Jean-Louis Dufresne
email: dufresne@lmd.jussieu.fr

Sandrine Bony

An assessment of the primary sources of spread of global warming estimates from coupled atmosphere-ocean models

ABSTRACT

Climate feedback analysis constitutes a useful framework to compare the global mean surface temperature responses to an external forcing predicted by general circulation models (GCMs). Nevertheless, the contributions of the different radiative feedbacks to global warming (in equilibrium or transient conditions) and their comparison with the contribution of other processes (e.g. the ocean heat uptake) have not been quantified explicitly. Here we define these contributions from the classical feedback analysis framework, and we quantify them for an ensemble of 12 CMIP3/IPCC-AR4 coupled atmosphere-ocean GCMs. In transient simulations, the multi-model mean contributions to global warming associated with the combined water vapor -lapse rate feedback, cloud feedback and ocean heat uptake are comparable. However, inter-model differences in cloud feedbacks constitute by far the primary source of spread of both equilibrium and transient climate responses simulated by GCMs. The spread associated with inter-model differences in cloud feedbacks appears to be roughly three times larger than that associated either with the combined water vapor -lapse rate feedback, the ocean heat uptake or the radiative forcing.

Introduction

The spread of the equilibirum or transient surface temperature response to a CO 2 doubling as predicted by atmosphere-ocean coupled models is still large [START_REF] Meehl | Global climate projections[END_REF]) and an open question is to identify the primary sources of this spread. Global warming estimates depend on radiative forcing, on feedback processes that may amplify or dampen the climate response and, in the transient case, on the ocean heat uptake. For individual models, it has been suggested that atmospheric processes were the most critical factors for estimating global temperature changes in transient simulations (e.g. [START_REF] Williams | Transient climate change in the Hadley Centre models: The role of physical processes[END_REF][START_REF] Meehl | Factors affecting climate sensitivity in global coupled models[END_REF][START_REF] Collins | The sensitivity of the rate of transient climate change to ocean physics perturbations[END_REF]). Here our purpose is to investigate whether these results extend to multi-model ensembles, and how much the various feedbacks and the ocean heat uptake contribute to the multi-model mean and spread of global warming estimates. The main radiative feedbacks are associated with changes in water vapor, temperature lapse rate, clouds and surface albedo. The associated feedback parameters have been diagnosed for some multi-model ensembles (e.g. [START_REF] Colman | A comparison of climate feedbacks in general circulation models[END_REF][START_REF] Soden | An assessment of climate feedbacks in coupled ocean atmosphere models[END_REF][START_REF] Webb | On the contribution of local feedback mechanisms to the range of climate sensitivity in two GCM ensembles[END_REF]) but they have not been translated into temperature changes. It makes it difficult to compare the temperature change associated with each feedback with that from other processes such as the ocean heat uptake.

In this paper we show that it is possible to decompose, and thus to compare, the contributions of the different climate feedbacks, and eventually of the ocean heat uptake, to the global temperature response to a specified forcing. After a brief presentation of the feedback analysis framework (section 2), the decomposition methodology is presented (section 3) and, after gathering the required data (feedback parameters, radiative forcing and ocean heat uptake) (section 4), this methodology is applied to an ensemble of models that participated in the World Climate Research Programme's (WCRP's) Coupled Model Intercomparison Project phase 3 (CMIP3) in support of the IPCC AR4 (section 5). There is very little in this paper that is entirely new. Rather we propose a new presentation of existing results that allows us to quantify in a more straightforward way the relative contribution of different processes to inter-model differences in global mean temperature changes.

The feedback analysis framework

Let us consider a steady state climate, with a time average value F o t = 0 of the global mean net flux at the top of the atmosphere (TOA) and a time average value T o s of the global mean surface temperature. Let us impose to the climate system a radiative forcing, such as a change in the greenhouse gas concentration or in the TOA incoming solar radiation. In the absence of surface temperature change, this forcing translates into a radiative flux perturbation ∆Q t at the TOA, called radiative forcing. In response to this disequilibrium, the surface temperature changes. It appears that at any time, the anomalies ∆T s and ∆F t of the surface temperature and the TOA flux from their unperturbed initial steady state are approximatively related through the following equation:

∆T s = ∆F t -∆Q t λ . ( 1
)
where λ is called the "climate feedback parameter", and the fluxes are positive downward. This relationship holds both for transient and equilibrium conditions. If the temperature changes until a new equilibrium is reached, the TOA net flux reaches its steady state value (∆F t = 0) and the equilibrium temperature change is

∆T e s = -∆Q t λ . (2)
The total feedback parameter is commonly split as the sum of 5 terms

λ = λ P + λ w + λ L + λ c + λ α (3)
which are respectively the Planck (P ), water vapor (w), lapse rate (L), cloud (c) and surface albedo (α) feedback parameters. In this approach it is assumed that everything is linear (see for instance the appendix of [START_REF] Bony | How well do we understand and evaluate climate change feedback processes?[END_REF] for more details on this approach and for a discussion of the approximations).

In climate feedback studies, temperature responses are often compared to the basic equilibrium temperature response ∆T s,P that would be obtained if the temperature change was horizontally and vertically uniform and was only modifying the infrared emission through a change in the Planck function (e.g. [START_REF] Hansen | Climate sensitivity: Analysis of feedback mechanisms[END_REF]:

∆T s,P = - ∆Q t λ P (4)
As the total feedback parameter may be decomposed as λ = λ P +

x =P λ x (cf. Eq. 3), at equilibrium (∆F t = 0) equation ( 1) reads:

∆T s = 1 1 - x =P g x ∆T s,P (5)
where g x = -λ x λ P is called the feedback gain for the variable x. If the total feedback gain

g = x =P g x (6)
is positive (negative), the temperature change ∆T s is larger (smaller, respectively) than the temperature change ∆T s,P associated with the Planck response.

3. Relative contribution of each feedback to the global temperature change a. Equilibrium temperature change

When only one feedback loop x is active in addition to the Planck response, the equilibrium temperature change due to this feedback is simply and uniquely defined from Eq. 5 as the difference δ 1 T s,x between the temperature change with and without this feedback x:

δ 1 T s,x = 1 1 -g x ∆T s,P -∆T s,P (7)
When several feedbacks are active, various approaches may be used. A first one is to quantify, as previously, the effect of each feedback as the difference between the temperature change with and without this feedback x (Eq. 7). A second possibility is to quantify this effect as the difference δ 2 T s,x between the temperature change when all the feedbacks are active and when all the feedbacks but x are active:

δ 2 T s,x = 1 1 -g - 1 1 -(g -g x ) ∆T s,P (8)
In this definition, the effect of a feedback loop x on the temperature change depends both on its gain g x and on the gain g of all feedbacks (e.g. [START_REF] Hansen | Climate sensitivity: Analysis of feedback mechanisms[END_REF][START_REF] Hall | The role of water vapour feedback in unperturbed climate variability and global warming[END_REF] and thus it can not be defined independently of the rest of the system. The temperature change obtained with these two definitions may be very different.

As there is no unique way to define the effect of individual feedbacks on the temperature change, we reformulate the question as: knowing the global temperature change, what is the part of this temperature change that is due to each feedback? In other words, we impose that the sum of the different temperature changes ∆T s,x associated with each feedback x plus the temperature change ∆T s,P associated with the Planck response equals the total temperature change ∆T s :

∆T s = ∆T s,P +

x =P ∆T s,x . (9)
From Eq. 5, it follows that:

∆T s,x = g x 1 -g ∆T s,P = g x ∆T s for x = P. (10)
This expression can also be directly obtained by noting that ∆T s (Eq. 5) can not be directly decomposed into additive contributions associated with each feedback, whereas the difference ∆T s -∆T s,P can. This new definition leads to partial temperature changes that have some interesting properties. If the feedback parameter λ x of a feedback x is multiplied by a factor α and the total gain g is unchanged (in this case, other feedback parameters have also to be modified), the temperature change ∆T s,x associated with this feedback x is multiplied by α.

If the feedback parameters of two feedbacks x and y are both multiplied by a factor α, the ratio ∆Ts,x ∆Ts,y is not modified. If the feedback parameters of all the feedbacks are multiplied by a same factor α, the ratio ∆Ts,x P y =P ∆Ts,y , i.e. the relative fraction of the temperature change due to each feedback x is not modified. Therefore this definition of the partial temperature change allow us to compare and to add the contribution of the various feedbacks to the temperature response.

