domain: info.info-au math spi sdv

text:
Terence Bayen

Francis Mairet

Marc Mazade

Minimal time problem for a fed-batch bioreactor with a non admissible singular arc

In this paper, we consider an optimal control problem for a system describing a fed-batch bioreactor with one species and one substrate. Our aim is to find an optimal feedback control in order to steer the system to a given target in minimal time. The growth function is of Haldane type implying the existence of a singular arc. Unlike many studies on the minimal time problem governed by an affine system w.r.t. the control with one input, we assume that the singular arc is non-necessary controllable. This brings interesting issues in terms of optimal synthesis. Thanks to the Pontryagin Maximum Principle, we provide the optimal synthesis of the problem, It turns out that singular extremal trajectories are no longer optimal on a subset of the singular arc.

I. INTRODUCTION

The study of the minimal time control problem for affine systems with one input:

ẋ = f (x) + ug(x), x ∈ R n , |u| ≤ 1, (1)
has been investigated a lot in the literature, see [START_REF] Boscain | Optimal Syntheses for Control Systems on 2-D Manifolds[END_REF] for n = 2 and references herein. One often encounters singular trajectories which appear when the switching function of the system is vanishing on a sub-interval. In order to find an issue to an optimal control problem governed by [START_REF] Bayen | Optimal synthesis for the minimal time control problems of fed-batch processes for growth functions with two maxima[END_REF], studies often require that the singular arc is controllable which means that the singular control u s allowing the trajectory to stay on the singular arc is supposed to verify the inequality:

|u s | ≤ 1. (2)
However, one cannot in general show that this assumption holds. In fact, if the singular arc is of first order, the expression of the singular control u s is in general complicated as it depends both on the state and the adjoint state. The objective of this work is to study a minimal time control problem in the plane, that is n = 2 where the singular control is not always admissible i.e. ( 2) is not always satisfied. We consider a fed-batch bioreactor with one species and one substrate. Our aim is to find an optimal feedback control that steers the system in minimal time to a given target where the substrate concentration is less than a prescribed value, see [START_REF] Moreno | Optimal time control of bioreactors for the wastewater treatment[END_REF]. Finding an optimal feeding strategy can significantly increase the performance of the system and has tbayen@math.univ-montp2.fr, mmazade@math.univ-montp2.fr francis.mairet@inria.fr several advantages from a practical point of view (see e.g. [START_REF] Bayen | Optimal synthesis for the minimal time control problems of fed-batch processes for growth functions with two maxima[END_REF], [START_REF] Bayen | Minimal time control of fed-batch bioreactor with product inhibition[END_REF], [START_REF] Dochain | Minimal time control of fed-batch processes for growth functions with several maxima[END_REF], [START_REF] Gajardo | Minimal time sequential batch reactors with bounded and impulse controls for one or more species[END_REF], [START_REF] Lee | Control of fedbatch fermentations[END_REF], [START_REF] Moreno | Optimal time control of bioreactors for the wastewater treatment[END_REF]). Whenever the growth function is of Monod type, then one can prove that the optimal feeding strategy is bang-bang [START_REF] Moreno | Optimal time control of bioreactors for the wastewater treatment[END_REF]. This means that the reactor is filled until its maximum volume with the maximum input flow rate. Then, microorganisms consume the substrate until reaching a reference value. In the case where the growth function is of Haldane type (in case of substrate inhibition), this strategy is not optimal. In fact, one can prove (see [START_REF] Moreno | Optimal time control of bioreactors for the wastewater treatment[END_REF]) that the optimal strategy is singular. It consists in reaching a substrate concentration s (corresponding to the maximum of the growth rate function) in minimal time. Then, the substrate concentration is kept constant to this value until reaching the maximal volume. This can be proved by using either the Pontryagin Maximum Principle (PMP) or the clock form [START_REF] Bonnard | Singular Trajectories and their role in Control Theory[END_REF], [START_REF] Miele | Application of Green's Theorem to the extremization of linear integrals[END_REF].

In the present work, we are interested in studying the optimal synthesis for Haldane-type growth function whenever the singular arc is no longer admissible from a certain volume value. This happens when the singular control is larger than the maximal input flow rate which is allowed in the system. It follows that there exists a volume value above which singular extremal trajectories are no longer optimal. Whereas in [START_REF] Moreno | Optimal time control of bioreactors for the wastewater treatment[END_REF], [START_REF] Bayen | Minimal time control of fed-batch bioreactor with product inhibition[END_REF], [START_REF] Bayen | Optimal synthesis for the minimal time control problems of fed-batch processes for growth functions with two maxima[END_REF], we suppose that the maximal volume value can be reached by the singular arc, there exists a volume value above which it is not possible to keep the substrate concentration equal to s in the system. The main issue of the paper is the following. We aim at determining an optimal feedback control whenever the the singular arc becomes non admissible (i.e. the singular control saturates one of the two bounds on the control).

