Claire Loupias
email: claire.loupias@univ-evry.fr

Bertrand Wigniolle
email: wignioll@univ-paris1.fr

Technological changes and population growth: the role of land in England 1

Keywords: endogenous fertility, land. JEL Classification: D9, J1, O11, R21

diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

During the industrial revolution, England has experienced a significant increase in total population, associated with a decrease in mortality. The outstanding growth rate was driven by a technical progress biased in favor of capital that generated an unbalanced growth process. The value added produced by capital increased dramatically with respect to the one produced by land (see [START_REF] Allen | Engels' pause: Technical change, capital accumulation, and inequality in the british industrial revolution[END_REF]. At the same time, the expected life at birth rose and infant mortality decreased (see [START_REF] Matteo | Human Capital Formation, Life Expectancy, and the Process of Development[END_REF][START_REF] Matteo | Human Capital Formation, Life Expectancy, and the Process of Development[END_REF][START_REF] Maddison | The Maddison-Project[END_REF].

In this paper, we build a model able to reproduce the actual data on population since 1760. This model incorporates capital, land, and endogenous fertility. Two main mechanisms will determine population dynamics: for households, the trade-off between housing and fertility; for firms, the trade-off between land and physical capital as production factors. Moreover, two external forces will drive the dynamics of capital and population: the capital biased technical progress and mortality rates. The model is able to mimic the historical evolution of population. Moreover, it allows to make a counterfactual analysis, and to disentangle the relative effect of technical changes and mortality fall on population dynamics.

Many articles have tried to provide explanations of the historical dynamics of population, growth, and industrialization. [START_REF] Kremer | Population Growth and Technological Change: one million B.C. to 1990[END_REF] is interested in the empirical relation between technology growth and population. Aggregate relations are assumed without microeconomic foundations; technological progress depends on population size and technology limits population growth. Combining these assumptions leads to the prediction that the growth rate and the size of population are positively related. [START_REF] Galor | Population, Technology and Growth: From the Malthusian Regime to the Demographic Transition[END_REF] propose a unified growth theory to explain the qualitative features of the demographic evolution. The main mechanisms are the quantity quality trade-off in fertility (as described in [START_REF] Doepke | Gary Becker on the Quantity and Quality of Children[END_REF] and a human capital accumulation technology that depends negatively on the growth rate of the economy. [START_REF] Kongsamut | Beyond Balanced Growth[END_REF] propose a theoretical explanation of the unbalanced growth of different sectors (agriculture, manufacturing, and services), using non constant consumption elasticities that vary with the level of consumption in each sector. [START_REF] Hansen | Malthus to Solow[END_REF] replicate fertility behaviors during the industrialization process, driven by the substitution of capital to land in production, which is induced by biased technical progress. Fertility behaviors are assumed to follow an ad hoc function of consumption. [START_REF] Matteo | Human Capital Formation, Life Expectancy, and the Process of Development[END_REF] provide an explanation of the development process that is based on the interplay between human capital formation, technological progress, and life expectancy, all endogenous in the model. But, fertility is not taken into account, neither land. Leukhina and Turnovsky (2016) investigate the roles of technology and trade in the structural transformation from farming to manufacturing of England. Population is taken as exogenous in their model.

All these contributions investigate the role of some particular variables in the development process. Our contribution is to emphasize the role of land, life expectancy, and biased technical progress in the population growth. We adopt a perspective close to [START_REF] Hansen | Malthus to Solow[END_REF], with three improvements: a microfoundation of the fertility behavior, an explicit land market allocation, and a confrontation of the model with historical data. We build on [START_REF] Loupias | Population, land, and growth[END_REF] which have developed a theoretical model on the same topic. The present paper adopts a very different perspective. Its aim is to reproduce historical data of population in England. To do that, we simplify the technology in taking the technical progress as exogenous. The model is fully calibrated using historical data and succeeds in reproducing the historical population growth.

The present paper develops an overlapping generations model in which fertility is endogenous. The utility of the parents is a function of good consumptions, of the number of their children, and of the consumption of a fixed asset: land. Under the form of housing services, land provides utility to households. Moreover, as the demand for housing services depends on the number of children, land is also related to fertility behaviors. Each child implies a financial cost and induces a congestion effect on the utility of land. These assumptions introduce a trade-off between housing and fertility.

To complement our model we introduce two types of survival probabilities: a child survival rate and an adult survival rate. As shown in [START_REF] Aghion | The relationship between health and growth: when Lucas meets Nelson-Phelps[END_REF], improvement in life expectancy has a significant positive impact on per capita GDP growth.

In our analysis, land can be also used as a production factor by firms. Production uses three factors: labor, capital, and land. Capital and land are both affected by a specific technical progress term. These two technical progresses generate a GDP growth at aggregate level and a shift in the relative income shares of capital and land in GDP.

The model is calibrated using historical data for mortality rates, GDP growth rates, and the shares of capital and land incomes in GDP.

The model is able to reproduce the dynamics of population since 1760. Moreover, it is possible to disentangle the relative effect of technical changes and mortality fall on the evolution of population. We conduct a counterfactual analysis eliminating successively the increase in life expectancy and the technological bias. With no increase in life expectancy, population would have been respectively 10% and 30% lower in 1910 and in the long run. The figures would have been respectively 40% and 60% lower, with no bias in the technical progress. Finally, population would have been 45% smaller in 1910 and 70% smaller in the long run, neutralizing both the effect of life expectancy and technological bias. According to our model, the major part of population increase is due to the technological bias evolution between land and capital.

The dynamics of the model is driven by two mechanisms. First, land and capital are assumed to be substitutable. An increase in technical progress in favor of capital allows firms to substitute capital to land, which leaves more land services available for households. Since cost of child rearing depends partly on land rent rates, more supply of land services for households decreases the effective cost of child rearing. This leads to an increase in fertility, which coupled with falling mortality, leads to a higher population growth rate.

Secondly, as utility depends on land with a congestion effect, the model generates a long run equilibrium value for population density. Starting with a population density lower than the long run value leads to a higher fertility rate in the short run. These two mechanisms add up and drive an increase in population.

In the long run, the trade-off between housing and fertility for the households leads, via the congestion effect, to a constant population level. Removing the congestion effect assumption would lead to a constant population growth rate, but not to a constant population level. The actual level of population in England in the long run is assumed to stabilize at 58 millions of inhabitants. The model is calibrated to match this data and to reproduce the historical population dynamics. Thanks to our calibration, the model is able to reproduce population dynamics. Nevertheless, we have neglected a lot of other possible mechanisms that have been studied in the literature, as the quantity quality trade-off or the introduction of an old age insurance.

Section Two presents the model. Section Three analyzes the dynamics of the intertemporal equilibrium. Section Four describes the calibration. Section Five compares simulation results to the stylized facts and gives counterfactual analysis. Section Six concludes and section Seven gives references. A last section of appendix provides the numerical results obtained through counterfactual analysis.

The Model

We develop a two-period overlapping generations model à la [START_REF] Diamond | Nominal Debt in a Neoclassical Growth Model[END_REF] where fertility is endogenous. The life cycle of agents consists of one working period and one retirement period. Childhood implicitly exists as an initial period of life during which agents have a probability η to survive. The number of units of labor is equal to the number of young people and thus determined by households' fertility decisions in the previous period. In every period the economy produces a single homogenous good, using land, labor, and capital as inputs. Production benefits from two biased technical progress in favor of capital and land. The single good is used both for consumption and capital accumulation. Land is a fixed factor that includes agricultural land, business building, and housing. Services of land may be used both by firms as input in the production process and by households as housing. For the sake of simplicity, its supply is assumed to be constant and exogenous.

The first subsection is devoted to the firm, the second to the households, and the last one to market equilibrium.

The firm

Production occurs according to a constant-returns-to-scale technology that is subject to technological progress. The output produced at time t, Y t , is: The aggregate formulation of the production function (1) can be interpreted as a reduced form of a two-sector model, such as [START_REF] Hansen | Malthus to Solow[END_REF], Galor et alii (2009) or [START_REF] Matsuyama | Agricultural productivity, comparative advantage, and economic growth[END_REF]. To show this, let us assume a sector one more intensive in land:

Y t = λ(AS t K t ) 1-1 ε + (1 -λ) (AM t X t ) 1-1 ε εα ε-1 L 1-α t (1) with 0 < α < 1, 0 < λ < 1, ε > 1, where K t , L t ,
F 1 (L 1 t , X t ) = [AM t X t ] α L 1 t 1-α
and a sector two more intensive in capital:

F 2 (A t K t , L 2 t ) = (AS t K t ) α L 2 t 1-α
where L 1 t is the quantity of labor in sector one and L 2 t the quantity of labor in sector two. The final good is a CES aggregation of the goods produced by the two sectors:

Y t = a (AS t K t ) α L 2 t 1-α δ-1 δ + b [AM t X t ] α L 1 t 1-α δ-1 δ δ δ-1
where a and b are positive coefficients. The elasticity of substitution δ is assumed to be higher than 1. The case considered in [START_REF] Hansen | Malthus to Solow[END_REF] is the limit case δ → +∞. Labor is shared between the two sectors:

L t = L 1 t + L 2 t
In a competitive economy, labor is allocated optimally among the two sectors, and it is possible to derive an aggregate production given by (1) for appropriate values of δ, a and b. See Appendix 1 for more details.

The capital is fully depreciated in one period. The number of units of labor is determined by households' decisions in the preceding period regarding the number of their children. Households have property rights over land. The land used as an input by the firm is rented from households. The rent rate is taken as given by the firm.

The firm maximizes its profit, taking the wage rate w t , the interest rate (R t -1), and the rent rate π t as given.