It is important to note that the temperature change associated with the Planck response (Eq. 4) and the one associated with each feedback x (Eq. 10) are of different nature owing to the very specific role of the Planck response (the "basic" response on which the others are feedbacks). Equation 10may also be written as follow:

λ P ∆T s,x = -λ x ∆T s for x = P (11)
In this equation, the left hand side is the change of the TOA flux due to the partial temperature change ∆T s,x if the temperature change was uniform and affecting only the thermal emission. The right hand side is the change of the TOA flux ∆F t,x due to the total temperature change ∆T s through the feedback x. The temperature change ∆T s,x associated with a feedback x is the temperature change that would be necessary to produce the same perturbation ∆F t,x of the TOA flux through thermal emission. This illustrates how the Planck response compensates the flux disequilibrium associated with each feedback.

b. Transient temperature change

Without dealing with the complexity of the feedback analysis under transient conditions (e.g. [START_REF] Hallegatte | An elicitation of the dynamic nature of water vapor feedback in climate change using a 1D model[END_REF], we now consider the ocean response in a very simple way in order to quantify the feedback processes in transient runs using the same feedback framework as above. Following [START_REF] Gregory | The climate response to CO 2 of the Hadley Centre coupled AOGCM with and without flux adjustment[END_REF], we assume that in transient experiments in which the forcing increases regularly with time, the disequilibrium ∆F t of the net flux at the TOA is equal to the ocean heat uptake and is related to the surface temperature change ∆T s by: ∆F

t = -κ∆T s ( 12
)
where κ is the ocean heat uptake efficiency (< 0). This assumption is common and useful despite its limited validity. For instance, it is valid neither when the climate tends toward equilibrium (∆T s increases slowly whereas ∆F t decreases to zero) nor immediately after applying an abrupt forcing (∆T s ≈ 0 whereas ∆F t ≈ ∆Q t ). Using Eq. 1 and 12, the transient temperature change (also called the transient climate response, TCR) can be expressed as

∆T t s = - ∆Q t λ + κ (13)
Although the ocean heat uptake is not a feedback, the only difference between the expression of the equilibrium (Eq. 2) and transient (Eq. 13) temperature changes is that in the later one, the ocean heat uptake efficiency κ is added to the total feedback parameter λ . Using the same approach as for the equilibrium temperature, we thus require the total temperature change ∆T s to be the sum of the temperature change due to the Planck response, climate feedbacks and ocean heat uptake. We obtain the same equation as for the equilibrium temperature, except that the ocean uptake efficiency has to be added to the sum over x in Eq. 6 and 9. The contribution ∆T s,x of a feedback x to the global temperature change is then given by Eq. 10 where the gain g is replaced by g = g + g o with g o = -κ λ P and the contribution of the ocean heat uptake is given by ∆T s,o = go 1-g ∆T s,P . Because of the ocean heat uptake, g differs from g and the transient temperature change ∆T t s,x associated with a feedback x differs from that at equilibrium ∆T e s,x . The transient temperature change ∆T t s also differs from that at equilibrium ∆T e s , and a direct consequence of Eq. 10 is that the contribution of a feedback x to the global temperature change is the same in both equilibrium and transient conditions:

∆T e s,x

CMIP3/AR4 AOGCMs

We now apply the above decomposition to the global surface temperature response to a CO 2 doubling predicted by an ensemble of 12 coupled atmosphere-ocean GCMs (AOGCMs) participating in the third Coupled Model Intercomparison Project (CMIP3-AR4) [START_REF] Meehl | Overview of the Coupled Model Intercomparison Project[END_REF]Randall et al. 2007a). For this purpose, we need for each model the global mean values of the radiative forcing, of the climate feedback parameters and of the ocean heat uptake efficiency.

a. 2×CO 2 radiative forcing

In this study we use the radiative forcing for a CO 2 doubling reported by [START_REF] Forster | Climate forcings and climate sensitivities diagnosed from coupled climate model integrations[END_REF] and Randall et al. (2007b). These forcings have been computed after stratospheric adjustment, in all sky conditions and are averaged over the globe and over a year (Table 1). For the 12 GCMs considered here, the multi-model average of the net radiative forcing (3.71W.m -2 ) is very close to previous [START_REF] Myhre | New estimates of radiative forcing due to well mixed greenhouse gases[END_REF] results, and the relative inter-model standard deviation is about 6% (Table 2).

In another intercomparison study, [START_REF] Collins | Radiative forcing by well-mixed greenhouse gases: Estimates from climate models in the Intergovernmental Panel on Climate Change (IPCC) Fourth Assessment Report (AR4)[END_REF] obtained for 16 GCMs an intermodel spread of the net radiative forcing as large as 15% (Randall et al. 2007a). These forcing have been computed at 200hPa, for a unique atmospheric profile (mid latitude summer climatological conditions), in clear sky conditions and without any stratospheric adjustment. When compared with [START_REF] Forster | Climate forcings and climate sensitivities diagnosed from coupled climate model integrations[END_REF] results, the relative values of the inter-model standard deviation of the longwave (LW) forcing are similar in both studies (8%, Table 1). This is not the case in the shortwave (SW) domain and the difference is even larger for the net radiative forcing. In the results of [START_REF] Collins | Radiative forcing by well-mixed greenhouse gases: Estimates from climate models in the Intergovernmental Panel on Climate Change (IPCC) Fourth Assessment Report (AR4)[END_REF], as reported by Randall et al. (2007a), the standard deviation of the net forcing is larger than the quadratic sum of the standard deviation of the SW and LW forcings, which indicates that the SW and LW inter-model differences are positively correlated. The opposite is found in [START_REF] Forster | Climate forcings and climate sensitivities diagnosed from coupled climate model integrations[END_REF], which indicates that the error in the SW and LW domains are anti-correlated, and the stratospheric adjustment can explain part of it. We believe that the inter-model spread of the forcing reported by [START_REF] Forster | Climate forcings and climate sensitivities diagnosed from coupled climate model integrations[END_REF] is the most relevant for our study because the global warming estimates are derived from global simulations including clouds and a stratospheric temperature response.

All contributions to the global warming ∆T s are proportional to ∆T s,P (Eq. 10), and therefore to the forcing ∆Q t (Eq. 4). Part of inter-model differences in these contributions may thus arise from inter-model differences in the radiative forcing. To quantify this part, for each model we compute ∆T s,P for a reference forcing value ∆Q r t (set to the multi-model mean forcing estimate, namely 3.71W.m -2 , Table 1) and we add a term which represents the impact on ∆T s of the discrepancy δQ t between the actual forcing of each model and the reference value:

∆T s = 1 1 -g -∆Q r t λ P + 1 1 -g -δQ t λ P (15)
b. Feedback parameters

As reviewed by different authors (e.g. [START_REF] Soden | On the use of cloud forcing to estimate cloud feedback[END_REF][START_REF] Stephens | Cloud feedbacks in the climate system: a critical review[END_REF][START_REF] Bony | How well do we understand and evaluate climate change feedback processes?[END_REF]), several approaches have been followed to decompose the total feedback parameter into its several components (water vapor, clouds, surface albedo...), each method having its own strengths and weaknesses. [START_REF] Soden | An assessment of climate feedbacks in coupled ocean atmosphere models[END_REF] computed these feedback parameters for 12 CMIP3/AR4 models (Table 1), using the SRES-A1B simulations, and their results are fairly consistent with previous results obtained by [START_REF] Colman | A comparison of climate feedbacks in general circulation models[END_REF] with older GCMs (cf. [START_REF] Bony | How well do we understand and evaluate climate change feedback processes?[END_REF]. The multi-model mean and standard deviation of the total feedback parameters ( λ = -1.3 W.m -2 , σ λ = 0.3 W.m -2 , Table 3) are consistent with the values obtained by [START_REF] Forster | Climate forcings and climate sensitivities diagnosed from coupled climate model integrations[END_REF] for a larger set of CMIP3/AR4 models and for different ensembles of runs: When analyzing the 1%yr -1 increase of CO 2 simulations performed by 20 AOGCMs, they found a multi-model mean value of the total feedback parameter λ = -1.4 W.m -2 and a standard deviation σ λ = 0.3 W.m -2 . When considering another set of experiments, namely doubled CO 2 equilibrium runs from 11 atmospheric GCMs coupled to slab oceans, they found a mean value λ = -1.2 W.m -2 and a standard deviation σ λ = 0.3 W.m -2 .

c. Ocean heat uptake efficiency

We computed the ocean heat uptake efficiency κ using Eq. 12. For each model, the TOA flux F t and the surface air temperature T s were averaged over the 20-year period centered at the time of CO 2 doubling, that is year 70 for the 1%yr -1 simulation. The differences with the corresponding period of the control simulation were performed and the values of κ reported in Table 1.

d. Representativity of the ensemble of models considered

Using the values reported in Table 1, the equilibrium and transient temperature changes are computed for each of the 12 models as ∆T e s = -∆Q t /λ and ∆T t s = -∆Q t /(λ + κ) respectively. This leads to a multi-model mean ± 1 standard deviation of the equilibrium temperature change of 3.1 ± 0.7 • C. These numbers are comparable with those of the AR4 equilibrium climate sensitivity estimates derived from 18 atmospheric GCMs coupled to slab oceans (3.3 ± 0.7 • C, [START_REF] Meehl | Global climate projections[END_REF]). For the transient temperature change, we obtain 2.0 ± 0.3 • C, which is closed to the AR4 values reported on the basis of 19 coupled atmosphere-ocean GCMs: (1.8 ± 0.3 • C, [START_REF] Meehl | Global climate projections[END_REF]). As far as global temperature change is concerned, the sub-set of 12 models considered here is therefore representative of the larger set of CMIP3/AR4 models.

Results

a. Decomposition of equilibrium temperature changes

The multi-model mean of the equilibrium temperature change and the contributions associated with the Planck response (Eq. 4) and each feedback (Eq. 10), computed for a reference radiative forcing, are shown in Fig. 1-a and reported in Table 3. On average, for the set of models considered here, the Planck response represents about a third of the total temperature response (1.2 • C vs 3.1 • C), whilst climate feedbacks account for two thirds of it.

The increase of water vapor with warming enhances the absorption of longwave radiation and enhances the warming by 1.7 • C. Lapse rate changes are associated with a negative feedback, owing to the moist adiabatic structure of the tropical atmosphere. Due to the strong anticorrelation between these two feedbacks, it is convenient to consider the sum of both of them (WV+LR) [START_REF] Soden | An assessment of climate feedbacks in coupled ocean atmosphere models[END_REF]. This combined feedback increases the temperature by 0.9 • C, slightly less than the Planck response. The cloud feedback's contribution to the warming is, on average, slightly weaker than that of the WV+LR feedback, and the surface albedo feedback's contribution is the smallest.