First, one cannot apply the clock form [START_REF] Bonnard | Singular Trajectories and their role in Control Theory[END_REF] (that allows a direct comparison between the cost of two trajectories) as we do not have a natural candidate for optimality in this case. In particular, optimal strategies developed in [START_REF] Bayen | Minimal time control of fed-batch bioreactor with product inhibition[END_REF], [START_REF] Bayen | Optimal synthesis for the minimal time control problems of fed-batch processes for growth functions with two maxima[END_REF] are no longer admissible. Thanks to Pontryagin's principle, we show that the optimal synthesis of the problem is rather different as in [START_REF] Moreno | Optimal time control of bioreactors for the wastewater treatment[END_REF]. We introduce solutions of the system backward in time with the maximum control that allow to determine where the switching time occurs for optimal trajectories. Our main result is that a singular extremal trajectory stops to be optimal before reaching the maximal admissible value (this is rather non-intuitive and is slightly different as the controllable case). We show that there exists a maximal volume value above which a singular trajectory is not optimal. We also provide numerical switching curves for the control which allow to determine an optimal feedback control of the problem.

The paper is organized as follows. The first section states the optimal control problem. We also recall the optimality result of [START_REF] Moreno | Optimal time control of bioreactors for the wastewater treatment[END_REF] and we apply the PMP. The second section is devoted to the optimal synthesis of the problem when the singular is not admissible everywhere. We first describe the curves that are solution of the system backward in time and that allow to determine properties of optimal trajectories. Then, we state our main results in the case where the singular arc is never controllable (Proposition 2) and whenever it is controllable until a certain volume (Proposition 3). The last section is devoted to numerical simulations. We exhibit a numerical switching curve for the study that allows to provide an optimal feedback control of the problem.

II. GENERAL RESULTS

In this section, we state the optimal control problem, we make a review of the standard optimality result in [START_REF] Moreno | Optimal time control of bioreactors for the wastewater treatment[END_REF] when the singular arc is controllable, and we apply the Pontryagin Maximum Principle (see [START_REF] Pontryagin | The Mathematical Theory of Optimal Processes[END_REF]) that will be used in the next section.

A. Statement of the problem

We consider a system describing a fed-batch bioreactor with one species and one substrate:

     ẋ = x µ(s) -u v , ṡ = -µ(s)x + u v (s in -s), v = u, (3)
where x represents the concentration of micro-organisms, s the concentration of substrate, and v is the volume of the tank. The input substrate concentration is denoted by s in > 0, and u is the input flow rate in the system. For convenience, we have taken yield coefficient equal to one (by rescaling the equation). The function s -→ µ(s) is the growth function of Monod or Haldane type (see [START_REF] Smith | The theory of the chemostat, Dynamics of microbial competition[END_REF]). In the following, we consider that u takes values within the set:

U := {u : [0, +∞) → [0, u max ] | u meas.}. (4)
Here u max denotes the maximum input flow rate in the system. In the following, we can take by time scaling u max = 1. The target we consider is defined by:

T = R * + × [0, s ref ] × {v m }. (5)
For u ∈ U, let t ξ0 (u) the time to steer (3) from an initial condition

ξ 0 := (x 0 , s 0 , v 0 ) ∈ R * + × [0, s in ] × [0, v m ].
The optimal control problem becomes:

inf u∈U t ξ0 (u), s.t. ξ(t(u)) ∈ T , (6)
where ξ(•) denotes the unique solution of (3) for the control u that starts at ξ 0 . One essential feature in the system (3) is that the quantity

M := v(x + s -s in ), (7)
is conserved along any trajectory of (3), hence M is constant and equal to v 0 (x 0 + s 0 -s in ). From [START_REF] Dochain | Minimal time control of fed-batch processes for growth functions with several maxima[END_REF], we obtain:

x = M v + s in -s, (8)
and system (3) can be put into a two-dimensional system:

ṡ = -µ(s)( M v + s in -s) + u v (s in -s), v = u, (9)
On can easily show that the set [0, s in ] × R * + is invariant by [START_REF] Lee | Control of fedbatch fermentations[END_REF]. Notice that if we define x by ( 8), the micro-organisms concentration may not be positive. This may happen when M ≤ 0 which means that initial conditions of microorganisms and substrate are low. Therefore, we consider initial conditions for [START_REF] Lee | Control of fedbatch fermentations[END_REF] in the domain D defined by:

D := (s, v) ∈ [0, s in ] × (0, v m ] | M v + s in -s > 0 . (10

B. Optimal synthesis when the singular arc is admissible

In this part, we review the result of [START_REF] Moreno | Optimal time control of bioreactors for the wastewater treatment[END_REF] on optimal trajectories for problem (6) that will allow to introduce the problem without supposing that the singular arc is admissible.