First order conditions for the optimization problem are derived below. All markets are perfectly competitive. On the labor market the quantity of labor used in production L t is equal to N t the number of young households at period t. Defining, k t ≡ Kt Nt and x t ≡ Xt Nt , the competitive wage, the interest factor, and the rent rate are:

w t = (1 -α) λ(AS t k t ) 1-1 ε + (1 -λ) (AM t x t ) 1-1 ε εα ε-1
(2)

R t = αλ(AS t ) 1-1 ε k -1 ε t λ(AS t k t ) 1-1 ε + (1 -λ) (AM t x t ) 1-1 ε εα ε-1 -1
(3)

π t = α(1 -λ)(AM t ) 1-1 ε x -1 ε t λ(AS t k t ) 1-1 ε + (1 -λ) (AM t x t ) 1-1 ε εα ε-1 -1 (4)

Households

Households are behaving as in [START_REF] Loupias | Population, land, and growth[END_REF]. In each period t a generation consists of N t identical adult individuals. Members of generation t live with probability p t for two periods and die with probability (1 -p t ) at the end of the first period. p t is taken as exogenous, as it will be calibrated following historical data. Generation t agents work in the first period and are retired during the second one. Members of generation t choose at date t consumption while young (c t ) and old (d t+1 ), as well as the number of their children per adult (m t ), and their use of land (v t ). Only a fraction η t of the children m t survives. Individuals of generation t implicitly live for three periods: childhood (in t -1), young adult (in t), and old adult (in t + 1). The preferences of members of generation t are represented by the utility function:

U (c t , d t+1 , m t , v t ) = Γ 1 ln c t + p t Γ 2 ln d t+1 + Γ 3 ln η t m t + Γ 4 ln(v t -ξη t m t ) (5)
where ξ is a positive parameter and

Γ 1 + Γ 2 + Γ 3 + Γ 4 = 1.
Households maximize their expected utility taking into account the probability of reaching the second period. One can define ζ t ≡ v t -ξη t m t that measures the services of land per adult. It is increasing with the total amount of land per adult and decreasing with the number of surviving children per adult. For tractability, it is assumed that households value the land services only when young adults.

Since [START_REF] Dusansky | The Demand for Housing: Considerations[END_REF], it is a standard assumption to consider that land services are an argument of the utility function. What is new here is the congestion effect due to children introduced by [START_REF] Loupias | Population, land, and growth[END_REF].

Land plays two roles for households. The first role is housing for which they pay the rent π t v t when young adult. Secondly, land is a portfolio asset that is bought in period t, that yields rents in t + 1, and that is sold in t + 1 to the next generation. In t + 1, rents are paid both by households and firms to owners.

Each newborn child entails a rearing cost of φ 1 w t . Moreover, for each surviving child, an additional cost of φ 2 w t is borne: the costs of rearing children are proportional to the standard of living of their parents. Through the paper φ 1 and φ 2 are assumed to be constant parameters. The total cost of children in consumption good (housing not included) is thus:

(φ 1 + φ 2 η t )w t m t
The number of surviving children per adult is m t ≡ η t m t . The corresponding cost is φ t w t m t with:

φ t = φ 1 η t + φ 2
The agent saves an amount s t that is shared between two assets: productive capital and land. As agents can arbitrate between the two assets, the non arbitrage condition implies that land offers the same return as capital. The gross return on capital is R t+1 . One unit of land has a price q t in period t and is resold q t+1 in t + 1. Moreover, it allows to earn a rent π t+1 . The non arbitrage condition is written as follows:

R t+1 = q t+1 + π t+1 q t (6)
Members of generation t maximize their intertemporal utility function under the following budget constraints:

c t + s t + φ t w t m t + π t v t = w t (7) d t+1 = R t+1 p t s t (8)
The actual return on savings is ρ t+1 ≡ R t+1 pt as the savings of the dead agents are redistributed to the surviving ones. This is equivalent to assume the existence of a perfect annuity market. Note that using ζ t (the services of land per adult), one can easily make clear the real cost of one surviving child (φ t w t +ξπ t ) which can be broken down as the sum of the cost in consumption good and the cost in land:

c t + s t + (φ t w t + ξπ t ) m t + π t ζ t = w t (9)
The intertemporal budget constraint may be rewritten as:

c t + d t+1 ρ t+1 + (φ t w t + ξπ t ) m t + π t ζ t = w t (10)
In macroeconomics models with representative agents, it is usual to consider that fertility and cost of children are continuous variables, even if at a microeconomic level, the number of children is discrete and the cost of children may jump because of indivisibilities. The assumption of continuous variables is a simplifying assumption that may be viewed as a consequence of aggregation.

In the model, the total cost of children (φ t w t + ξπ t ) m t is linear in the number of surviving children. The cost per child φ t w t + ξπ t is the sum of the cost of land renting ξπ t and the rearing cost φ t w t which depends on the standard of living. The assumption of linear costs with respect to the number of children has been used in some macroeconomic models (see for instance Eckstein et alii 1988, Becker et alii 1990[START_REF] Galor | Population, Technology and Growth: From the Malthusian Regime to the Demographic Transition[END_REF][START_REF] De La Croix | Inequality and Growth: Why Differential Fertility Matters[END_REF]. It is a simplification with respect to microeconomic analysis that points out the possible economies of scale. In Appendix 2, we show that it is possible to add non linearities in the cost functions related to economies of scale. This more general framework would require a change in the calibration of the model, but the main qualitative properties would remain.

First order conditions for the optimization problem lead to the following solutions:

c t = γ 1,t w t (11) s t = γ 2,t w t (12) d t+1 = γ 2,t w t ρ t+1 (13) m t = γ 3,t w t (φ t w t + ξπ t ) (14) v t = ξγ 3,t w t (φ t w t + ξπ t ) + γ 4,t w t π t (15) with γ 1,t = Γ 1 Γ 1 + p t Γ 2 + Γ 3 + Γ 4 (16) γ 2,t = p t Γ 2 Γ 1 + p t Γ 2 + Γ 3 + Γ 4 (17) γ 3,t = Γ 3 Γ 1 + p t Γ 2 + Γ 3 + Γ 4 (18) γ 4,t = Γ 4 Γ 1 + p t Γ 2 + Γ 3 + Γ 4 (19)
As shown in equations ( 11) to ( 19), a rise in life expectancy (p t ) increases γ 2,t , and savings s t . It decreases first period consumption c t , fertility m t , and demand for land v t .

The number of young households at date t + 1 is by definition equal to:

N t+1 ≡ m t N t (20)
Total population at date t can be written as:

N tot t = p t-1 N t-1 + N t + N t+1 (21)
Thus, the survival probability at old age p has a direct effect on total population (via the number of old individuals) and indirect effects via m t-1 and m t as γ 3,t-1 and γ 3,t are respectively depending on p t-1 and p t .

From now on, the lower case designates the upper case variable divided by the number of young individuals. For instance, x t is defined as X Nt the quantity of land available per young living agent. The evolution of land per young alive can thus be described by the following equation:

x t+1 = x t m t (22)

Market equilibrium

Land has two prices: the rent rate π t and the price for sale q t . 1 There are thus two markets: one for land services and one for ownership. It is the rent

1 These two prices are not similar to the two prices for durable goods as in M. Morishima (1995) : i ) land is not produced: ii ) contrarily to capital, there are no vintage effects on land; iii ) the existence of the two markets come from the two uses of land (land services -either for firms as production factor or for households as housing-, and allocation of savings). rate π t that determines the allocation of rented land between firms (x t ) and consumers (v t ). The equilibrium on the rent market expressed per head of young household is:

v t + x t = x t ( 23
)
The price of land for sale q t depends on the global equilibrium on savings market. Household savings have to be split into physical capital and land.

γ 2,t w t = m t k t+1 + q t x t ( 24
)
where k t+1 stands for the capital per young household at date t + 1. The amount of physical capital per young household available in the economy in t + 1 is thus depending on the value of land q t x t . Agents are indifferent in investing in capital or land as long as the non arbitrage condition in portfolios holds (6).

Dynamics

In this section, we characterize the dynamics and transform the model in a way that makes it comparable to historical data. The first subsection defines the intertemporal equilibrium. In the second subsection variables are deflated with respect to technological progress parameters. The third subsection replaces some unobservable variables by observable ones, and the fourth conducts a theoretical analysis of the dynamics.

Intertemporal equilibrium

The dynamics of the economy is characterized by the set of the nine previous equations:

-( 2), (3), and (4), the equilibrium prices of production factors w t , R t , π t , -( 14), and (15), the optimal behavior of households for fertility m t , and housing v t , -( 22), the evolution of land per young alive, x t , -( 23), the equilibrium allocation of rented land between firms and households, -(24), the equilibrium allocation of savings between land and capital, -(6), the non arbitrage condition between the yields of land and capital. These equations determine the nine endogenous variables k t , x t , v t , m t , R t , π t , w t , q t , and x t .

Deflated model

Variables are deflated in order to be stationary in the long run.

We define G t and A t as follows:

G t = AS t+1 AS t A t = AM t (AS t ) 1/(1-α)
G t is the growth factor of the capital productivity level and A t is a measure of the technological bias between the capital and the land factor. Defining the deflated variables kt , πt , wt , and qt , as:

ht = h t (AS t ) α/(1-α)
we rewrite the model of the previous section as a system of nine equations with nine endogenous variables ( kt , x t , v t , m t , R t , πt , wt , qt , and x t ) and two exogenous variables (G t and A t ).

Substituting in the model of the previous section, one has:

wt = (1 -α) λ( kt ) 1-1 ε + (1 -λ) (A t x t ) 1-1 ε εα ε-1 (25) R t = αλ k-1 ε t λ( kt ) 1-1 ε + (1 -λ) (A t x t ) 1-1 ε εα ε-1 -1 (26) πt = α(1 -λ)(A t ) 1-1 ε x -1 ε t λ( kt ) 1-1 ε + (1 -λ) (A t x t ) 1-1 ε εα ε-1 -1 (27)
m t = γ 3,t wt (φ t wt + ξπ t ) (28) v t = ξγ 3,t wt (φ t wt + ξπ t ) + γ 4,t wt πt (29) x t+1 = x t m t (30) v t + x t = x t (31) γ 2,t wt = m t kt+1 G α/(1-α) t + qt x t (32) R t+1 = qt+1 + πt+1 qt G α/(1-α) t (33)
So we have a system of nine equations with nine endogenous variables ( kt , x t , v t , m t , R t , πt , wt , qt , and x t ) and two exogenous variables (G t and A t ).

Unfortunately, G t and A t are not directly observable. In the next subsection we find a way to replace G t and A t by observable exogenous variables.