However Fig. 2 shows that for each feedback there are some inter-model differencies, especially for the cloud feedback contribution, and that the amplitude of the equilibrium temperature change is primarily driven by the cloud feedback component. This appears also clearly when considering the inter-model standard deviation of the temperature change due to each feedback normalized by the inter-model standard deviation of the total temperature change (Fig. 1-b). The standard deviation due to cloud feedback represents nearly 70% the standard deviation of the total temperature change. The temperature spread due to the radiative forcing is comparable to the spread due to the WV+LR feedback and the spread due to the surface albedo feedback is the smallest.

b. Decomposition of transient temperature changes

The transient temperature changes (or transient climate responses, TCR) from individual GCMs, as well as the contribution of the various feeedbacks are displayed in Fig. 3. The multi-model mean and standard deviation are displayed in Fig. 4 and reported in Table 3. The temperature damping due to the ocean heat uptake is about -0.4 • C and its absolute value is comparable to the multi-model contributions of the WV+LR (0.6 • C) and cloud (0.4 • C) feedback. The mean transient temperature change is nearly 2/3 of that at equilibrium, therefore the transient temperature changes associated with each feedback scale with it (cf. Eq. 14). The inter-model standard deviation of the temperature change due to cloud feedback represents nearly 90% the standard deviation of the total temperature change (Fig. 4-b). Like for the equilibrium case, cloud feedbacks thus constitute the main source of spread of the transient temperature response among GCMs. The WV+LR feedback, the ocean heat uptake and the radiative forcing constitute secondary and roughly comparable sources of spread and the surface albedo feedback constitutes the smallest one.

The inter-model standard deviation of the global temperature change may also be normalized with the multi-model mean global temperature change. This relative standard deviation is comparable in both equilibrium and transient conditions, the spread in equilibrium being slightly larger (23 % vs 16%). The same olds for the relative standard deviation of the temperature change associated with each feedback. Therefore the contribution of the various feedbacks to the total spread is, in relative terms, as important in the transient case than in the equilibrium case.

Summary and conclusion

In this paper we propose a simple decomposition of the equilibrium and transient global temperature responses to an external forcing into a sum of contributions associated with the Planck response, the different climate feedbacks, and eventually the ocean heat uptake. This allows us to quantify how the various processes contribute to the multi-model mean and inter-model spread of the global temperature change. This is illustrated (Figures 1 to 4) using published results for the feedback parameters and the radiative forcings [START_REF] Soden | An assessment of climate feedbacks in coupled ocean atmosphere models[END_REF][START_REF] Forster | Climate forcings and climate sensitivities diagnosed from coupled climate model integrations[END_REF]Randall et al. 2007b) and diagnosing the ocean heat uptake efficiency from model outputs. In transient simulations, the absolute values of the contributions of the WV+LR feedback, the cloud feedbacks and the ocean heat uptake to the global temperature response appears to be comparable (Fig. 4-a). However, for the ensemble of models considered here, the spread of the transient temperature change due to inter-model differences appears to be primarily due to cloud feedback. The spread due to WV+LR feedback, ocean heat uptake or radiative forcing appears to be of the same order of magnitude and roughly one third of the spread due to the cloud feedback (Fig. 4-b). Note that the radiative forcing associated with non-CO 2 greenhouse gases and aerosols is more uncertain than that associated with CO 2 [START_REF] Forster | Changes in atmospheric constituents and in radiative forcing[END_REF]. Therefore, the intermodel spread of radiative forcing estimates might be larger for 20 th century simulations or for climate change simulations based on emission scenarios that include changes in aerosol concentrations than in this study. This difference is mitigated, however, by the fact that the relative contribution of aerosols vs greenhouse gases is likely to decrease in the future [START_REF] Dufresne | Contrasts in the effects on climate of anthropogenic sulfate aerosols between the 20 th and the21 st century[END_REF].

Our analysis shows that the contribution of each feedback and of the radiative forcing to inter-model differences in temperature change is roughly similar, in a normalized sense, in equilibrium and transient simulations (Fig. 1-b and4-b). In particular, cloud feedbacks appear to be the main source of spread in both cases. Inter-model differences in cloud feedbacks have been shown to arise primarily from the response of low level clouds [START_REF] Bony | Marine boundary layer clouds at the heart of cloud feedback uncertainties in climate models[END_REF][START_REF] Webb | On the contribution of local feedback mechanisms to the range of climate sensitivity in two GCM ensembles[END_REF][START_REF] Wyant | A comparison of low-latitude cloud properties and their response to climate change in three AGCMs sorted into regimes using mid-tropospheric vertical velocity[END_REF]. Understanding and evaluating the physical processes that control these cloud responses thus appears to be of primary importance to better assess the relative credibilty of climate projections from the different models.

List of Figures

1Fig. 1 .Fig. 2 .Fig. 3 .Fig. 4 .

 1234 Fig. 1. For a CO 2 doubling, (a) multi-model mean ± 1 standard deviation (thick line) and 5%-95% interval (thin line) of the equilibrium temperature change (∆T e s ), and contributions to this temperature change associated with the Planck response, combined water vapor and lapse rate (WV+LR) feedback, surface albedo feedback and cloud feedback. (b) inter-model standard deviation of the temperature change estimates associated with the radiative forcing, the Planck response and the various feedbacks normalized by the inter-model standard deviation of the equilibrium temperature change ∆T e s reported in (a).

Table 1 .

 1 ∆T t s .-2 ) (W.m -2 .K -1 ) (W.m -2 .K -1 ) 2×CO 2 radiative forcing ∆Q t , total feedback parameter λ and ocean heat uptake efficiency κ estimates of the 12 CMIP3/AR4 models used in this paper, and their multi-model mean and standard deviation. LW SW Net mean std. dev. mean std. dev. mean std. dev.

		∆Q t	λ	κ
	model (W.m CNRM-CM3 3.71	-1.17	-0.80
	GFDL-CM2.0	3.50	-1.18	-0.53
	GFDL-CM2.1	3.50	-1.37	-0.81
	GISS-ER	4.06	-1.64	-0.92
	INM-CM3.0	3.71	-1.46	-0.56
	IPSL-CM4	3.48	-0.98	-0.79
	MIROC3.2(medres)	3.60	-0.91	-0.77
	MRI-CGCM2.3.2	3.47	-1.50	-0.61
	ECHAM5/MPI-OM	4.01	-0.88	-0.57
	CCSM3	3.95	-1.62	-0.70
	PCM	3.71	-1.53	-0.62
	UKMO-HadCM3	3.81	-0.97	-0.59
	multi-model mean	3.71	-1.27	-0.69
	inter-model RMS	0.20	0.27	0.12

Table 2 .

 2 Multi-model mean and inter-model standard deviation of the longwave (LW), shortwave (SW) and net radiative forcing (W.m -2 ) for a CO 2 doubling computed by GCMs in two inter-comparison studies, with two different numerical set-up (see text). In parenthesis, the standard deviation is computed relative to the mean.

					associated equilibrium associated transient
		Variables		temperature change	temperature change
			mean std. dev. mean	std. dev.	mean	std. dev.
	Feedback parameter
	all	λ	-1.27	0.30	3.1	0.7	2.4	0.4
	Planck	λ P	-3.2	0.05	1.2	0.0	1.2	0.0
	water vapor λ W	1.80	0.18	1.7	0.4	1.1	0.2
	lapse rate	λ L	-0.84	0.26	-0.8	0.3	-0.5	0.2
	WV+LR	λ W L 0.96	0.11	0.9	0.2	0.6	0.1
	albedo	λ a	0.26	0.08	0.3	0.1	0.2	0.05
	clouds	λ c	0.69	0.38	0.7	0.5	0.4	0.3
	OHU efficiency	κ	-0.67	0.12	-	-	-0.4	0.1
	Rad. forcing	∆Q t 3.71	0.20	0	0.2	0.0	0.1
	∆T e s = ∆Q t /λ ∆T t s = ∆Q t /(λ + κ)	3.1 2.0	0.7 0.3

Table 3 .

 3 Multi-model mean and inter-model standard deviation of total feedback parameter λ and its components λ x (W.m -2 .K -1 ), the ocean heat uptake efficiency κ (W.m -2 .K -1 ) and the 2×CO 2 radiative forcing ∆Q t (W.m -2 ), and their associated equilibrium and transient temperature changes ( • C). The multi-model mean and standard deviation of the equilibrium (∆T e s ) and transient (∆T t s ) temperature changes ( • C) are also given

Acknowledgments.

We thank Jean-Yves Grandpeix for frequent and useful discussions on feedbacks. We are grateful to Jonathan Gregory and an anonymous reviewer for useful comments on this study, and acknowledge the modeling groups for making their simulations available for analysis, the Program for Climate Model Diagnosis and Intercomparison (PCMDI) for collecting and archiving the CMIP3 model output, and the WCRP's Working Group on Coupled Modelling (WGCM) for organizing the model data analysis activity. The WCRP CMIP3 multi-model dataset is supported by the Office of Science, U.S. Department of Energy. This research was supported by CNRS, by INSU-EVE French Program under the project MissTerre and by the European Commission under the project ENSEMBLES (contract number GOCE-CT-2003-505539).

List of Tables

Multi-model mean and inter-model standard deviation of total feedback parameter λ and its components λ x (W.m -2 .K -1 ), the ocean heat uptake efficiency κ (W.m -2 .K -1 ) and the 2×CO 2 radiative forcing ∆Q t (W.m -2 ), and their associated equilibrium and transient temperature changes (


Jose Marie Orellana
email: jose.orellana@cnam.fr

Optimal Control for HIV Multitherapy Enhancement

Keywords: Fixed-end-time optimization, HIV, mathematical model, multi-drug therapy, Scheduled Interrupted Treatment. 2000 MSC: 92D30, 49K15, 34B15

The purpose of the paper is to use numerical analysis and optimization tools developped for applied mechanic research to suggest improved therapies to try and cure HIV infection. The evolution of the infection is modelled by an ordinary differential equation system which includes both immune response and multi-drug effects. For a fixed time, one looks for a two drugs treatment strategy based on Pontryagine's minimum Principle. Basically, the method studied in this paper can be considered as an optimal control one where drug doses are regarded as control inputs. The quadratic objective function considered takes into account three contributions: the viral load, the transient evolution of infection and the quantities of drug used. Simulations are carried out using an indirect optimization method. At each step the differential system is solved using Runge-Kutta five order scheme. Results highlight that a progressive reduction of Reverse Transcriptase Inhibitor drug dose on the one hand along with on the other hand a progressive increase of Protease Inhibitor one is needed for optimality.

