. Following production of the desired bis-phosphorylated oligosaccharide, attention turned to its conjugation to b-CD by use of our reported method for the in situ formation of glucosyl azides of unprotected oligosaccharides using 2-azido-1,3dimethylimidazolinium hexafluorophosphate (ADMP), and their immediate Cu-catalysed Huisgen cycloaddition to species In order to directly visualise b-CD in a cellular environment, the production of a fluorescent derivative was required. Importantly the associated fluorophore must emit at a wavelength of light that did not overlap with cellular labels that were to be used to visualise the location of the lysosomes (e.g. Lysotracker TM ). Rather than undertaking the covalent attachment of a fluorophore directly to the b-CD framework, 32 which would have required re-synthesis of a new differently functionalised b-CD, and which additionally may interfere with binding to the CI-M6PR, an approach taking advantage of the well-established host-guest properties of b-CD was adopted. Thus, following a reported procedure, 33 adamantane, a known guest which forms high affinity inclusion complexes with b-CD, was selected as the scaffold to which a fluorescent tag was covalently attached; achieved by reaction of adamantylamine 20 with the commercially available N-hydroxysuccinimide ester of Cy5 21, a widely used fluorophore (Scheme 3). The resultant amide, Ad-Cy5 22, fluoresced strongly in the far-red. Next an inclusion complex was made simply by mixing unmodified b-CD 15 and Ad-Cy5 amide 22, giving complex 22⊂b-CD. An NMR study of complex 22⊂b-CD by ROESY showed correlations between both H-5 and H-3 of the b-CD and adamantane protons, demonstrating the formation of an inclusion complex between the adamantane of 22 and b-CD 15 (see Supporting Information, Figure S1). Finally, the fluorescent inclusion complex of Ad-Cy5 amide 22 and M6Pconjugated b-CD 19, termed 22⊂19 was formed analogously, and again complex formation demonstrated by ROESY. After 2 h, only limited amounts of the unmodified 22⊂b-CD complex had been endocytosed. In contrast, after 2 h incubation fluorescence arising from the 22⊂19 complex was approximately two-fold higher. In both cases the red fluorescence was higher than the control (Fig 1B ), indicating that complex formation with either modified or unmodified b-CD also increased the rate of endocytosis of 22. After 4 h incubation, whilst more of the unmodified 22⊂b-CD complex had been endocytosed, the amount of red fluorescence observed from the 22⊂19 complex was still considerably higher. Experiments were repeated three times and similar results obtained. Next the co-localisation of Cy5 and Lysotracker fluorescence was assessed (Fig. 2) by measurement of both Cy5 and LysoTracker fluorescence at varying positions along lines (marked as red in Fig. 2) across the cells. Firstly, no correlation between red and yellow fluorescence was observed for control 22 after either 2 h or 4 h. After 2 h, unmodified 22⊂b-CD produced limited levels of red fluorescence, the location of which only weakly correlated with the lysosomes (Fig. 2A). In contrast, complex 22⊂19 was both more effectively endocytosed, and more effectively transported to the lysosome, as shown by excellent correlation between red and yellow fluorescence (Graph, Fig2A (iv). After 4 h more of the unmodified 22⊂b-CD was endocytosed and some lysosomal transport was observed. However, endocytosis and lysosomal transport of complex 22⊂19 were still significantly higher. A second series of experiments was performed in HepG2 cells, and similar results were obtained (see Supporting Information; Figs S2 andS3). It is known that b-CD undergoes both endocytosis and lysosomal transport, as demonstrated by previous studies and attempted applications for the treatment of lysosomal storage disorders and Stagardt's disease. [11][START_REF] Rosenbaum | Proc. Natl. Acad. Sci[END_REF][START_REF] Ottinger | [END_REF][14][15][START_REF] Nociari | Proc. Natl. Acad. Sci[END_REF] However, this study indicates that the attachment of a bis-M6P-containing glycan to b-CD not only increased the rate of endocytosis, but also of lysosomal transport. Presumably this arose as a result of interaction of the M6P residues attached to the b-CD with the CI-M6PR, though definitive evidence of this mechanism has not yet been obtained. This work reports the first synthesis of a b-CD derivative decorated with a glycan that contains mannose-6-phosphate residues. The use of a functionalised adamantane derivative, comprising a Cy5 fluorescent label, allowed comparative in vitro studies of the localisation of both modified and unmodified b-CDs by way of the formation of fluorescent host-guest complexes. Fluorescence microscopy using mammalian cell lines demonstrated that the b-CD modified with the M6P-containing glycan was both more rapidly internalised into the cell and transported to the lysosome. These results indicate that M6P-glycan attachment augments the efficiency of trafficking of b-CD to the lysosome, which in turn may improve the efficacy of b-CD-based therapies for lysosomal storage disorders, such as Niemann-Pick type C, and other disease states which may be ameliorated by the reduction of an accumulated lipophilic metabolite in the lysosome. A logical next step is an in vivo study of the biological effects of applications of M6P-modified b-CD in appropriate disease models. Furthermore, the effectiveness of the host-guest complexes in these in vitro localization studies implies that different ligands, for example comprising other oligosaccharides, may be usefully attached to b-CD in order to assess the effects of such ligands on the rate of endocytosis and the destination of intracellular transport.