Capital share and growth rate

From the theoretical model we can compute the three factor shares in production:

β X t = πt x t wt + R t kt + πt x t β K t = R t kt wt + R t kt + πt x t β W t = wt wt + R t kt + πt x t
We define Z t as the growth factor of production:

Z t = Y t+1 Y t
Our aim is to calibrate the model using historical data. As G t (the growth factor of the capital productivity level) and A t (a measure of the technological bias) are unobservable, we replace them in the equations of the model by Z t and β K t , which are observable in the data. Computations are given in appendix 3. Two key equations allow understanding how it is possible to identify A t and G t from Z t , β K t , and the other endogenous variables of the model:

A t = λ kt x t α β K t -1 ε ε-1 1 1 -λ ε ε-1 (34) G t = Z t m t (1-α) α β K t+1 β K t ε(1-α) ε-1 kt kt+1 (1-α) (35)
(34) shows the relation between technical bias A t and the share of capital income in total production β K t . When A t becomes close to zero, the bias in favor of capital is huge, the share of capital income in total production β K t becomes close to α, and the share of land income in total production β X t close to zero.

(35) shows that the technical progress on capital G t is the main determinant of production growth Z t .

Using historical data for β K t and Z t , the model allows to recover the values for A t and G t through equations ( 34) and ( 35). In other words, these two observable variables β K t and Z t are substituted to the two exogenous variables A t and G t , as they are functions of A t and G t and the three endogenous variables m t , kt , and x t .

In the end, the dynamics of the economy can be written as:

wt = (1 -α) kα t λα β K t εα ε-1 (36) R t = β K t kα-1 t λα β K t εα ε-1 (37) πt = α -β K t x t kα t λα β K t εα ε-1 (
x t+1 = x t m t (41) x t = v t + x t (42) γ 2,t wt = Z t β K t+1 β K t εα ε-1 kα t k1-α t+1 + qt x t (43) R t+1 = qt+1 + πt+1 qt Z t m t β K t+1 β K t εα ε-1 kt kt+1 α (44)
So we have a system of nine equations with nine endogenous variables ( kt , x t , v t , m t , R t , πt , wt , qt , and x t ) and two observable variables β K t and Z t .

Theoretical analysis of the dynamics

The dynamics of the variables x t , v t , m t , and x t can be studied as an autonomous subsystem as:

wt πt = (1 -α) (α -β K t )
x t and thus only depends on the quantity of land used by firms x t , and not on kt .

Using this property, equation ( 57) can be written:

m t = γ 3,t (1 -α)x t φ t (1 -α)x t + ξ(α -β K t ) (45)
Equation ( 58) can be written: 42), we obtain a relation between x t and x t :

v t = ξγ 3,t (1 -α)x t φ t (1 -α)x t + ξ(α -β K t ) + γ 4,t (1 -α)x t (α -β K t ) Replacing in (
x t = x t + ξγ 3,t (1 -α)x t φ t (1 -α)x t + ξ(α -β K t ) + γ 4,t (1 -α)x t (α -β K t ) (46)
Thus, one can get x t from x t as x t is monotonically increasing in x t . Finally, equation ( 41) with (45) determines the dynamics of x t :

x t+1 = x t φ t (1 -α)x t + ξ(α -β K t ) γ 3,t (1 -α)x t (47)
In the end, the dynamics of N t does not depend on kt due to the homothetic assumptions on the utility and the production functions combined with a child cost proportional to wages.

Z t has no effect on population. β K t , η t (via φ t ), and p t (via γ 3,t ) are the exogenous shocks that determine N t .

Equations ( 45), ( 46) and ( 47) allow to understand how the technological progress affects fertility and population growth. The bias of technological progress in favor of capital induces an increase in β K t , which increases the net fertility factor m t , all other things being equal. Firms substitute capital to land, wt /π t increases, fertility increases as relative cost of land is cheaper for households. As long as population increases, both x t and x t decrease. The decrease of the quantity of land per adult used by firms x t leads to a decrease in fertility m t . These two antagonistic effects on m t lead to an inverse U-shaped evolution of fertility.

The two equations ( 43) and (44) determine the dynamics of kt and qt , with the prices R t , wt , and πt , given by ( 37), (36), and (38). The other variables, m t and x t , have been determined by the autonomous system analyzed above.

Introducing the variable, χ t = qt kα t the system of the two equations ( 43) and ( 44) becomes:

γ 2,t (1 -α) λα β K t εα ε-1 = Z t β K t+1 β K t εα ε-1 k1-α t+1 + χ t x t (48) β K t+1 kα-1 t+1 λα β K t+1 εα ε-1 = χ t+1 + α-β K t+1 x t+1 λα β K t+1 εα ε-1 χ t Z t m t β K t+1 β K t εα ε-1 (49)
Eliminating k1-α t+1 between these two equations, an autonomous dynamic equation in χ t is obtained. χ t is a forward looking variable determined by the terminal condition. As χ t is determined, equation ( 48) allows to find kt+1 . Thus, k0 has no impact on the dynamics, as kt+1 does not depend on kt . This is a usual property in endogenous fertility models with Cobb-Douglas production function and log-linear preferences.

Calibration

Subsection 1 is devoted to the value of parameters and exogenous variables and subsection 2 to the simulation strategy.

Parameters and exogenous variables

Parameters

The model incorporates ten parameters:

• ε, λ, and α for technology,

• Γ 1 , Γ 2 , Γ 3 , Γ 4 , and ξ for households' preferences,

• φ 1 and φ 2 for child costs.

The parameters used to simulate the dynamics are given in table 1. α is taken from historical data from the share of labor in GDP (1 -α), see the next subsection.

Parameters Technology λ = 0.5 ε = 10 α = 0.45 Utility Γ 1 = 0.35 Γ 2 = 0.25 Γ 3 = 0.3 Γ 4 = 0.1 ξ = 1 Cost of a child φ 1 = 0.08 φ 2 = 0.07

Table 1: Calibration

Exogenous variables

There are four exogenous variables, taken from historical data: -η t and p t for surviving probabilities, -β K t and Z t for the share of capital in production and the growth factor. Details on parameters and historical data are given below.

Technology

We recall that the production function is:

Y t = λ(AS t K t ) 1-1 ε + (1 -λ) (AM t X t ) 1-1 ε εα ε-1 L 1-α t ( 50
)
Parameters ε and λ are taken as ε = 10 and λ = 0.5. We assume a high substitutability between capital and land. As mentioned previously, our technology might be viewed as the aggregation of two sectors as in [START_REF] Hansen | Malthus to Solow[END_REF]. In their model, perfect substitutability is assumed (thus ε → ∞). We have chosen a high degree of substitutability (ε = 10), but not infinite.

The impact of the two technological progresses AS t and AM t is measured indirectly by Z t and β K t , where Z t is the growth factor and β K t the share of capital in production. Z t is measured as the England production growth factor on 30 years for periods between 1730 and 1910, and β K t is the share of capital incomes in production for England at the same dates.

The share of capital income in production β K t is taken from [START_REF] Allen | Engels' pause: Technical change, capital accumulation, and inequality in the british industrial revolution[END_REF], see Figure 1 below. From these data, the share of labor income β W t = 1 -α can be considered as constant over the period and equal to 0.55. Therefore, α = 0.45. The rest of the income is shared between land β X t and capital β K t . The figure for 1730 is not available, so we have taken β K t = 0.21 assuming that the evolution is the same between 1730 and 1760 than between 1790 and 1760. The share of capital income in production β K t is bounded by 0.45 as β K t + β X t = 0.45. As the share of agricultural land income in GDP for UK is around 1% in 2000 (from the World Bank database), we report 0.44 for 2000. Note that the data from [START_REF] Allen | Engels' pause: Technical change, capital accumulation, and inequality in the british industrial revolution[END_REF] about the share of land in income β X t correspond to the geographical area of England and include neither the rest of the United Kingdom nor the colonies.

The growth factor Z t reported in the 1760 column is the one from 1730 to 1760, and so on. The figures come from the Historical Statistics of the World Economy: 1-2006 AD from [START_REF] Maddison | The Maddison-Project[END_REF]. Details at the beginning of the 18th century are inferred from Craft (2004). Data for the growth factor Z t have been also reported after 1910 from [START_REF] Maddison | The Maddison-Project[END_REF] for U.K. in order to be consistent with demographic data (see above).

Preferences and costs

As mentioned above, utility is written as (5):

U (c t , d t+1 , m t , v t ) = Γ 1 ln c t + p t Γ 2 ln d t+1 + Γ 3 ln η t m t + Γ 4 ln(v t -ξη t m t )
The parameters are fixed to Γ 1 = 0.35, Γ 2 = 0.25, Γ 3 = 0.3, Γ 4 = 0.1, and ξ = 1. With this choice, the rate of time preference δ is such that (1 + δ) 30 = p t Γ 2 /Γ 1 . The rate of time preference δ is thus decreasing from 6.7% per year to 1.3%, thanks to the increase in the surviving probability p t . Γ 3 determines fertility and is chosen to replicate the evolution of population. Population also crucially depends on φ t , and we can find several combinations of Γ 3 and φ t able to match data on population growth.

The cost in consumption good of one surviving child is φ t = φ 1 η t + φ 2 . We have chosen φ 1 = 0.08 and φ 2 = 0.07. The total cost of one surviving child including housing, expressed as a fraction of time per adult, is:

φ t + ξ π t w t
In the long run, according to equation ( 57), as m t = 1, we get ξ π∞ w∞ = γ 3,∞ -φ ∞ . Thus the total cost of one surviving child (φ ∞ + γ 3,∞ -φ ∞ ) per adult including housing in the long run is γ 3,∞ = 0.354 which is in line with the calculations of [START_REF] Apps | Household production, full consumption and the costs of children[END_REF], and [START_REF] Bargain | Expenditure on children: A Rothbarth-type method consistent with scale economies and parents' bargaining[END_REF]. Some sensitivity analysis have shown that what matters for the results is mainly the relative values of γ 3,∞ and φ ∞ , and not their level.