Introduction

During the two last decades, immunodeficiency virus treatments did improve. Despite the fact that preventative vaccine is still unavailable, more accurate assays and new regimen help improving and prolonging the patient's life. Highly Active Anti Retroviral Therapies (HAART) consisting in a combination of Reverse Transcriptase Inhibitors (RTIs) and Protease Inhibitors (PIs) drugs, have proven to be extremely effective. Patients maintain low viral load and have their immunity system restored so that not to be vulnerable to any opportunistic potentially lethal infection. Though, it is still impossible to eradicate the virus because an undetectable virus load in blood doesn't mean infection is no longer present. Virus could hide out in one part of organism such as the lymph nodes, the brain, testes or the retina in quiescent state waiting any opportunity to bounce back and therefore is not targeted by immune defense neither treatments. Another concern is that a long term drug intake induces many sides effects. Among them, strong side effects like nervous breakdown, anaemia, pain, weakness, fat redistribution are common not to mention others serious illness like insulin resistance, cardiovascular pathologies, hepatitis or myopathies due to the toxicity of treatment. Other concerns also arise from viral rebound due to mutation of the virus. Indeed on the one hand, virus replicates at extremely high rates and have many opportunities to mutate. On the other hand, reverse transcritptase process leads to frequent DNA virus mistakes producing mutant which are likely to resist treatments. Regularly drugs changes are needed and in some cases inability to find any appropriate pharmaceutical drug combinations is noted since virus is no longer reactive to any therapy. Hence poor compliance with drug prescription is currently noticed in the patient's behavior. Up to now, the solution proposed by the World Health Organisation is to administrate a constant antiviral drug doses which efficiency is assumed to be stable in time and which can be changed from time to time according patient's condition. Idealistic solution will be to lower and maintain the virus load at such a level that the immune system controls with low amount of drugs over short spells. Although mathematical studies were first ignored by the experimental community, the disease has become the subject of intense theoretical modelling efforts. Many important papers investigate dynamic models of host-drug-virus interactions [START_REF] Ho | Rapid turnover of plasma virions and CD4 lymphocytes in HIV-1 infection[END_REF][START_REF] Murray | Mathematical biology[END_REF][START_REF] Nowak | Anti-viral drug treatment: dynamics of resistance in free virus and infected cell populations[END_REF][START_REF] Perelson | Dynamics of HIV Infection in CD4ˆ+ T cells[END_REF]. Mathematical models have become essential tools to make assumptions, suggest new experiments or help one explaining easily complex processes. Different aspects are encompassed in each model which are by the time more and more comprehensive and accurate [START_REF] Callaway | HIV-1 infection and low steady state viral loads[END_REF][START_REF] Culshaw | A delay-differential equation model of HIV infection of CD4+ T-cells[END_REF][START_REF] Karrakchou | Optimal control and Infectiology: Application to an HIV/AIDS Model[END_REF][START_REF] Kirschner | Immunotherapy of HIV-1 infection[END_REF][START_REF] Kwon | Optimal treatment strategies derived from a HIV model with drug-resistant mutants[END_REF][START_REF] Snedecor | Comparison of three kinetic models of HIV-1 infection: implications for optimization of treatment[END_REF][START_REF] Stengel | Mutation and control of the human immunodeficiency virus[END_REF][START_REF] Velichenko | Numerical methods of optimal control of the HIV-infection dynamics[END_REF][START_REF] Wodarz | Mathematical models of HIV pathogenesis and treatment[END_REF]. But complete analysis can hardly be achieved with involved models and simple conclusions are difficult to deduce. Also all new parameters introduced, which values must be known to carry out any simulation, may not be accurately fit by experimental data or they require new measurement method [START_REF] Adams | HIV dynamics: modeling, data analysis, and optimal treatment protocols[END_REF][START_REF] Bortz | Model selection and mixed-effects modeling of HIV infection dynamics[END_REF][START_REF] Perelson | Mathematical analysis of HIV-I: dynamics in vivo[END_REF]. Most of the models are deterministic prey-predator systems of non linear differential equations. Sometimes stochastic terms are included to address the random behavior of features of disease process. Typically, dynamic changes are modelled considering cell numbers progression of CD4+T cells, infected cells and virus population under drugs effects [START_REF] Joshi | Optimal control of an hiv immunology model[END_REF][START_REF] Kirschner | optimal control of the chemotherapy of hiv[END_REF][START_REF] Nowak | Anti-viral drug treatment: dynamics of resistance in free virus and infected cell populations[END_REF][START_REF] Perelson | Mathematical analysis of hiv-1 dynamics in vivo[END_REF]. At the same time, optimal control have received much attention especially in mechanical and aerospace engineering for example to the reentry shuttle problems. The main idea is to use optimization techniques and theories to propose an alternative treatement based on administrating continually adjustable antiviral drug doses once a proper model is obtained. The remainder of this paper is organized as follows : in the second section, the deterministic model chosen with its specific aspects is presented and explained. In the third section an optimal control is derived by using Pontryaguine's minimum principle and the adjoint method. Numerical results are illustrated and commented in the last section.

Methods

Mathematical model

The present model accounts for multi-drug therapy combination and also include specific immune response to HIV. The infection mechanism is described by the system of non-linear ordinary differential equations with six compartments. Namely, the state variables are T the concentration of uninfected CD4+T cells, L, the concentration of latently infected T-cells, I, the concentration of actively infected cells, V , the concentration of infectious viruses, N , the concentration of non infectious viruses by the action of protease inhibitor, E, the concentration of cytotoxic lymphocytes effector. These state variables are the key-compartments commonly observed in clinical data and obviously must be positive along our calculus. Drugs efficiency is represented by the controls u 1 and u 2 , (0 ≤ u 1 ≤ 1, 0 ≤ u 2 ≤ 1) which accounts respectively for reverse transcriptase and protease inhibitors actions.

Ṫ = rT (1 - T + L + I T max ) -µ T T -(1 -u 1 )k 1 V T + s 1 . (1)
L = ω(1 -u 1 )k 1 V T -µ T L -k 2 I. (2) İ = (1 -ω)(1 -u 1 )k 1 V T + k 2 I -µ I I -k 3 IE. (3) V = a(1 -u 2 )I -k 1 V T -µ V V. (4) Ṅ = au 2 I -µ V N. ( 5
) Ė = k 4 IT E -µ E E + s 2 . (6)
or in vector form ẋ = F (x(t, u), u(t)),

(0) = t (T 0 , 0, 0, V 0 , 0, 0) (7) x
where x(t) = t (T, L, I, V, N, E). t denotes the transposition. The concentration N and the ode ( 5) can be omitted since they are decoupled and do not affect the remaining system. Source and death rates of cells population s and µ terms respectively, are not commented in order to focus on non linear terms introduced by cells interactions. Nevertheless definitions and numerical values for the parameters are summarized in Table 1 and taken from from [START_REF] Culshaw | A delay-differential equation model of HIV infection of CD4+ T-cells[END_REF][START_REF] Culshaw | Optimal HIV treatment by maximising immune response[END_REF][START_REF] Garira | Optimal control of combined therapy in a single strain HIV-1 model[END_REF]. As we assume the system is well mixed, mass action law is used to account for them at a first approximation. For example, the viral particle is tightly linked to receptors on the lymphocyte membrane enabling fusion with the cell membrane. Therefore each time a cell is infected one virion is lost and we add the term -k 1 V T to (4). Once viral RNA and enzymes enter the host cell's cytoplasm, reverse transcriptase reads and transcribes the sequence of viral genome into a complementary DNA sequence. Reverse transcriptase inhibitor blocks the recoding process and viral RNA is eventually degraded. The cell is only temporarily infected. Then to model drug action we include (1 -u 1 )k 1 V T term to (1) to account for uninfected T cells loss. Viral DNA once integrated in cell nucleus, may remain dormant, in the latent stage. Thus we introduce ω to represent fraction of latently infected CD4+T cells in infected cells production and then we add (1 -ω)(1 -u 1 )k 1 V T to (2) and ω(1 -u 1 )k 1 V T to (3). We assume that latently infected cells which haven't yet produce virus, switch to productively infected cell with rate k 2 in (2). The virus genome, provirus, is transcribed into new RNA and new enzymes. HIV protease inhibits cleavage of viral polyprotein and new virions will lack functional enzymes. The viruses produced are defective. We add a(1 -u 2 ) to (4) and au 2 to (5) to model protease inhibitor effect. Among immune responses, cytotoxic T lymphocytes action is known to be particularly effective. CTLs response had been investigated under different assumptions in various papers [START_REF] Adams | HIV dynamics: modeling, data analysis, and optimal treatment protocols[END_REF][START_REF] Bonhoeffer | Risks and benefits of structured antiretroviral drug therapy interruptions in HIV-1 infection[END_REF][START_REF] Callaway | HIV-1 infection and low steady state viral loads[END_REF][START_REF] Perera | Deterministic and stochastic models of virus dynamics[END_REF][START_REF] Wodarz | Dynamics of cytotoxic Tlymphocyte exhaustion[END_REF]. CTLs kill infected cells without being targeted by HIV virus and prevent uninfected cells from being contaminated by the chemokines they release [START_REF] Garira | Optimal control of combined therapy in a single strain HIV-1 model[END_REF]. The reduction of viral infectivity is not examined and only effector CTL population is considered in this paper. CTLs proliferate proportionately with the number of infectious CD4+T cells since it is an immune specific response to HIV. They are also dependent on CD4+ T cell helper and of CTLs, hence the trilinear term in ( 6) is introduced [START_REF] Culshaw | Optimal HIV treatment by maximising immune response[END_REF][START_REF] Garira | Optimal control of combined therapy in a single strain HIV-1 model[END_REF][START_REF] Wodarz | Dynamics of cytotoxic Tlymphocyte exhaustion[END_REF][START_REF] Wodarz | Mathematical models of HIV pathogenesis and treatment[END_REF]. Elimination of infectious CD4+T cells by CTLs are added by -k 3 IE term in (4). Early infection is simulated by introducing one virus particles per ml of blood. We assumed that half of infected cells becomes latently before actively infected. Natural response of the system is shown in Fig. 1 and Fig. 2. Further analysis shows that system has two biologically significative steady states: an uninfected steady state x1 = (1000, 0, 0, 0, 0, 50) which is unstable and an infected steady state x2 = (254, 221, 13, 1644, 0, 77) which is locally stable. The first corresponds with healthy state and the last with HIV seropositive state. When full treatement is administered, the system drives toward the uninfected state which is an evidence of medication efficiency.