Scheme 1 .

 1 Scheme 1. Synthesis of M6P-containing hexasaccharide 1 (a) TTBP, MeOTf, MS4Å, CH2Cl2, 0 °C to r.t., 97%; (b) H2NNH2•AcOH, MeOH, CH2Cl2, 98%; (c) (i) Tf2O, pyridine, CH2Cl2; (ii) BuN4AOAc, toluene, sonication, 84% over 2 steps; (d) [Ir(cod)(PPh2Me) 2]PF6, NIS, H2O, THF, 91%; (e) 8, NIS, TfOH, MS4Å, Et2O, -20 °C, 74%; (f) Et3SiH, PhBCl2, MS4Å, CH2Cl2, -78 °C, 77%; (g) 8, NIS, TfOH, MS4Å, Et2O, -20 °C, 62%; (h) (i) ethylenediamine, MeOH, 70 °C, ii) Ac2O, pyridine, 68% over 2 steps; (i) BF2•OEt2, CH2Cl2, 0 °C, 63%; (j) (BnO)2PNiPr2, 1H-tetrazole, CH2Cl2, then H2O2(aq), 92%; (k) Na, NH3(l), THF, -78 °C, 61%.

Scheme 2 .

 2 Scheme 2. Synthesis of a propargyl-functionalised b-CD, and Click conjugation to unprotected glycan 1. (a) (i) BnCl, NaH, DMSO; (ii) DIBAL-H; (iii) MsCl; (iv) NaN3 (see Ref 29); (b) H2, Pd/C, TFA, H2O/THF, r.t., 24 h; (c) TBTU, pent-4-ynoic acid, DIPEA, DMSO, r.t., 24 h; (d) 1, ADMP, Et3N, D2O, MeCN, 0 o C, 1.5 h, then 18, CuSO4, L-ascorbic acid, 50 o C, 14 h, 42%.

Scheme 3

 3 Scheme 3 Synthesis of fluorescent probes for in vitro studies of b-CD localisation in mammalian cells. (a) 20, 21, DMF, NaHCO3(aq), rt, 16 h; (b) 22, b-CD or 19, H2O, rt, 30 min.

Figure 1 .

 1 Figure 1. Fluorescence microscopy studies. A) HEK293 cells were incubated with 5 µM with 22 (as control), 22⊂b-CD, or 22⊂19 for either (a) 2h or (b) 4 h. After washing, cells were treated with LysoTracker TM Yellow and then Hoechst 33342. B) Graph shows corrected total cell fluorescence (CTCF) of Cy5 in cells; analysis was performed using ImageJ software. Mean ± SD, n = 3.

Figure 2 .

 2 Figure 2. Cell images obtained after: A) 2 h incubation; B) 4 h incubation. (i) Cy5 (red), white scale bar = 10 µm; (ii) LysoTracker TM (yellow); (iii) merged with Hoechst 33342 (blue); (iv) graphs of fluorescence intensity (FI) of LysoTracker (yellow) and Cy5 (red) at positions along the red line across cells. Analysis was performed using ImageJ software.

Acknowledgements

The authors thank MBIE (Grant UOCX1710, awarded to AJF), the Marsden Fund (Grant UOC1608, awarded to AJF) for financial support, Cyclolab for generous gift of CD and the China Scholarship Council for PhD grant (JC).

Conflict of Interest

The authors declare no conflict of interest