Demographics

Population for England before 1870 is taken from [START_REF] Wrigley | The population history of England: 1541-1871[END_REF]. Other figures are taken from University of Portsmouth (2015). The figures are reported below. All figures are expressed in millions. Note that we focus on population of England and not of United Kingdom. The surviving probability at 50 years old p t are computed in the following way. We assume that the childhood period is of 20 years, and that the two periods of adulthood last both for 30 years. Thus, children born in period t arrive on the eleventh year of this period. The three stages of life are then 0-20 years, 21-50 years, and 51-80 years. Observed expected life at birth is taken from [START_REF] Matteo | Human Capital Formation, Life Expectancy, and the Process of Development[END_REF]. Theoretical expected life at birth in our model is equal to (20 years) η t + (30 years) η t +(30 years) η t p t ; this allow us to compute p t in a way that is consistent with the model. Computed values are reported in the above table. Table 5: Model chronology

Simulation strategy

The model has two state variables (backward looking) x t and kt . For kt , the initial condition k0 has no impact on the dynamics, as shown in section 3.4, as kt+1 does not depend on kt . x 0 is chosen in order to reproduce the historical dynamics of population.

Using equations ( 46) and ( 47), the limit value of xt can be determined:

x ∞ = ξ (1 -α)(γ 3,∞ -φ ∞ ) (1 -α)(γ 3,∞ -φ ∞ ) + (α -β K ∞ ) + γ 4,∞ (1 -α)
The limit value of the size of the young adult generation tends to N ∞ = X x∞ . The value of X is chosen such that the limit value of population is 58 Moreover, it is possible to disentangle the relative effect of technical changes and mortality fall on the evolution of population. We conduct a counterfactual analysis eliminating successively the increase in life expectancy, the technological bias, and both of them.

Life expectancy

In this section, we successively neutralize the impact of the increase in life expectancy at 50 years old and the decrease in child mortality. Results are presented in Figure 3. Ntot pinitial is the computed total population for a surviving probability at 50 years p t that keeps its value of 1730. Ntot pinitial etatinitial is the computed total population for both the surviving probability of young children η t and the surviving probability at 50 years p t that keep their values of 1730. With no increase in life expectancy, neither during childhood nor at 50 years old, the population would have been 10% lower in 1910 and 30% lower in the long run, according to our model.

Technological Bias

In this section, we neutralize the impact of the technological bias: β K t keeps its 1730 value.

Figure 4 displays the evolution of the computed population without the technological bias (Ntot technoinit).

Life expectancy and Technological Bias

In this section, we neutralize successively the impact of surviving probabilities, the impact of the technological bias, and both of them. Results are all depicted in Figure 5 where Ntot technoinit pinitial etainitial stands for total computed population without any increase in surviving probabilities and no technological bias. We observe that the major part of population increase from 1730 is due to the technological bias evolution between land and capital.

Conclusion

In this paper, we reproduce the dynamics of population in England since 1760, using an overlapping generations model with endogenous fertility and land. The population growth is driven by a bias technological progress and life expectancy improvement. It is possible to disentangle the relative effect of technical changes and mortality fall on the evolution of population. We conduct a counterfactual analysis eliminating successively the increase in life expectancy and the technological bias. With no increase in life expectancy, population would have been respectively 10% and 30% lower in 1910 and in the long run. The figures would have been respectively 40% and 60% lower, with no bias in the technical progress. Finally, population would have been 45% smaller in 1910 and 70% smaller in the long run, neutralizing both the effect of life expectancy and technological bias. So the major part of population increase is due to the technological bias evolution between land and capital.

Appendix

Appendix 1

Along a competitive equilibrium, L 1 t and L 2 t are allocated optimally in such a way that total production is maximized. Total production is equal to:

Y t = a (AS t K t ) α L 2 t 1-α δ-1 δ + b [AM t X t ] α L 1 t 1-α δ-1 δ δ δ-1
Therefore, the optimal allocation of labor results from the program:

   max (L 1 t ,L 2 t ) a (AS t K t ) α (L 2 t ) 1-α δ-1 δ + b [AM t X t ] α (L 1 t ) 1-α δ-1 δ s. t. L 1 t + L 2 t = L t
To simplify calculations, the power δ δ-1 has been removed without any consequence on the result of the program as δ > 1. The Lagrangian of this program is

L = a (AS t K t ) α L 2 t 1-α δ-1 δ + b [AM t X t ] α L 1 t 1-α δ-1 δ + z L t -L 1 t + L 2 t
with z the shadow price. The optimality conditions are:

b (AM t X t ) (δ-1)α δ L 1 t (δ-1)(1-α) δ -1 (δ -1) (1 -α) δ = z (51) a (AS t K t ) (δ-1)α δ L 2 t (δ-1)(1-α) δ -1 (δ -1) (1 -α) δ = z (52) We set: z = zδ (1 -α) (δ -1)
From ( 51) and ( 52) we obtain L 1 t and L 2 t :

L 1 t = b z δ 1-α+δα (AM t X t ) (δ-1)α 1-α-δα L 2 t = a z δ 1-α+δα (AS t K t ) (δ-1)α 1-α-δα
The constraint L t = L 1 t + L 2 t allows to calculate the value of z with respect to L t :

L t = b z δ 1-α+δα (AM t X t ) (δ-1)α 1-α+δα + a z δ 1-α+δα (AS t K t ) (δ-1)α 1-α+δα 29 or z = L -1+α-δα δ t b δ 1-α+δα (AM t X t ) (δ-1)α 1-α+δα + a δ 1-α+δα (AS t K t ) (δ-1)α 1-α+δα 1-α+δα δ
Finally, the resulting total production can be written, using optimality conditions ( 51) and ( 52):

Y δ-1 δ t = z L 1 t + L 2 t = zL t or Y t = z δ δ-1 L δ δ-1 t
Replacing z by its value, one gets:

Y t = b δ 1-α+δα (AM t X t ) (δ-1)α 1-α+δα + a δ 1-α+δα (AS t K t ) (δ-1)α 1-α+δα 1-α+δα δ-1 L 1-α t δ is chosen such that: δ = α + ε -1 α which is equivalent to: 1 -α + δα δ -1 = εα ε -1 a and b are chosen such that: a δ 1-α+δα = λ b δ 1-α+δα = 1 -λ
This allows to recover (1).

Appendix 2: Economies of scale and non linear cost of children

This appendix considers the case in which the cost of children is non linear. This cost has two direct components (the rearing cost and the cost of surviving children), and one indirect (the congestion effect on land in the utility function). First we consider the two direct costs. The first one, the rearing cost of m t children is now φ 1 w t m ψ t , with 0 < ψ < 1. The second one associated to the η t m t surviving children becomes: φ 1 w t (η t m t ) ψ . Then, the total direct cost can be written:

φ 1 η ψ t + φ 2 w t (η t m t ) ψ
The number of surviving children per adult is m t ≡ η t m t . The corresponding cost is φt w t (m t ) ψ with:

φt = φ 1 η ψ t + φ 2
The parameter ψ < 1 implies a cost of children that is less than proportianal, or some economies of scale.

We now turn to the indirect congestion cost of children, and we assume that some economies of scale may exist. The preferences of members of generation t are represented by the utility function:

U (c t , d t+1 , m t , v t ) = Γ 1 ln c t + p t Γ 2 ln d t+1 + Γ 3 ln η t m t + Γ 4 ln(v t -ξ (η t m t ) ψ )

In this new formulation, the cost of surviving children for land is less than proportional. This translate the idea that part of land surfaces may be shared by children. With this new assumption, introducing the variable ζt ≡ v t -ξ (η t m t ) ψ , the intertemporal budget constraint of an household may be rewritten as: (62)

c
The number of young households at date t + 1 becomes equal to:

N t+1 ≡ (n t ) 1/ψ N t ( 63
)
The evolution of land per young alive is:

x t+1 = x t (n t ) 1/ψ (64)
The equilibrium conditions are slightly modified. The equilibrium on the rent market remains the same, but for different equilibrium values:

v t + x t = x t ( 65
)
The equilibrium on the capital market becomes: γ2,t w t = (n t ) 1/ψ k t+1 + q t x t (66)

Our basic formulation appears as the special case with ψ = 1. Calibrating the model with ψ < 1 would modify the parameters values. But the qualitative properties would remain the same.

Appendix 3

As the production technology is Cobb-Douglas between L t and the other factors, β W t = 1 -α and β K t + β X t = α. Using equations ( 26), ( 27), and (25), We can also rewrite R t and πt given by ( 26) and (27), using (67), thus

β K t = αλ k1-1 ε t λ k1-1 ε t + (1 -λ) (A t x t ) 1-1 ε then λ k1-1 ε t + (1 -λ) (A t x t ) 1-1 ε = αλ k1-1 ε t β K t (
R t = β K t kα-1 t λα β K t εα ε-1 πt = α -β K t x t kα t λα β K t εα ε-1
Using equation ( 69 67) and ( 69), it is possible to recover A t and G t from β K t and Z t .

A t = λ kt x t α β K t -1 ε ε-1 1 1 -λ ε ε-1 (70) G t = Z t m t (1-α) α β K t+1 β K t ε(1-α) ε-1 kt kt+1
(1-α) (71)

Appendix 4

Figure 1 :

 1 Figure 1: Historical factor shares 1770-1913, Source: Allen (2009)

Figure 2 :

 2 Figure 2: Historical and Computed Total Population for England

Figure 3 :

 3 Figure 3: Counterfactual Analysis With no Improvement in Life Expectancy

Figure 4 :

 4 Figure 4: Counterfactual Analysis with No Technological Bias

Figure 5 :

 5 Figure 5: Total Decomposition: Life Expectancy, Technological Bias, and Both

  and X t are the quantities of capital, labor, and land used in production at time t. AS t > 0 is a capital augmenting technical progress and AM t a land augmenting technical progress.

Table 2 :

 2 Share of capital incomes and growth

		1730 1760 1790 1820 1850 1880 1910 1940 1970 2000
	β K t	0.21 0.23 0.25 0.30 0.35 0.40 0.43 0.43 0.44 0.44
	Z t	-	1.31 1.20 1.66 1.75 1.90 1.72 1.60 1.81 2.02

Table 3 :

 3 Total population in England (in millions)We use the surviving probability of young children (from birth to seven years old included) η t , and the surviving probability at 50 years old p t .