Optimal control

The model described by ( 1)-( 6) can simulate the course of the disease under a prescribed treatment along with immune response but the issue of finding a control law such as a certain optimality criterion would be achieved hasn't been yet addressed. Many studies have already been carried out using a control theoretical approach to design an optimal drug therapy. But investigations were based on other type of mathematical models and mainly different objective functionals [START_REF] Culshaw | Optimal HIV treatment by maximising immune response[END_REF][START_REF] Fister | Optimizing chemotherapy in an HIV model[END_REF][START_REF] Garira | Optimal control of combined therapy in a single strain HIV-1 model[END_REF][START_REF] Joshi | Optimal control of an hiv immunology model[END_REF][START_REF] Kirschner | optimal control of the chemotherapy of hiv[END_REF][START_REF] Kirschner | Immunotherapy of HIV-1 infection[END_REF][START_REF] Stengel | Mutation and control of the human immunodeficiency virus[END_REF]. As we don't intend to model any mutation leading to drug resistance or to avoid side effects, control is applied in a finite short time interval [t 0 , t f ] with t f -t 0 < 100 days. The end-time control problem is considered with the cost function given by with

J ε,α,β (x, u) = φ(x t f ) + t f t 0 L(x, u, t)dt (9)
L(x, u, t) = α 2 V 2 + β 2 V 2 + ε 2 u 1 2 + u 2 2 (10) φ(x t f ) = α 2 V (t f ) 2 . ( 11
)
where

u ∈ ϑ : u = (u i ) /u i ∈ L 2 ]t 0 , t f [ : R 2 , 0 ≤ u i ≤ 1, i = 1, 2 (12)
and L(x, u, t), the Lagrangian of the problem. Unlike most papers, all parameters are squared in the objective functional. Existence of the result can be verified since all sufficient conditions are gathered but uniqueness can be debated [START_REF] Pontryagin | The mathematical theory of optimal processes[END_REF]. Scalar cost function includes a terminal cost associated to values of virus load at the end of the treatment as well as an integral cost of state and control along the period. Parameters α, β, ε are respectively weight constants for the virus, for the virus velocity and for the control inputs. They allow the balancing of size for each term and thus cost function can address various goals. First of all, one's target is to find a regimen which reduces high values of virus population both at the end time and during the period of the treatment. Also, we do not only want to minimize systemic cost of drugs in the aim to prevent from side effects and mutation of virus but also to slow down virus progression. An optimal control u * (t) , t 0 ≤ t ≤ t f is sought such as u * (t) minimizes the cost function J ε,α,β ,

u * = arg min u∈ϑ J ε,α,β (V, u) (13)
with the corresponding state x * (t) solution of state system subject to initial condition x * (t 0 ) ẋ * = F (x * , u * ),

x * (t 0 ) given. ( 14)

Methods for solving optimal control problem (OCPs) can be roughly classified in two different types -direct and indirect approaches [START_REF] Von Stryk | Direct and indirect methods for trajectory optimization[END_REF]. Direct methods optimise directly the cost functional using the parametrisation of control by approximating control and state vector with a sum of function expansion. The advantage is a good numerical robustness with respect to the initial guess but low accuracy of results is noticed. Indirect methods are based on Pontryaguine's minimum principle. Numerical convergence is fast and solutions are accurate if one starts with a good initial guess [START_REF] Bryson | Applied optimal control[END_REF].The optimality system obtained is a two-point boundary value problem, where initial conditions are specified for the state system ( 14) and terminal conditions are identified after calculations for the adjoint system ( 16), [START_REF] Kirschner | optimal control of the chemotherapy of hiv[END_REF]. Namely, the Hamiltonian of the problem is introduced

H ε,α,β,ω (x, u, p) = J ε,α,β (x, u) + t pF (x, u), ( 15
)
where p is the costate vector which components are called adjoint variables or more commonly Lagrange multipliers. According the Pontryaguine's minimum principle [START_REF] Adams | HIV dynamics: modeling, data analysis, and optimal treatment protocols[END_REF][START_REF] Callaway | HIV-1 infection and low steady state viral loads[END_REF][START_REF] Culshaw | A delay-differential equation model of HIV infection of CD4+ T-cells[END_REF][START_REF] Fister | Optimizing chemotherapy in an HIV model[END_REF][START_REF] Garira | Optimal control of combined therapy in a single strain HIV-1 model[END_REF][START_REF] Joshi | Optimal control of an hiv immunology model[END_REF][START_REF] Kirschner | Immunotherapy of HIV-1 infection[END_REF][START_REF] Stengel | Mutation and control of the human immunodeficiency virus[END_REF], the optimality conditions are,

ṗ = -t ∂H ε,α,β ∂x (x * , u * , p) (16)
or rather

ṗ = -t ∂F ∂x (x * , u * )p -t ∂L ∂x (x * , u * ) ( 17
)
with transversality condition

p(t f ) = ∂φ(x t f ) ∂x t f = t (0, 0, 0, αV (t f ), 0, 0) (18)
and

t ∂H ε,α,β ∂u (x * , u * ) = 0, ( 19
)
or another way

t ∂F ∂u (x * , u * )p + t ∂L ∂u (x * , u * ) = 0. ( 20
)

Numerical results and discussion

Analytical solution for optimal control is difficult to obtain since the systems are non-linear. One should proceed with an iterative gradient descent method. The dynamic systems response is exactly computed with adjusted control history from one iteration to the next in order to reduce cost function at each step. A starting guess for the control history u 0 1 and u 0 2 is made for the two controls on [t 0 , t f ]. Commonly zero or a constant control is chosen to initiate calculus. The state system ( 14) is solved forward from t 0 to t f using a variable step-size Runge-Kutta's algorithm taking into account initial state conditions. Cost function is then evaluated. Using the state values one solves the adjoint system backward integrating back from the end condition specified by [START_REF] Kirschner | optimal control of the chemotherapy of hiv[END_REF]. Unlike most papers the terminal state condition isn't reduced to zero due to end time first quadratic term and derivative term V under the integral in the cost function. Cost function gradient is then computed and condition [START_REF] Perelson | Mathematical analysis of hiv-1 dynamics in vivo[END_REF] is checked. Since condition is generally not satisfied a steepest-descent method is used to update the control,

u k+1 = u k -ρ k ∂J ε,α,β ∂u (u k ) 0 ≤ ρ k ≤ ρ 0 (21)
or according to control constraints

u k+1 = P ϑ (u k -ρ k ∂J ε,α,β ∂u (u k )) (22)
where P ϑ denotes projection operator on ϑ, ρ k is a small positive constant and k an iteration index. Proper selection of the step size ρ k is critical to get rapid convergence. In this paper, ρ k is set in order to minimize Minimizing f isn't an easy task due to the non linearity of the system. High instability of the optimality system in the adjoint variables can also be noticed. Variations of the adjoint variables even in controlled case can be very large. At each testing value ρ, a corresponding u ρ must be calculated and introduced to system ( 7)-( 8) to compute corresponding solution x ρ . A Nelder-Mead simplex algorithm is applied. This method is convenient because analytical or numerical gradient are not to be supplied and it is robust enough to handle non linear problems.

ρ ∈ R → f (ρ) = J ε,α,β (P ϑ (u k -ρ ∂J ε,α,β ∂u (u k ))). (23)
Noting thats it depends mainly on the initial values of control u 0 whether the method succed in finding the minimum point for the cost index. It may sometimes get to a saddle or even get lost. All calculus were carried out in a MATLAB environment. Results were also verified by employing a commercial package to solve continuous time optimal control problem for nonlinear dynamics, PROPT. Simulations are run with t f = 30, t f = 60 and t f = 100 days. In Fig. 3 to Fig. 8, results of population progression are presented in the case of optimal therapy for 1mm 3 of blood. Weighting constants must be chosen according to the relative significance between size of virus population and drug cost we want to establish in the objective function. As a drastic reduction of short term treatments eliminates totally infected sources and thereafter in- fection can resume. From a mathematical point of view, we force the model to drive toward the unstable healthy state by applying controls. Though optimal treatment allows to maintain constant CD4+ T cells almost the entire length of therapy without giving whole drug quantities. An acceptable approximation of the optimal control is able to make virus load decrease to values smaller than 100 copies per ml of blood at the end of treatement and therefore maintains a high healthy white cell count more than 800 copies. Treatment windows correlate with the period where greater oscillation takes place. The phenomenon is delayed and is drastically reduced in magnitude. In all cases, infected cell peaks of small magnitude and corresponding healthy cell falls occur at the end of therapie. RTI and PI dosage graphs are very different. Especially at each starting and end period where RTI drug dosage and PI drug dosage behaves in an opposite way. Corresponding values of cost functional for optimal control are given in Table [START_REF] Murray | Mathematical biology[END_REF]. For comparison, fully treated patient results are also provided. As expected, optimal cost is always smaller than fully treatment cost. As a matter of fact continuous variation of drug doses is difficult to apply in a real treatement of patient. For a pratical implementation drug dosages are sought in quantized levels. According to optimal control curves, treatment is divided in three steps. For RTIs medication, we notice a full dose at the beginning during the first week, a 15% to 20% of dose during main part of treatement and finally a gradually reduction to no dose throughout the last days. For PIs medication , we observe a gradually increasing dose at the beginning during the first week, a 15% to 20% of dose during main part of treatement and finally a full dose of drug throughout the last days. A bang-bang control can be applied as illustrated in Fig. 9 to Fig. 11 but is far from optimal in our problem since the cost index is J ε,α,β = 1.191 in sixty days treatment. It implies to put on and put off patient from treatment. It can be interpreted as scheduled interrupted treatments and has appeared to be a solution to bring relief from all those long term using drug complications. days full control optimal control final state 30 1 0.08 (999.9,3e-3,2e-3,9e-2,7e-2,48) 60 2 0.18 (999.9,3e-3,1e-3,9e-2,7e-2,49.9) 100 3 0.67 (999.9,3e-3,1e-3,5e-2,0.11,50)

Table 2: Values of the cost function J ε,α,β with ǫ = 1,α = 1e6 and β = 1e5.