	1730 1760 1790 1820 1850 1880 1910 1940 1970 2000
	η t 0.64 0.66 0.67 0.68 0.69 0.73 0.83 0.94 0.98 0.99
	p t 0.20 0.21 0.23 0.24 0.35 0.33 0.43 0.57 0.78 0.95

Table 4 :

 4 Surviving probabilities η t is computed from the death rates of England and Wales from the Human Mortality Database (2015) of the University of California (USA) and the Max Planck Institute for Demographic Research (Germany) that gives mortality per age from 1841. Figures for previous years are taken from Maddison (2013) on England.

  t + d t+1 ρ t+1 + φt w t + ξπ t (m t ) ψ + π t ζt = w t (53) To simplify, it is assumed that each type of children cost implies the same degree of economies of scale: all costs become proportional to (m t ) ψ . Let us define n t = (m t ) ψ . The household problem can be written: max (ct,d t+1 ,n t , ζt) Γ 1 ln c t + p t Γ 2 ln d t+1 + Γ 3 ψ ln n t + Γ 4 ln ζt s. t. c t + d t+1 ρ t+1 + φt w t + ξπ t n t + π t ζt = w t + p t Γ 2 + (Γ 3 /ψ) + Γ 4

	with
	γ1,t =	Γ 1 Γ 1 + p t Γ 2 + (Γ 3 /ψ) + Γ 4	(59)
	γ2,t =	p t Γ 2 Γ 1 + p t Γ 2 + (Γ 3 /ψ) + Γ 4	(60)
	γ3,t =	(Γ 3 /ψ) Γ 1 + p t Γ 2 + (Γ 3 /ψ) + Γ 4	(61)
	γ4,t =	Γ 1	Γ 4
	First order conditions for this optimization problem lead to the following
	solutions:
	c t = γ1,t w t				(54)
	s t = γ2,t w t				(55)
	d t+1 = γ2,t w t ρ t+1				(56)
	n t =	γ3,t w t φt w t + ξπ t			(57)
	v t =	ξγ 3,t w t φt w t + ξπ t	+ γ4,t	w t π t	(58)

  As the share of wagesβ W t = 1 -α, wt = (1 -α) wt + R t kt + πt x t , thus Z t = m t G

	then from equation (25),
									εα
			wt = (1 -α)	αλ t β K	ε-1 kα t	(68)
	Thus, we write Z t as:
	Z t =	N t+1 N t	α 1-α α 1-α (AS t+1 ) (AS t )	wt+1 + R t+1 kt+1 + πt+1 x t+1 wt + R t kt + πt x t
						α/(1-α)	wt+1
						t			wt
	and using (68), we get
		t Z t = m t G α/(1-α)	β K t+1 β K t		εα ε-1	kt kt+1	α
	and so
		G α/(1-α) t	=	Z t m t	β K t+1 β K t	εα ε-1	kt kt+1	α	(69)
									67)

Table 6 :

 6 counterfactual Analysis on Total Population for England

million, where the total population tends to N ∞ + N ∞ + p ∞ N ∞ . Thus, X = 58

x ∞ 2 + p ∞ The value of x 0 is chosen in order that the computed value for population in our model for date t = 2 fits the observed value in 1790. Indeed, population at date t = 2 is equal to N 2 + N 1 + p 0 N 0 , thus it is the first computation that depends only on one initial condition N 0 .

Total population in dates t = 0 and t = 1 in our model are taken from historical values. It is consistent with the model as population in t = 0 depends on N -1 and on N -2 , and population in t = 1 depends on N 0 and on N -1 . Thus, N -1 and N -2 are chosen in order to get the historical values for population in t = 0 and t = 1.

Simulations and stylized facts

This section presents different results obtained through simulations with Dynare (cf. [START_REF] Adjemian | Dynare: Reference Manual, Version 4[END_REF]. The central scenario tries to reproduce historical data. Then, different counterfactual analyses are computed.

The different scenario focus on the period 1760-1910, although graphics are shown for 1730-2000 for historical data and to the end of the convergence process for counterfactual analysis. The initial condition in 1730 is due to the availability of data and allows encompassing the pre-industrial revolution period. We interpret the results from 1760, since this is the first simulated point. To avoid the effects of the two world wars, we restrict interpretations to the period 1760-1910.

Appendix 4 provides all computed data corresponding to the figures for all subsections.

The central scenario

The model is able to reproduce the dynamics of population on the period 1760-1910 as shown by Figure 2, where Nhist is the historical value for total population in England and Ntot model is the value computed from the model. All figures are expressed in millions. Note that we focus on population of England and not of United Kingdom.


domain: sdv chim

text:
Mélanie L'enfant

Pricillia Kutudila

Catherine Rayon

Jean-Marc Domon

Woong-Hee Shin

Daisuke Kihara

Anne Wadouachi

Jérome Pelloux

Gwladys Pourceau

Corinne Pau-Roblot
email: corinne.pau-roblot@u-picardie.fr

Mélanie L'enfant A♯

Pricillia Kutudila B♯

Dr Corinne Pau

Lactose derivatives as potential inhibitors of pectin methylesterases

Keywords: PME, Pectin methylesterase, HG, homogalacturonan Pectin methylesterase, chemical inhibitors, lactoside derivatives, docking analysis

à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Introduction

Considering the deleterious effects of chemical pesticides on the environment, alternatives are needed to reduce the infection of plants by phytopathogens. As an example, a number of strategies were envisaged to limit the infection of Botrytis cinerea, one of the most widespread fungus attacking numerous plant species of agronomic interest. This include the use of antifungal protein from Aspergillus giganteus [START_REF] Moreno | Activity of the antifungal protein from Aspergillus giganteus against Botrytis cinerea[END_REF], essential oil as eugenol [START_REF] Wang | Antifungal activity of eugenol against Botrytis cinerea[END_REF][START_REF] Salimena | Extraction of essential oil from inflorescences of Dysphania ambrosioides and its activity against Botrytis cinerea[END_REF], or oxygenated monoterpenes [START_REF] Bilal | Macromolecular agents with antimicrobial potentialities: a drive to combat antimicrobial resistance[END_REF]. Most of bacterial or fungal phytophathogens secrete plant cell wall degrading enzymes that play a central role in the infection. The plant primary cell wall is highly complex, composed of polysaccharides (cellulose, hemicelluloses and pectins) and structural proteins that are linked to form a cohesive but highly dynamic structure outside of the plant cell. Pectins, and in particular homogalacturonan (HG), a homopolymer of α-(1→4)linked-D-galacturonic acid units, are major components of the plant cell wall and play a central role in regulating the chemistry and mechanics of the cell wall during development and plant pathogen interactions. Pectin remodeling enzymes, which play a key role in determining the phytopathogenicity of the various fungi [START_REF] Van Den | Fungal enzyme sets for plant polysaccharide degradation[END_REF], could thus be considered as a case study to engineer chemical compounds that could inhibit specifically a number of these enzymes. Pectin methylesterases (PME), enzymes that fine-tune the degree of methylesterification of homogalacturonans (HG), are secreted by fungi as Fusarium asiaticum or Rhizoctomia solani [START_REF] Glinka | Purification and partial characterisation of pectin methylesterase produced by Fusarium asiaticum[END_REF][START_REF] Wibberg | Transcriptome analysis of the phytopathogenic fungus Rhizoctonia solani AG1-IB 7/3/14 applying high-throughput sequencing of expressed sequence tags (ESTs)[END_REF] during infection, but are as well present in plant cell wall where they can regulate a number of developmental process including primordia emergence at the shoot apical meristem [START_REF] Peaucelle | Arabidopsis phyllotaxis is controlled by the methyl-esterification status of cell-wall pectins[END_REF][START_REF] Peaucelle | Pectininduced changes in cell wall mechanics underlie organ initiation in Arabidopsis[END_REF][START_REF] Peaucelle | The transcription factor BELLRINGER modulates phyllotaxis by regulating the expression of pectin methylesterase in Arabidopsis[END_REF], adventitious root formation [START_REF] Guénin | Identification of pectin methylesteras 3 as a basic pectin methylesterase isoform involved in adventitious rooting in Arabidopsis thaliana[END_REF], pollen tube elongation [START_REF] Leroux | PECTINMETHYLESTERASE 48 is involved in Arabidopsis pollen grain germination[END_REF][START_REF] Jiang | VANGUARD1 encodes a pectin methylesterase that enhances pollen tube growth in the Arabidopsis style and transmitting tract[END_REF], seed development [START_REF] Müller | Demethylesterification of cell wall pectins in Arabidopsis plays a role in seed germination[END_REF] and dark-grown hypocotyl elongation [START_REF] Derbyshire | Cell elongation in Arabidopsis hypocotyles involves dynamic changes in cel wall thickness[END_REF][START_REF] Wolf | Growth control and cell wall signaling in plants[END_REF]. PMEs from plants and pathogens differ in terms of substrate specificity and sensitivity to inhibitors. For instance, it was reported that proteinaceous inhibitors (PMEIs) are effective against plant PMEs but ineffective against pathogen PMEs [START_REF] Giovane | Pectin methylesterase inhibitor[END_REF]. Molecules such as catechins, phenolics extracts from green tea leaves, as well as aromatics have been identified as inhibitors of plant PME activity, although no inhibition was measured on fungal PME [START_REF] L'enfant | Characterization of the substrate specificity of plant and fungi pectin methylesterases. Identification of novel chemical inhibitors of PMEs[END_REF][START_REF] Lewis | Inhibition of pectin methyl esterase activity by green tea catechins[END_REF]. A recent report identified phenylephrine as inhibitor of a number of Arabidopsis pectin methylesterases, with consequent effects on root development [START_REF] Cheong | Phenylephrine, a small molecule, inhibits pectin methylesterase[END_REF]. As an alternative of chemical library screen, one of the methods to produce new inhibitors is to synthesize compounds mimicking the structure of the substrates of the enzyme [START_REF] Wadood | Selective glycosidase inhibitors: A patent review (2012-present)[END_REF]. Due to the recent advances in the chemistry and biochemistry, glycosidase inhibitors have already been synthesized or extracted from natural sources. In particular, some disaccharides were used as glycosidases inhibitors: kojibiose and nigerose were indeed identified as inhibitors of glucosidases and was used as therapeutic agents [START_REF] Uglade | Microsomal glucosidases of rat liver. Partial purification and inhibition by disaccharides[END_REF]. α-(1→5)-linked arabinofuranose disaccharides were reported to inhibit arabinosyltransferases of Mycobacterium tuberculosis that are induced in tuberculosis disease [START_REF] Pathak | Synthesis of deoxygenated α(1→5)-linked arabinofuranose disaccharides as substrates and inhibitors of arabinosyltransferase of Mycobacterium tuberculosis[END_REF]. The chemical synthesis of oligosaccharides related to plant cell wall were also reported, paving the way for potential modifications [START_REF] Kinnaert | Chemical synthesis of oligosaccharides related to the cell walls of plants and algae[END_REF]. As pectin methylesterase act on methylated homogalacturonan whose monomer unit is GalA methylated at C-6 position, carbohydrate analog with other group than natural COOMe could be envisaged as potential inhibitor of PME activity. Here we report the synthesis of lactose derivatives as bioisoster of PME substrates and the inhibition of plant or fungal pectin methylesterases, produced in vitro or commercially available, by the synthesized lactoside derivatives. Lactose was chosen, firstly because it is an abundant sugar, and secondly, because it includes a galactose unit that is structurally very close to the galacturonic acid present in homogalacturonan, natural substrate of PME enzymes. The results are discussed in the light of docking analysis of the lactosides derivatives with modelled structures of pectin methyesterases, showing the potential of this approach.

Materials and methods

Source of enzymes

Pectin methylesterase (PME) from orange peel (Citrus sinensis), referred to as CsPME, was used as a reference (Sigma, P5400).

Escherichia coli strain JM101, carrying the plasmid pREP4 [START_REF] Dedeurwaerder | Activity of an atypical Arabidopsis thaliana pectin methylesterase[END_REF] overexpressing PME-At3g29090, was used for the expression of the Arabidopsis protein and referred to as AtPME31. To express and produce recombinant AtPME31, recombinant E. coli was grown at 30 °C in Lysogeny Broth (LB) medium. 0.5 mM isopropyl β-D-1-thiogalactopyranoside (IPTG) was used as inducer of protein production. Cells were harvested by centrifugation and bacteria were lysed by 4x30 s sonication on ice and the lysate was centrifuged. The clarified culture supernatant was concentrated using a PelliconXL device with a Biomax10 membrane on a Labscale TFF pump (Millipore) and buffer was changed to 0.3 M NaCl, 50 mM sodium phosphate, pH 7.5. The supernatant was applied to an AKTA prime FPLC fitted with a HisTrap FF column at a flow rate of 0.25 ml.min -1 and eluted in a linear gradient of 20 to 500 mM imidazole (0.5 ml.min -1 ) over 10 ml.

Transgenic Nicotiana tabacum plants (var. PBD6) overexpressing Arabidopsis PME-At3g14310 was used for the expression of Arabidopsis protein and referred to as AtPME3 [START_REF] Sénéchal | Tuning of pectin methylesterification: PECTIN METHYLESTERASE INHIBITOR 7 modulates the processive activity of co-expressed PECTIN METHYLESTERASE 3 in a pH-dependent manner[END_REF]. Recombinant PME3 (PME3-6xHis) was purified from N. tabacum frozen leaf powder homogenized in 50 mM NaH2PO4 containing 2 M NaCl (pH 7). The protein extract was centrifuged and supernatant was diluted 6-fold in a NaH2PO4 50 mM buffer (pH 8) containing 40 mM imidazole. The supernatant was loaded onto a Ni-affinity column (HisTrap FF, GE Healthcare, Little Chalfont, United Kingdom) and washed with 10 column volumes of 40 mM imidazole in NaH2PO4 50 mM buffer, 500 mM NaCl (pH 8). The recombinant protein was eluted from the affinity resin with 500 mM imidazole in NaH2PO4 50 mM buffer, 500 mM NaCl (pH 8). The purified protein was loaded onto a spin column (Microcon YM-10, Merck Millipore, Tullagreen Carrigtwohill, Ireland) and desalted with 50 mM sodium phosphate buffer (pH 7.5).

Pichia pastoris strain GS115, carrying the plasmid pPIC3.5 expressing Botrytis cinerea BcPME1 was referred to as BcPME1 [START_REF] Kars | Functional analysis of Botrytis cinerea pectin methylesterase genes by PCR-based targeted mutagenesis: Bcpme1 and Bcpme2 are dispensable for virulence of strain B05.10[END_REF]. To express and produce BcPME1, recombinant P. pastoris were grown overnight at 30 °C on Yeast Extract Peptone Dextrose (YPD) medium plate. One colony was transferred into 2 ml liquid Buffered Methanol-complex Medium Yeast (BMMY) maintained at 30 °C for 15 h. Methanol was added every 24 h to a final concentration of 0.5% (v/v). After 4 days of expression, yeasts were removed by centrifugation. The supernatant containing proteins was precipitated using ammonium sulfate at 90% for 6 h at 4 °C. Proteins were harvested by centrifugation and dialyzed in GEBAflex tubes (6-8 kDa). Extracts were concentrated using 15 ml amicon tubes (10 kDa) and then purified as follows: the material was applied twice to a DEAE-Sepharose FF column (GE Healthcare, Sweden). Proteins were eluted with sodium-phosphate buffer (50 mM; pH 6.0).

Lactoside derivatives synthesis

Methyl β-D-lactoside 1 (OH)

The methyl lactoside was synthesized following a four-step sequence as described in literature [START_REF] Schämann | TEMPO-Mediated Anodic Oxidation of Methyl Glycosides and 1-Methyl and 1-Azido Disaccharides[END_REF]: a) peracetylation of lactose (quant. yield); b) activation of anomer position via a 2,3,6,2',3',4',6'-hepta-O-acetyl-α-D-lactosyl bromide (quant. yield); c) substitution of lactosyl bromide by a methoxy group (74% yield) and deacetylation (71% yield). Methyl β-Dlactoside 1 (OH) was obtained in 52% overall yield and the characterization ( 1 H, 13 C) was in accordance with literature.

Sodium (methyl β-D-lactosid)uronate 2 (Na)

To a solution of 2.39 g of methyl β-D-lactoside 1 (6.8 mmol, 1 eq) in 25 ml of acetonitrile and 20 ml of phosphate buffer (0.67M, pH 6.7) were added 544 mg of TEMPO (3.6 mmol, 0.5 eq), 34 ml of 2 M aqueous NaClO2 solution (67.9 mmol, 10 eq) and 323 µl of 2.1M aqueous NaOCl solution (0.679 mmol, 0.1 eq) portion wise. The reaction mixture was heated at 35 °C and monitored by TLC. After completion, the reaction mixture was evaporated to dryness, and then water and dichloromethane were added. Aqueous phase was extracted three times with dichloromethane (3 x 25 ml) and concentrated. The crude residue was diluted in water and dialyzed against water during 24h. After freeze-drying, sodium (methyl β-Dlactosid)uronate 2 (Na) was obtained as a white solid (1.15 g, 2.67 mmol, 40%)

1 H NMR (300 MHz, D2O) : δ (ppm) 4.45 (d, J H-1',H-2'= 7.8 Hz, 1H, H-1'), 4.43 (d, JH-1,H-2=

Methyl (methyl β-D-lactosid)uronate 3 (Me)

To a solution of 204 mg of sodium (methyl β-D-lactosid)uronate 2 (0.47 mmol, 1 eq) in 16 ml of MeOH were added 1.6 ml of 2,2-dimethoxypropane (0.013 mol, 28 eq) and a catalytic quantity of an aqueous 37% HCl solution. The reaction mixture was irradiated by microwaves at 63 °C during 40 min then neutralized by addition of Et3N and evaporated to dryness. The crude product was then purified by flash chromatography (EtOAc/MeOH) to obtain methyl (methyl β-D-lactosid)uronate 3 (Me) as white powder (36.9 mg, 0.087 mmol, 19%).

Butyl (methyl β-D-lactosid)uronate 4 (Bu)

To a solution of 99 mg of sodium (methyl β-D-lactosid)uronate 2 (0.23 mmol, 1 eq) in 3 ml of butanol were added 500 mg of Na2SO4 (32 mmol, 139 eq) and a catalytic quantity of an aqueous 37% HCl solution. The reaction mixture was irradiated by micro-waves at 70 °C during 40 min then neutralized by addition of Et3N, filtrated to remove Na2SO4 and evaporated to dryness. The crude product was purified by flash chromatography (CH2Cl2/MeOH) to obtain butyl (methyl β-D-lactosid)uronate 4 (Bu) as white powder (109,7 mg, 0.22 mmol, 95%).

Quantitative assay of PME activity and inhibition

Inhibition of CsPME, AtPME3 and AtPME31 activity was quantified using a colorimetric microassay adapted from Klavons and Bennett [START_REF] Klavons | Determination of methanol using alcohol oxidase and its application to methyl ester content of pectins[END_REF] on Citrus pectin >85% methylesterified (Sigma P9561) at 20 mg.ml -1 10 µM to 1500 µM of lactoside derivatives. The mixture was incubated at 30 °C for 30 min. Results were expressed as nmol MeOH.min -1 .µg -1 of protein using the methanol standard curve.

Considering the optimal pH activity of BcPME1 (pH 6) and the limitations of the alcohol oxidase assay with regards to pH, the inhibition of BcPME1 activity was measured using the Anthon and Barrett method [START_REF] Anthon | Comparison of three colorimetric reagents in the determination of methanol with alcohol oxidase. Application to the assay of pectin methylesterase[END_REF] on Citrus pectin >85% methylesterified (Sigma aldrich P9561) at 20 mg.ml -1 and with 10 µM to 1500 µM of lactoside. PME activity was calculated as the difference in absorbance between saponified and non-saponified samples and expressed, using the standard curve, as nmol MeOH.min -1 .µg -1 of proteins.

Protein-lactosides derivatives docking

The structural homology modeling of AtPME3 and at AtPME31 was established by Dedeurwaerder et al [START_REF] Dedeurwaerder | Activity of an atypical Arabidopsis thaliana pectin methylesterase[END_REF] and Sénéchal et al [START_REF] Sénéchal | Tuning of pectin methylesterification: PECTIN METHYLESTERASE INHIBITOR 7 modulates the processive activity of co-expressed PECTIN METHYLESTERASE 3 in a pH-dependent manner[END_REF] respectively.