Conclusions

In this paper, a deterministic model including immune response and multi drug effects is introduced to model HIV infection evolution. We use optimization theories in order to derive optimal control solution and design improved treatments. We proved that a reduced dosage of drugs can achieved similar goals than a constant level therapy and then can replace it. The possibility of a scheduled interrupted treatment was also considered through a bangbang control. It was not kept in our case since less optimal. Dynamic of infection is certainly far more complicated than the one captured by this simple mathematical model but this work illustrates the possibilities and difficulties of applying numerical methods of optimization to design future treatments. Above all, it mustn't be forgotten that non linear programming problems are to be addressed in most of the cases. Optimal control problems have received much attention and researches are still in progress to overcome solution convergence matters.

Figure 1 :

 1 Figure 1: Numerical solutions to the ordinary differential system with no treatment. Controls are fixed to u 1 = 0 and u 2 = 0. T 0 = 1000 cells and V 0 = 1e -3 virus for 1mm 3 of blood. The figures show history of : (a), the uninfected CD4+T cell population; (b), the latently infected CD4+T cells; (c),the infected CD4+T cells.

Figure 2 :

 2 Figure 2: Numerical solutions to the ordinary differential system with no treatment. Controls are fixed to u 1 = 0 and u 2 = 0. T 0 = 1000 cells and V 0 = 1e -3 virus for 1mm 3 of blood. The figures show history of : (d), active virus; (e), inactive virus; (f ), cytotoxic T lymphocytes.

Figure 3 :

 3 Figure 3: Numerical solutions to the optimality system showing reverse transcriptase inhibitor (a) and protease inhibitor (b) history when treatment is administered for 60 days.

Figure 4 :

 4 Figure 4: Numerical solutions to the optimality system when treatment is administered for 60 days.

Figure 5 :

 5 Figure 5: Numerical solutions to the optimality system when treatment is administered for 60 days.

Figure 6 :

 6 Figure 6: Numerical solutions to the optimality system showing reverse transcriptase inhibitor (a) and protease inhibitor (b) history when treatment is administered for 100 days.

Figure 7 :

 7 Figure 7: Numerical solutions to the optimality system when treatment is administered for 100 days.

  virus load is sought, we set α = 1e6, ǫ = 1 and β = 1e5. None of these

Figure 8 :

 8 Figure 8: Numerical solutions to the optimality system when treatment is administered for 100 days.

Figure 9 :

 9 Figure 9: Numerical solutions to the optimality system showing reverse transcriptase inhibitor (a) and protease inhibitor (b) history when treatment is administered for 60 days.

Figure 10 :

 10 Figure 10: Numerical solutions to the optimality system when treatment is administered for 60 days.

Figure 11 :

 11 Figure 11: Numerical solutions to the optimality system when treatment is administered for 60 days.

Table 1 :

 1 Parameters and constants used in the model.


Cong Wang

Jingwei Li

Erwan Paineau

Hynd Remita

Mohamed Nawfal Ghazzal
email: mohamed-nawfal.ghazzal@universite-paris-saclay.fr

Pt Atomically Dispersed in Black TiO 2-x /Cu x O with Chiral-Like Nanostructure for Visible-Light H 2 Generation

Keywords: black titania, chiral nematic structures, cellulose, single atom, hydrogen evolution

1.Introduction

Hydrogen is widely considered as one of the most promising clean energy to realize a decarbonized society. [START_REF] Van Renssen | [END_REF] However, the hydrogen currently produced is mostly obtained by reforming methane, an energy-consuming process that emits large amounts of carbon dioxide. [2] To face this so called "grey hydrogen", photocatalytic H 2 generation can realize a direct conversion from solar energy into "green hydrogen", which is beneficial for alleviating the energy crisis and environmental issues. [3] The photocatalytic efficiency depends on three main key factors, that are, solar light harvesting, photogenerated charge lifetime and surface reactivity. [4] The major driving force to enhance the light harvesting and then the photocatalytic performances is material design. Photonic crystal with periodic dielectric structure provides a unique approach to improve solar light harvesting. The periodic dielectric structure will generate a photonic stop band, where the velocity of photons slows down near the blue and red edges, thereby improving the photochemistry. [5] The confinement of the light in a periodically structured optical medium (due to multiple scattering) enables the control of the light propagation, in particular, can improve the light scattering to enhance the light absorption in films. In this context, cellulose nanocrystals (CNCs), bioderived material, possess the ability to self-assemble into chiral nematic (Ch) structure, which can be replicated into inorganic materials. [6] Structuring inorganic materials on the mesoscale using a self-biotemplate assembly process appears as a sustainable method for the synthesis of functional materials. For that purpose, impregnation method of free-standing cellulose photonic films, sol-gel self-biotemplating approach and microemulsion were proposed to structure TiO 2 in the films and spheres morphologies for improved photocatalytic hydrogen generation. [7] Even if these strategies provided an improvement in the photocatalytic efficiency, the activity of nanostructured TiO 2 is still limited to the UV range. To alleviate this restriction, metal nanoparticles are usually used as co-catalysts to promote charge separation by forming an interfacial Schottky barrier. [8] Notably, the size of the nanoparticles can influence markedly the photoactivity of photocatalysts in hydrogen evolution reaction (HER). [9] Downsizing the metallic cocatalyst to the ultimate single atomic site can maximize the atom-utilization efficiency and provide new features that are relevant to construct active interfaces. [10] Single atom (SA) co-catalysts inherit the advantages of heterogeneous and homogeneous catalysts, insuring a high dispersion at the photocatalyst surface and strong-metal support interaction. Moreover, SA exhibit a d-band depletion, which facilitates electronic hybridization and triggers the charge carriers transfer between the photocatalyst and the metal species. One of the major issues facing SAs is their stability against agglomeration. Defects-engineering has been proposed to stabilize SAs. Also, defect-engineering of TiO 2 into black TiO 2-x by introducing oxygen vacancy (O v ) is a promising and effective strategy to strengthen light absorption. [11] The defects and low oxidation state in TiO 2-x can narrow the bandgap and enable to broaden the spectrum absorption to visible light, [12] but the light harvesting capability remains far from expectation and needs to be further improved. Inspired by this phenomenon, we speculate that the construction of black TiO 2-x with chiral nematic like structure coupled with nanoparticles can simultaneously enlarge the absorption range and enhance the light harvesting capability of TiO 2 .

Herein, we successfully synthesized black TiO 2-x /Cu x O composite films with Ch nanostructure decorated with Pt SAs. One-pot self-assembly approach was employed to produce TiO 2 /Cu x O films using CNCs as structuring biotemplate. [7a] Subsequently, black TiO 2-x /Cu x O composites were achieved through hydrothermal reduction in the presence of ascorbic acid. The generation of surface oxygen vacancies enables the deposition of Pt SAs, which can enhance the separation of charge carriers. This novel construction showed combined synergetic behaviors: (i) the preserved Ch nanostructure pro-vides enhanced light harvesting capability; (ii) oxygen defects enable the broadband absorption under visible light, stabilize the Pt SAs and facilitates electronic hybridization to improve the charge separation. Such unique combination endows the resulting black TiO 2-x with chiral nematic like nanostructure with good visible-light H 2 evolution performance.

Results

Figure 1a depicts the schematic illustration of fabricating black chiral nematic TiO 2-x /Cu x O films decorated with Pt SAs. The mixture containing CNCs suspension, ethanolic Ti(acac) 2 (OiPr) 2 and Cu(OAc) 2 solution was cast into polystyrene Petri dish. CNCs rods self-organize into helical Ch structures by evaporation induced self-assembly (EISA) process, [7] yielding iridescent CNC/TiO 2 /Cu(OAc) 2 hybrid film. The Ch structure is evidenced from its fingerprint texture observed under cross-polarized optical microscopy (POM, Figure 1b and Figure S1) that corresponds to the helical pitch of the helix (Figure 1a). [13] The hybrid films were calcined at 500 ℃ under air to convert the amorphous TiO 2 to anatase, and formed heterojunctions with copper oxides, thus generating freestanding mesoporous TiO 2 /Cu x O films with Ch structures (named Ch-TiO 2 /Cu x O). The Ch structures are well preserved in the inorganic films after the removal of CNCs templates, as evidenced from SEM images showing alternated chiral like multilayers. In addition, the corresponding POM images dis-play strong birefringence where distinctive fingerprint textures are observed (Figure S2).