Binding site loop modeling for AtPME31 was modeled by GalaxyWeb server [START_REF] Ko | GalaxyWEB server for protein structure prediction and refinement[END_REF].

Interactions between the proteins and lactosides derivatives were predicted by AutoDock4 [START_REF] Morris | AutoDock4 and AutoDockTools4: Automated docking with selective receptor flexibility[END_REF]. For a regular ligand-protein docking analysis, Autodock4 (considering flexibility of the ligands) would be suitable because the method is developed and tuned specifically for ligandprotein interactions and practically has a good balance between accuracy and computational speed.

Results

Lactosides derivatives synthesis

Three lactosides derivatives, modified at C-6 position by replacing primary alcohol (1) (OH) with carboxylate (2) (Na), carboxymethyl (3) (Me) or carboxybutyl (4) (Bu) groups, were synthesized as bioisostere substrates (Scheme 1). These derivatives were obtained following two-steps procedure starting from methyl β-Dlactoside 1 (OH), obtained as described in literature [START_REF] Schämann | TEMPO-Mediated Anodic Oxidation of Methyl Glycosides and 1-Methyl and 1-Azido Disaccharides[END_REF] using TEMPO-mediated oxidation followed by acid-catalyzed esterification [START_REF] Lu | Catalytic Synthesis of a New Series of Alkyl Uronates and Evaluation of Their Physicochemical Properties[END_REF]. Sodium (methyl β-D-lactosid)uronate 2 (Na), methyl (methyl β-D-lactosid)uronate 3 (Me) and butyl (methyl β-D-lactosid)uronate 4 (Bu) were obtained with moderate to high yields (19-95%).

Inhibition of PME by lactosides derivatives

First, four PMEs previously identified for their various substrates specificities [START_REF] L'enfant | Characterization of the substrate specificity of plant and fungi pectin methylesterases. Identification of novel chemical inhibitors of PMEs[END_REF] were used to test the inhibitory potential of lactosides derivatives. Commercial PME from orange peel (Citrus sinensis: CsPME), two PMEs from Arabidopsis thaliana (AtPME3 and AtPME31) previously reported to differ in their structure [START_REF] Sénéchal | Tuning of pectin methylesterification: PECTIN METHYLESTERASE INHIBITOR 7 modulates the processive activity of co-expressed PECTIN METHYLESTERASE 3 in a pH-dependent manner[END_REF] and in their inhibition by PMEI and one fungal PME from Botrytis cinerea. The two PMEs from Arabidopsis thaliana were expressed in heterologous system (N. tabacum leaves or E. Coli) and purified by affinity binding on a His-Trap FF column [18;26]. The last PME, from B. cinerea, was produced in P. pastoris [START_REF] Kars | Functional analysis of Botrytis cinerea pectin methylesterase genes by PCR-based targeted mutagenesis: Bcpme1 and Bcpme2 are dispensable for virulence of strain B05.10[END_REF] and purified using a DEAE-Sepharose FF column.

We investigated the inhibition of enzyme activities from plant and fungi PMEs by lactoside derivatives using increasing concentrations of molecules (0 to 1500 µM). Results are presented in Fig. 1. Inhibition of CsPME was effective at a concentration of lactoside of 500 µM. The most effective derivative was the methyl (methyl β-D-lactosid)uronate 3 (Me) which induced a maximum decrease in CsPME activity of about 50%. Methyl β-D-lactoside 1 (OH) had also an effect on CsPME with a maximum reduction of 40% of the enzymatic activity at 1500 µM. In contrast, the other two lactoside derivatives had a weak effect on PME activity. The same effect was observed for another plant PME, AtPME3: the methyl (methyl β-Dlactosid)uronate 3 (Me) inhibited most strongly the enzyme with a maximal decrease of about 80% of the activity at a concentration of 1500 µM. This lactoside derivative had a significant effect from a concentration of 100 µM. Methyl β-D-lactoside 1 (OH) inhibited AtPME3 with a maximum decrease in the activity of 40% compared to the control. Similarly to what observed for CsPME, the other two lactoside derivatives had only a slight effect on PME activity. In contrast, the third plant PME, AtPME31, was only very weakly inhibited by any of the lactoside derivatives (maximum decrease of about 15% compared to the control), and whatever the concentration of the molecules. Lactoside derivatives showed also a weak inhibition of the fungal BcPME1.

Docking analysis revealed difference in the affinity of lactoside derivatives for PME

In order to understand the differences in the inhibiting capacity of the molecules on distinct PMEs, we used functional and catalytic actives sites prediction, as well as and docking analysis [START_REF] Thakuria | Functional and catalytic active sites prediction and docking analysis of azoreductase enzyme in Pseudomonas putida with a variety of commercially available azodyes[END_REF]. Only AtPME3 and AtPME31 were chosen for this study because of their plant origin and a large contrast in their response to lactoside derivatives. The 3D structure of AtPME3 was modeled in a previous study [START_REF] Sénéchal | Tuning of pectin methylesterification: PECTIN METHYLESTERASE INHIBITOR 7 modulates the processive activity of co-expressed PECTIN METHYLESTERASE 3 in a pH-dependent manner[END_REF]. AtPME31 was modeled using AtPME3 coordinates as template. As observed in Fig. 2B and2D, homology modeling showed the presence of a supplementary loop of amino acids in AtPME31, as compared to AtPME3 conformation (Fig. 2A and2C).

Results of docking of the lactoside derivatives into the active site of AtPME3 and AtPME31

are presented in Fig. 3. If all the lactoside derivatives were included in the active sites for AtPME3, this was not observed for AtPME31.

The binding affinity, number of hydrogen bonds and the binding residues calculated by docking for AtPME3 and AtPME31 are presented in table 1. The negative values observed for AtPME3 revealed the high interaction between the enzyme and the inhibitor. The docking analyses showed that H-bonds were formed between lactoside derivatives with catalytic sites residues Arg, Asp and Gln. Methyl β-D-lactoside 1 (OH) and methyl (methyl β-Dlactosid)uronate 3 (Me) presented the more H-bonds than the over derivatives. These analyses indicated that the inhibition could occur via the interaction of lactoside derivatives with the substrate-binding site of AtPME3. On the over hand, the predicted binding affinity were very positive for AtPME31 and no H-bonds were predicted to be present. Based on these elements, no interaction between lactoside derivatives and binding site can be achieved with AtPME31. This could notably be explained by the presence of a supplementary loop of amino acids in AtPME31, not present in AtPME3, which narrows binding site (Fig. 2), impairing the binding of ligand to the catalytic site.

Discussion

Pectin methylesterases from plants and pathogens play a central role in regulating plant development and pathogenicity, respectively [START_REF] Peaucelle | Arabidopsis phyllotaxis is controlled by the methyl-esterification status of cell-wall pectins[END_REF]. The discovery of a novel molecules that could inhibit PMEs is thus of a broad interest to either control i) plant growth and/or ii) plantpathogen interactions. Over the recent years, a number of proteinaceous (PMEI) or chemical inhibitors, were shown to be effective on PME activity [START_REF] Cheong | Phenylephrine, a small molecule, inhibits pectin methylesterase[END_REF]. However, no generic PME inhibitor could be identified as, for instance, PMEI are ineffective on bacterial and fungal enzymes and chemical inhibitors remain largely untested on non-plant enzymes [START_REF] Giovane | Pectin methylesterase inhibitor[END_REF][START_REF] L'enfant | Characterization of the substrate specificity of plant and fungi pectin methylesterases. Identification of novel chemical inhibitors of PMEs[END_REF][START_REF] Lewis | Inhibition of pectin methyl esterase activity by green tea catechins[END_REF]. In order to effectively inhibit PMEs, we designed a strategy based on the engineering of novel chemical inhibitors. Considering previous studies showing that the use of bioisoster substrates can effectively inhibit enzymes is one of the solutions [START_REF] Asano | Sugar-mimicking glycosidase inhibitors: bioactivity and application[END_REF], we synthesized molecules derived from lactose, and exhibiting oxidized groups such as carboxylate, carboxymethyl group (as the native substrate of the enzyme) and carboxybutyl group. These compounds were obtained from lactose following a multi-step procedure. Methyl β-D-lactoside 1 (OH) was obtained, as described in literature [START_REF] Schämann | TEMPO-Mediated Anodic Oxidation of Methyl Glycosides and 1-Methyl and 1-Azido Disaccharides[END_REF] by first peracetylation of lactose, then activation of the C1 position using bromide group, substitution of the anomer bromide by a methoxy group and final deacetylation. The primary hydroxyl groups (at C6 and C6' positions) of this compound were then oxidized using TEMPO-assisted procedure to lead to sodium (methyl β-Dlactosid)uronate 2 (Na) which was then esterified in acidic conditions using two different alcohols: MeOH (leading to methyl (methyl β-D-lactosid)uronate 3 (Me)) or BuOH (leading to butyl (methyl β-D-lactosid)uronate 4 (Bu)). TEMPO-catalyzed oxidation of carbohydrates is a well-known and powerful strategy when using protected carbohydrates. When free carbohydrates were used, the procedure can be very slow and tricky. Indeed, about 1 week was necessary to oxidize the methyl β-D-lactoside, with portion additions of TEMPO and cooxidants. Furthermore, esterification of uronic acids is challenging since, under acidic conditions, (trans)glycosylation and esterification were competitive, leading to degradation products such as shorter saccharides. In the procedure, a careful control of the conditions was necessary to avoid by-product formation.

Purified proteins, either from plant or fungal sources, were used to test the inhibition of their activity by lactoside derivatives. The inhibition of CsPME and AtPME3 activities by lactoside derivatives was dependent upon substitution: the most efficient being the methyl (methyl β-Dlactosid)uronate 3 (Me) with an inhibition of the activity of CsPME and AtPME3 of 50 and 80%, respectively. The other derivatives only decreased the activity by 20%. In contrast, the inhibition of AtPME31 and BcPME1 was not more that 20% whatever the molecules tested.