Ascorbic acid was reported as an efficient reductant of TiO 2 to some low-valence-state Ti species (Ti 3+ ), thereby forming defective black TiO 2-x forms. [14] In our procedure, Ch-TiO 2 /Cu x O films were impregnated in a 60 mM ascorbic acid aqueous solution and immediately followed by a hydrothermal treatment at 180 ℃ for 24 h. The resulting films turn black (named BCh-TiO 2-x /Cu x O), which attests to the reduction of Ti 4+ to Ti 3+ and the creation of O v . [15] It is worth mentioning that the iridescence of BCh-TiO 2-x /Cu x O films can be observed under natural light as shown in Figure 1e, confirming the conservation of the Ch nanostructures. The analysis of the films by POM shows typical fingerprint of the Ch patterns (Figure S3), further confirming the preservation of the Ch structure. The BCh-TiO 2-x /Cu x O films are stable under ambient atmosphere, and the O v -rich surface provides suitable trap sites to stabilize SAs. [16] The Pt SAs decoration was realized by immersing the BCh-TiO 2-x /Cu x O films in 1mM H 2 PtCl 6 solution (under anaerobic condition) in the dark for 24 h. The final BCh-TiO 2-x /Cu x O/Pt films displayed the same appearance and structural characteristics as the previous samples (Figure S4). Elemental analysis was performed using inductively coupled plasma-optical emission spectrometry (ICP-OES), revealing the mass content of Pt is 0.16 %. The morphology of the films were revealed by scanning electron microscopy (SEM). The cross-section SEM images of CNC/TiO 2 /Cu(OAc) 2 film display periodic multilayer with left-handed twist, indicating the formation of Ch structure (Figure 1c and Figure S5). Such kind of structure is preserved after the removal of CNCs template by calcination, but with a reduced helical pitch (Figure 1d, Figure S6a andS6b). and CuO ((112), 1.97 Å and (111), 2.35 Å) (Figure 2d and Figure S7a). [17] HRTEM further demonstrates that the reduction treatment does not affect the main crystal structure of TiO 2 , while introduced amorphous domains in the outer surface region (red dashed lines in Figure 2d). [18] The high-angle annular scanning TEM (HAA-STEM) under bright-field and dark-field were performed to confirm the successful Pt SAs anchor. More importantly, HAA-STEM images (Figure 2d and 2e, Figure S7b) reveal that Pt atoms exist as individual and isolated species substituted in TiO 2 (101, 3.5 Å), Cu x O (Cu 2 O (211, 1.72 Å) and CuO (111, 2.35 Å)) surface lattice. [19] These observations sustain the formation of single atom alloys (SAA). [20] The crystalline form of the films was further confirmed by X-ray scattering (XRS). All XRS diagrams are similar and display several well-resolved Bragg reflections related to nanosized TiO 2 crystallites. The indexation of the peaks is in good agreement with the anatase form of TiO 2 (Figure 3a). This indicates that the reduction guided by ascorbic acid only occurred on the outer surface of TiO 2 and has no significant effect on bulk crystalline structure. [11b] No diffraction peaks related to Cu or Pt are detected owing to the relatively small particle size and low loading. The UV-vis diffuse reflectance spectra (DRS) was performed to assess the optical properties of composite films (Figure 3b). The as-prepared TiO 2 film display typical absorption band of anatase, which is limited to the UV range. The addition of Cu x O induces a shift of the absorbance to higher wavelength. In comparison, BCh-TiO 2-

x /Cu x O exhibits a remarkable broadband absorption covering the entire visible region. The red-shift is attributed to the formation of O v defects and low-valence-state Ti 3+ , which are expected to introduce mid-gap states and narrow the bandgap. [11a, 18b] Moreover, the deposition of Pt SAs and formation of SAAs lead to a further enhancement in visible absorption. To confirm that the Ch like structure has an enhancement effect on the light harvesting, we analyzed the absorption spectrum of the films and the same materials, but crushed in a mortar to completely destroy the Ch like structures (Figure S8a). Compared to crushed samples, uncrushed samples exhibit enhanced light absorption, especially in the visible rang (Figure S8b). This result suggests that Ch like structure play an important role in the enhancement of visible light absorption due to the multiple light scattering. The optical bandgaps were estimated according to Kubelka-Munk (K-M) fitting. As shown in Figure S9, the bandgaps of BCh-TiO 2- illumination. We performed Raman spectroscopy characterization, whose results are depicted in Figure 3c,d. Both pristine TiO 2 and Ch-TiO 2 /Cu x O films present well-resolved peaks. In particular, the characteristic peak at ~ 143 cm -1 is shifted compared to BCh-TiO 2-x /Cu x O (≈ 149 cm -1 ) and BCh-TiO 2-

x /Cu x O/Pt (≈ 147 cm -1 ). This effect originates from the surface reduction of the TiO 2 structure by ascorbic acid, evidencing the reduction of the TiO 2 and the creation of surface defects and O v . [14a, 21] X-ray photoelectron spectroscopy (XPS) was employed to further investigate the surface composition of the asprepared films. The general survey of the films indicates the presence of C, O, Ti, Cu, and Pt species at the surface (Figure S10). The Ti 2p shows the typical characteristic peaks for Ti 2p 3/2 and 2p 1/2 at 458.9 eV and 464.6 eV, respectively, corresponding to the Ti 4+ in TiO 2 (Figure 4a). [22] The Ti 2p spectra are similar for all films, indicating that the bonding environment of Ti atoms is not modified after hydrothermal treatments and deposition of Pt. [11a] The high resolution Cu 2p spectra exhibits two major peaks located at 932.4 eV and 952.3 eV, respectively, which can be attributed to Cu 2p 3/2 and Cu 2p 1/2 (Figure 4b). In terms of the position, these two peaks mainly originate from Cu + species, rather than Cu 2+ species, which usually have higher binding energies. [23] However, the broad satellite peak can be observed in Ch-TiO 2 /Cu x O film, proving the existence of CuO. [24] Notably, the satellite peaks are not detected in black films after hydrothermal reduction using ascorbic acid, which suggests that part of the Cu 2+ is reduced to Cu/Cu + during this process. The O1s XPS spectra of TiO 2 can be deconvoluted into three peaks, i.e., lattice oxygen (O L , 530.1 eV), surface defects (O v , 531.6 eV) and adsorbed H 2 O (O ads ) with higher binding energy (Figure 4c). [25] Compared with the pristine TiO 2 , black TiO 2-x displays a significant increase in the concentration of O v , which is attributed to the formation of oxygen vacancy and/or surface defects in agreement with the results of UV-Vis and Raman spectroscopies. [18b, 26] Whereas the concentration of O v decreases again after loading Pt, which is probably due to the coverage of a large amount of O v by Pt atoms, suggesting the successful anchoring of Pt SAs. [27] According to the Pt 4f spectrum, Pt 0 and Pt 2+ are predominantly found in BCh-TiO 2-x /Cu x O/Pt film (Figure 4d). Consistent with previous studies, typical peaks of Pt 2+ /Pt 0 in Pt SA are located at 76.1/74.2 eV and 72.8/70.9 eV, respectively, corresponding to Pt 4f 5/2 and Pt 4f 7/2 . [28] The posi-tively charged Pt SAs are indicative of the existence of strong interactions between the metal and the support materials, allowing the two-way electron transfer to promote the photocatalytic reaction. [29] Mott-Schottky (MS) measurement were carried out to reveal the band structures and spatial band bending regions of composites. [30] All MS plots display positive slopes that possess n-type feature. Compared to Ch-TiO 2 /Cu x O, the flat-band potential of TiO 2 shifts from -0.64 to -0.59 eV due to the constructed internal electric fields in bandgap overlapped TiO 2 -Cu x O interfaces with band bending (Figure 5a). The introduction of O v in BCh-TiO 2-x /Cu x O enlarges the spatial charge layer region of internal electric fields on TiO 2-x -Cu x O interfaces, resulting in a positive shift of -0.56 V compared to Ch-TiO 2 /Cu x O. [31] Notably, the deposition of Pt SAs on O v of BCh-TiO 2-x /Cu x O induces a re-equilibrium of Fermi energy levels on BCh-TiO 2-x /Cu x O/Pt interfaces, which leads obviously to a positive shift to -0.35 V, with respect to BCh-TiO 2-

x /Cu x O. This suggests the formation of a significantly enlarged space charge layer, which is beneficial for the transfer and separation of photogenerated charge carriers. [31][32] Electrochemical impedance spectroscopy (EIS) was performed to provide insight into the photogenerated charge carriers transfer capability of composites. The smaller curvature radius on Nyquist plots relates to the lower charge transfer resistance (R ct ) at the interfaces of the photocatalyst. [24,33] A significant reduction of R ct is noticed in BCh-TiO 2-x /Cu x O/Pt films with respect to others, indicating that Pt SAs possess a strong metal-support interaction and play key role in improving charges transfer efficiency, in agreement with the MS plot (Figure 5b). In addition, the semicircle radius of all composites further decreases due to the generation of more carriers under light irradiation. The charge separation efficiency and redox ability of composite films are further analyzed by the cyclic voltammetry (CV) curves. In comparison, the BCh-TiO 2-x /Cu x O/Pt exhibits relatively higher redox peak intensity, which relates to the highest redox ability and charge transfer efficiency (Figure S11). [34] The photocatalytic activity of composite films was investigated through HER under visible light excitation (>420 nm). TiO 2 shows rather negligible amount of H 2 generation, while Ch-TiO 2-x /Cu x O develops interesting activity under visible light with kinetic rate of H 2 production of 12.7 μmol·g -1 ·h -1 . The reduced BCh-TiO 2-x /Cu x O films show significantly improved H 2 generation rate of 20.4 μmol·g -1 ·h -1 (Figure 5c and Figure S12) in good agreement with the absorption and photoelectrochemical measurements. The decoration of Pt SAs leads to an additional enhancement in photocatalytic activity with an H 2 evolution rate 46 times higher than that of TiO 2 under visible light. No significant photocatalyst deactivation was observed in the HER measurements after four consecutive cycles, indicating the robust stability of BCh-TiO 2-x /Cu x O/Pt (Figure S13). Based on the above investigations and characterisations, photocatalytic mechanism over BCh-TiO 2-

x /Cu x O/Pt films is proposed as summarized in Figure S14. The generated Ti 3+ -O v on the TiO 2-x surface leads to the formation of valence band (VB) and conduction band (CB) tailing and a consequent bandgap narrowing, which enables the absorption in visible region. [35] Upon visible light irradiation, photogenerated electrons could be excited from the VB of black TiO 2-x to its CB and subsequent migrate to Pt SAs deposited on the TiO x and/or Cu x O. As demonstrated above, Pt SAs significantly enlarge the space charge layer, facilitating both the transfer and separation of photogenerated charge carriers. As part of the process, Pt SAs and Pt/CuO x SAAs are suggested to act as reduction sites with prolonged charge carriers lifetime. The O v surface defects are able to increase the donor density and further participate in improving the charge transfer and reduce the charge carriers recombination. [18b] Moreover, Pt SAs can expand the spatial overlapped region and result in a further enhancement in charge carriers separation. BCh-TiO 2-

x /Cu x O/Pt benefits from multireduction sites producing higher H 2 under visible light illumination.

Conclusions

In summary, we used cellulose nanocrystals as biotemplates to fabricate a black TiO 2-x /Cu x O/Pt composite film with chiral nematic photonic nanostructure through simple one-pot self-assembly method and hydrothermal reduction. The combination of chiral nematic structure and abundant oxygen vacancies endows the black film with both enhanced light-harvesting capability and broadband absorption in the visible region. In addition, the deposition of Pt single atoms can significantly expand the spatial overlapped region for band bending, favoring the photogenerated charge separation and resulting in an excellent photocatalytic H 2 evolution efficiency under visible light. This unique coupling approach offers a promising strategy to meet the challenges of solar light utilization and charge separation in TiO 2 -based photocatalysts.

ASSOCIATED CONTENT Supporting Information

Synthetic details of BCh-TiO 2-x /Cu x O/Pt; associated characterizations include POM, SEM, HRTEM, XPS; electrochemical and photocatalytic experiments (PDF)

Figure 1 .

 1 Figure 1. (a) Schematic diagram for the fabrication of BCh-TiO 2-x /Cu x O/Pt films. (b) POM and (c) cross-section SEM images of CNC/TiO 2 /Cu(OAc) 2 . (d) Cross-section SEM image of Ch-TiO 2 /Cu x O. (e) Photograph of BCh-TiO 2-x /Cu x O films showing iridescent colour. There is no change in colour after Pt SAs deposition, and the iridescent colour are still present.The morphology of the films were revealed by scanning electron microscopy (SEM). The cross-section SEM images of CNC/TiO 2 /Cu(OAc) 2 film display periodic multilayer with left-handed twist, indicating the formation of Ch structure (Figure1cand FigureS5). Such kind of structure is preserved after the removal of CNCs template by calcination, but with a reduced helical pitch (Figure1d, FigureS6a and S6b). Further hydrothermal treatment and dark deposition only change the visual appearance of the films without affecting the

  Figure 1. (a) Schematic diagram for the fabrication of BCh-TiO 2-x /Cu x O/Pt films. (b) POM and (c) cross-section SEM images of CNC/TiO 2 /Cu(OAc) 2 . (d) Cross-section SEM image of Ch-TiO 2 /Cu x O. (e) Photograph of BCh-TiO 2-x /Cu x O films showing iridescent colour. There is no change in colour after Pt SAs deposition, and the iridescent colour are still present.The morphology of the films were revealed by scanning electron microscopy (SEM). The cross-section SEM images of CNC/TiO 2 /Cu(OAc) 2 film display periodic multilayer with left-handed twist, indicating the formation of Ch structure (Figure1cand FigureS5). Such kind of structure is preserved after the removal of CNCs template by calcination, but with a reduced helical pitch (Figure1d, FigureS6a and S6b). Further hydrothermal treatment and dark deposition only change the visual appearance of the films without affecting the overall Ch nanostructure. Consequently, well-defined multilayers and unique Ch structure remain visible for BCh-TiO 2- x /Cu x O/Pt films (Figure 2 a-c, Figure S6c and S6d). To go further, high-resolution transmission electron microscopy (HRTEM) was employed to characterize the nanostructure of BCh-TiO 2-x /Cu x O/Pt films. The interfaces between TiO 2 and Cu x O are in physical contact to form heterojunctions. The crystal lattices were identified as corresponding to anatase TiO 2 ((101), 3.5 Å) and the mixture of Cu 2 O ((200), 2.16 Å)

Figure 2 .

 2 Figure 2. (a, b) SEM images of BCh-TiO 2-x /Cu x O/Pt film observed along the fracture cross-section, showing periodical multilayers. (c) HRTEM image of BCh-TiO 2-x /Cu x O/Pt film. The red dashed line was created to reveal the amorphous region. (d, e) HAA-TEM images of BCh-TiO 2-x /Cu x O/Pt films under bright-field and dark-field showing the distribution of Pt SAs.The crystalline form of the films was further confirmed by X-ray scattering (XRS). All XRS diagrams are similar and display several well-resolved Bragg reflections related to nanosized TiO 2 crystallites. The indexation of the peaks is in good agreement with the anatase form of TiO 2 (Figure3a). This indicates that the reduction guided by ascorbic acid only occurred on the outer surface of TiO 2 and has no significant effect on bulk crystalline structure.[11b] No diffraction peaks related to Cu or Pt are detected owing to the relatively small particle size and low loading. The UV-vis diffuse reflectance spectra (DRS) was performed to assess the optical properties of composite films (Figure3b). The as-prepared TiO 2 film display typical absorption band of anatase, which is limited to the UV range. The addition of Cu x O induces a shift of the absorbance to higher wavelength. In comparison, BCh-TiO 2-

Figure 3 .

 3 Figure 3. (a) X-ray scattering patterns, (b) UV-vis diffuse reflectance spectra, and (c) Raman spectra of TiO 2 , Ch-TiO 2 /Cu x O, BCh-TiO 2-x /Cu x O and BCh-TiO 2-x /Cu x O/Pt composite films. (d) the local enlargement of (c).X-ray photoelectron spectroscopy (XPS) was employed to further investigate the surface composition of the asprepared films. The general survey of the films indicates the presence of C, O, Ti, Cu, and Pt species at the surface (FigureS10). The Ti 2p shows the typical characteristic peaks for Ti 2p 3/2 and 2p 1/2 at 458.9 eV and 464.6 eV, respectively, corresponding to the Ti 4+ in TiO 2 (Figure4a).[22] The Ti 2p spectra are similar for all films, indicating that the bonding environment of Ti atoms is not modified after hydrothermal treatments and deposition of Pt.[11a] The high resolution Cu 2p spectra exhibits two major peaks located at 932.4 eV and 952.3 eV, respectively, which can be attributed to Cu 2p 3/2 and Cu 2p 1/2 (Figure4b). In terms of the position, these two peaks mainly originate from Cu + species, rather than Cu 2+ species, which usually have higher binding energies.[23] However, the

Figure 4 .

 4 Figure 4. XPS spectra of (a) Ti 2p, (b) Cu 2p, (c) O 1s and (d) Pt 4f corresponding to TiO 2 , Ch-TiO 2 /Cu x O, BCh-TiO 2-x /Cu x O and BCh-TiO 2-x /Cu x O/Pt films.Mott-Schottky (MS) measurement were carried out to reveal the band structures and spatial band bending regions of composites.[30] All MS plots display positive slopes that possess n-type feature. Compared to Ch-TiO 2 /Cu x O, the flat-band potential of TiO 2 shifts from -0.64 to -0.59 eV due to the constructed internal electric fields in bandgap overlapped TiO 2 -Cu x O interfaces with band bending (Figure5a). The introduction of O v in BCh-TiO 2-x /Cu x O enlarges the spatial charge layer region of internal electric fields on TiO 2-x -Cu x O interfaces, resulting in a positive shift of -0.56 V compared to Ch-TiO 2 /Cu x O.[31] Notably, the deposition of Pt SAs on O v of BCh-TiO 2-x /Cu x O induces a re-equilibrium of Fermi energy levels on BCh-TiO 2-x /Cu x O/Pt interfaces, which leads obviously to a positive shift to -0.35 V, with respect to BCh-TiO 2-

Figure 5 .

 5 Figure 5. (a) Mott-Schottky plots of TiO 2 , Ch-TiO 2 /Cu x O, BCh-TiO 2-x /Cu x O and BCh-TiO 2-x /Cu x O/Pt composite films. Insets are the schematic of band bending. (b) EIS Nyquist plots of composite films under dark (solid line) and light illumination (dotted lines) situations. (c) H 2 evolution rate of composite films under visible light (λ > 420 nm).

x /Cu x O and BCh-TiO 2-x /Cu x O/Pt are 1.3 eV and 1.2 eV, respectively, which are smaller than Ch-TiO 2 /Cu x O (2.8 eV). The strong visible absorption and narrow bandgap make BCh-TiO 2-x /Cu x O/Pt films promising photocatalysts under visible

ACKNOWLEDGMENT

The authors thank Franc ois Brisset for collecting the SEM images, Paul Haghi-Ashtiani for the TEM images and Cyrille Hamon for his assistance in the Raman measurements.

AUTHOR INFORMATION Corresponding Author

Author Contributions

M.-N.G. and C.W. conceived the idea. J. W. L. conducted the photo-electro chemistry experiments. E.P. carried out the polarized optical microscopy, Raman spectroscopy and X-ray scattering experiments. C.W. curated the data. H.R. discussed the results. C.W., J. W. L. and M.-N.G. wrote the paper with the help from all authors.

Funding Sources

Cong Wang acknowledges the China Scholarship Council (CSC) for his fellow research position.

Notes

The authors declare no conflict of interest.