The methyl (methyl β-D-lactosid)uronate 3 (Me) was the substrate the most similar to the natural substrate : it has composed of two uronate derivatives esterified by a methyl group while the substrate of the PME is homogalacturonan methyl esterified in C-6 [START_REF] Pelloux | New insights into pectin methylesterase structure and function[END_REF]. The methyl group could probably interact with amino acid residues involved in the catalytic site. The butyl (methyl β-D-lactosid)uronate 4 showed no inhibitor activity probably due to the length of the ester group which was so not cleavable by the enzyme. We thus managed to design novel chemical inhibitors of PME, that had distinct effects depending on the PMEs considered (plant vs fungi). Early report showed that plant PMEs can be inhibited by green tea catechins such as polyphenon-60 and epigallocatechin-3-gallate [START_REF] Lewis | Inhibition of pectin methyl esterase activity by green tea catechins[END_REF]. This last compound was as well effective in inhibiting a PME from Aspergillus flavus when it was used at very high concentrations (20 mg ml -1 ) [START_REF] Jiang | Recombinant expression and inhibition mechanism analysis of pectin methylesterase from Aspergillus flavus[END_REF]. We previously reported the inhibition of another fungal PME, BcPME1 using EGCG concentrations above 1 mg ml -1 [START_REF] L'enfant | Characterization of the substrate specificity of plant and fungi pectin methylesterases. Identification of novel chemical inhibitors of PMEs[END_REF]. Although this appears in accordance with the above-mentioned report, the use of such high concentrations of chemicals questions the affinity of fungal PMEs towards EGCG. Application of this latter chemical on plant seedlings had drastic effects on root growth, showing the importance of PME-mediated tuning of pectins in the control of plant development [START_REF] Wolf | Plant cell homeostasis is mediated by brassinosteroid feedback signaling[END_REF]. Similarly, a chemical library screen recently identified phenylephrine, an amine, as an inhibitor of plant PME activity, with consequent effect on root development; Application of phenyledrine indeed induced a strong decrease in root length [START_REF] Cheong | Phenylephrine, a small molecule, inhibits pectin methylesterase[END_REF]. We previously showed that the screen of the LATCA chemical library allowed the identification of a number of compounds that differed in their inhibition potential of plant and fungal PMEs [START_REF] L'enfant | Characterization of the substrate specificity of plant and fungi pectin methylesterases. Identification of novel chemical inhibitors of PMEs[END_REF], suggesting that enzymes of distinct structures do not have the same binding affinity towards chemical compounds. This is similar to what was shown in our current work where the synthetized lactoside derivatives did not have the same impact on plant and fungal PME activities. To understand what could drive the interaction between the various lactoside derivatives and PME, we used structural homology modeling and docking analyses. These approaches were proven to be powerful tools to assess the potential binding of ligands to proteins [START_REF] Thakuria | Functional and catalytic active sites prediction and docking analysis of azoreductase enzyme in Pseudomonas putida with a variety of commercially available azodyes[END_REF]. Considering the fact that AtPME3 and AtPME31 are plant PMEs but differ in their inhibition by lactoside derivatives, we used the two proteins as case-studies. The 3D structure of AtPME3 was modeled in a previous study [START_REF] Sénéchal | Tuning of pectin methylesterification: PECTIN METHYLESTERASE INHIBITOR 7 modulates the processive activity of co-expressed PECTIN METHYLESTERASE 3 in a pH-dependent manner[END_REF]. The 3D structure of AtPME31 were investigated and showed that the overall structure of plant PMEs was well conserved, but a supplementary loop was present in the structure. Docking analysis was carried out using AtPME3 and AtPME31 and the different lactoside derivatives. This strongly suggests that inhibition of AtPME3 activity, as what previously shown for PMEI [START_REF] Sénéchal | Structural and dynamical characterization of th pH-dependence of the pectin methylesterase-pectin methylesterase inhibitor complex[END_REF][START_REF] Hocq | Combined experimental and computational approaches reveal distinct pH dependence of pectin methylesterase inhibitors[END_REF] occurs through the interaction of lactoside derivatives with the ligand-binding cleft structure within AtPME3. This interaction is dependent on the structure of the lactoside derivatives and the best interaction occurred with the bioisosteres with structure close to natural substrate of the enzyme. Docking interaction between AtPME3 and lactoside derivatives predicted number of stable hydrogen bonds, but no hydrogen bonds could be predicted for the interaction between AtPME31 and lactoside derivatives. This result was probably due to the presence of a supplementary loop above the ligand binding site.

Concluding remarks

In conclusion, we used purified proteins from plants and fungi to test some newly synthetized potential inhibitors of PME activity. Lactoside derivatives of distinct structure were shown to inhibit differentially plant and fungal PMEs. The difference in the inhibiting capacity of lactoside derivatives towards two plant PMEs of different structures could be modeled using docking analyses, showing good correlation with experimental data. Our study paves the way for the chemical synthesis of novel specific inhibitors of PME activity that could be used to target either plant or pathogen enzymes;

8. 1 13 C

 113 Hz, 1H, H-1), 4.22 (dd, J H-4',H-3'= 3.5 Hz, JH-4',H-5'=1.3 Hz, 1H, H-4'), 4.10 (d, J H-5', H-4'= 1.3 Hz, 1H, H-5'), 3.91 -3.86 (m, 1H, H-4), 3.77 -3.65 (m, 3H, H-3', H-3, H-5), 3.61 -3.53 (m, 4H, H-2', OCH3 ), 3.43 -3.33 (m, 1H, H-2). NMR (75 MHz, D2O): δ (ppm) 175.2 (C-6'), 174.6 (C-6), 103.0 (C-1), 102.8 (C-1'), 81.1 (C-5 or C-3 or C-3'), 75.7 (C-4), 75.5 (C-5'), 74.6 (C-5 or C-3 or C-3'), 72.6 (2s, C-2 & C-5 or C-3 or C-3'), 70.6 (C-2'), 70.1 (C-4'), 57.3 (OCH3).

1 H

 1 NMR (300 MHz, D2O): δ (ppm) 4.52 ( d, J H-1',H-2' = 8.1 Hz, 1H, H-1'), 4.52 (d, J H-1,H-2= 1.5 Hz, 1H, H-1) 4.43 (d, JH-4,H-5= 7.8 Hz, 1H, H-5), 4.30 (dd, J H-2,H-3= 3.4 Hz, JH-2,H-1=1.3 Hz, 1H, H-2), 4.27 (d, JH-4',H-5'= 9.7 Hz, 1H, H-5'), 3.92 -3.85 (m, 1H, H-4'), 3.85 (s, 6H, COOCH3), 3.76 (t, JH-3',H'2=JH-3',H-4'=9.0 Hz, 1H, H-3'), 3.75 (dd, JH-3,H-4=10.2 Hz, JH-3,H-2=3.3 Hz, H-3) 3.59 (s, 3H, OCH3), 3.54 (dd, JH-4,H-3 = 10.0 Hz, JH-4,H-5=7.8 Hz, 1H, H-4), 3.41 (dd, JH-2',H-3' = 9.3 Hz, JH-2',H-1'=8.0 Hz, 1H, H-2'). 13 C NMR (75 MHz, D2O): δ (ppm) 170.2-170.1 (2s, C-6, C-6'), 103.4 (C-1'), 102.8 (C-5), 80.6 (C-4'), 74.1 (C-1), 73.9 (C-3), 73.1 (C-5'), 72.4 (C-2'), 71.7 (C-3'), 69.9 (C-4), 69.3 (C-2), 57.6 (OCH3), 53.3-53.0 (2s, COOCH3)

9 66. 3 (

 93 1 H NMR (600 MHz, CD3OD): δ (ppm) 4.36 ( d, J H-4',H-5' = 1.4 Hz, 1H, H-5'), 4.33 (d, J H-1',H-2'= 7.4 Hz, 1H, H-1') 4.30 (d, JH-1,H-2= 7.8 Hz, 1H, H-1), 4.26 -4.12 (m, 5H, OCH2, H-4'), 4.02 (d, JH-4,H-5= 9.7 Hz, 1H, H-5), 3.77 (dd, JH-3,H-4=8.8 Hz, JH-4,H-5= 9.6 Hz, H-4), 3.60 (t, JH-2,H-3=JH-3,H-4=8.9 Hz, 1H, H-3), 3.55 (dd, JH-3',H-4'= 9.8 Hz, JH-2',H-3'=3.2 Hz, H-3') 3.51-3.50 (m, 4H, OCH3, H-2'), 3.28 (dd, JH-1,H-2 = 7.8 Hz, JH-2,H-3= 9.1 Hz, 1H, H-2), 1.76 -1.57 (m, 4H, OCH2CH2-), 1.50 -1.36 (m, 4H, OCH2CH2CH2), 0.96 (2t, J=7.4Hz, 6H, CH2CH3) 13 C NMR (150 MHz, CD3OD): δ (ppm) 170.0 (C-6), 169.7 (C-6'), 105.6 (C-1), 104.9 (C-1'), 82.7 (C-4), 75.8 (C-3), 75.5 (C-5'), 75.2 (C-5), 74.5 (C-2), 74.0 (C-3'), 71.6 (C-2'), 71.2 (C-4'), 66.8 -OCH2) 57.5 (OCH3), 31.7 -31.5 (2s, OCH2CH2), 20.1 (2s, OCH2CH2CH2), 14.0 (2s, CH2CH3).

Figures captions: Scheme 1 .

 1 Figures captions:

Table 1 :

 1 Binding affinity, number of hydrogen bonds and the binding residues calculated by docking for AtPME3 and AtPME31

		Lactoside	OH	Me	Na	Bu
		derivatives
	AtPME3	Predicted	-9.08	-5.16	-4.59	-0.32
		binding
		affinity
		(kcal/mol)
		N° of H-bonds 5	4	1	0
		Residues	Arg (2), Asp	Asp (2), Gln	Gln	-
			(2), Gln (1)	(2)
	AtPME31	Predicted	24.25	70.30	42.75	213.35
		binding
		affinity
		(kcal/mol)
		N° of H-bonds 0	0	0	0
		Positions	-	-	-	-
		residues

Funding information and acknowledgments

This work was supported by a grant from the Conseil Régional de Picardie, France and the European Regional Development Fund (PECTINHIB project) through a studentship awarded to ML' and PK. DK acknowledges supports from the National Institutes of Health (R01GM123055), the National Science Foundation (DMS1614777, CMMI1825941), and the Purdue Institute of Drug Discovery. We thank Jan A. L. Van Kan for the supply of the Pichia pastoris strain BcPME1.