Next, we say that the growth function µ is of Monod type if µ(s) = μs k+s with μ > 0 and k > 0.

Theorem 1: Assume that µ is of Monod type. Then, the optimal feedback control u M steering any initial condition in D to the target T is:

u M [s, v] := 1 if v < v m , 0 if v = v m . (11)
In other words, the optimal strategy is fill and wait, and it consists in filling the tank with maximum input flow rate until v = v m , and then we let u = 0 until s reaches the value s ref (if necessary).

In the rest of the paper, we will consider only the case where the growth function µ is of Haldane type i.e. µ(s) = μs gs 2 + s + k , with μ > 0, g > 0, and k > 0. In this case, µ has exactly one maximum over R + , that we denote s, and we suppose that s > s ref (which means that the reference concentration to achieve is sufficiently small). The optimal synthesis in this case is rather different as for Monod growth function.

Theorem 2: Assume that µ is of Haldane type and that the following assumption holds:

µ(s) M s in -s + v m ≤ 1. (12)
Then, the optimal feedback control u H to reach the target is given by

u H [s, v] :=      0 if v = v m or s > s, 1 if s < s and v < v m , u s (v) if s = s and v < v m , (13)
where

u s (v) := µ(s) M s in -s + v . (14)
This result can be proved via the clock form, see e.g. [START_REF] Moreno | Optimal time control of bioreactors for the wastewater treatment[END_REF]. This tool is based on Green's Theorem and allows a direct comparison of the cost between two trajectories steering an initial point to the same target point.

Here we have emphasized the assumption (12) ensuring the system to stay on the singular arc (see e.g. [START_REF] Dochain | Minimal time control of fed-batch processes for growth functions with several maxima[END_REF], [START_REF] Bayen | Optimal synthesis for the minimal time control problems of fed-batch processes for growth functions with two maxima[END_REF]). The control u s is singular (see section II-C). It allows to maintain the substrate concentration equal to s. It can be written u s (v) = µ(s)x v(sin-s) so that u s ≥ 0. Therefore, assumption ( 12) is essential to ensure that u s (v) satisfies the upper bound

u s (v) ≤ 1 for all v ≤ v m .
The objective of this paper is to provide an optimal synthesis of the problem whenever ( 12) is non-necessarily satisfied. Note that in practice, one should start the fed-batch with a high biomass concentration (i.e. high M ) in order to speed up the process, so that condition [START_REF] Pontryagin | The Mathematical Theory of Optimal Processes[END_REF] can no longer be satisfied.

C. Pontryagin maximum principle

In this part we apply Pontryagin maximum principle on [START_REF] Boscain | Optimal Syntheses for Control Systems on 2-D Manifolds[END_REF]. Let H := H(s, v, λ s , λ v , λ 0 , u) the Hamiltonian of the system defined by:

H := -λ s µ(s) M v -(s -s in ) + u λ s (s in -s) v + λ v + λ 0 . ( 15
)
If u is an optimal control and (s, v) the corresponding solution of ( 9), there exists t f > 0, λ 0 ≤ 0, and an absolutely

continuous map λ = (λ s , λ v ) : [0, t f ] → R 2 such that (λ 0 , λ) = 0, λs = -∂H ∂s , λv = -∂H ∂v , that is: λs = λ s µ ′ (s)x -µ(s) + u v , λv = λ s -µ(s)M+u(sin-s) v 2 , ( 16
)
and we have the maximality condition: 9)-( 16)-(17), and extremal control the control u associated to this extremal trajectory. As t f is free, the Hamiltonian is zero along an extremal trajectory. Following [START_REF] Bayen | Optimal synthesis for the minimal time control problems of fed-batch processes for growth functions with two maxima[END_REF], one can prove that λ s is always non-zero (it is therefore of constant sign from the adjoint equation), and that λ 0 < 0 (hence we take λ 0 = -1 in the following). Next, let us define the switching function φ associated to the control u by:

u(t) ∈ argmax ω∈[0,1] H(s(t), v(t), λ s (t), λ v (t), λ 0 , ω), ( 17
) for almost every t ∈ [0, t f ]. We call extremal trajectory a sextuplet (s(•), v(•), λ s (•), λ v (•), λ 0 , u(•)) satisfying (
φ := λ s (s in -s) v + λ v . (18)
We obtain from (17) that any extremal control satisfies the following control law: for a.e. t ∈ [0, t f ], we have

     φ(t) < 0 =⇒ u(t) = 0 (No feeding), φ(t) > 0 =⇒ u(t) = 1 (Maximal feeding), φ(t) = 0 =⇒ u(t) ∈ [0, 1].
We say that t 0 is a switching point if the control u is non-constant in any neighborhood of t 0 which implies that φ(t 0 ) = 0. In this case, the control is bang-bang around t 0 , that is u switches either from 0 to 1 or from 1 to 0 at time t 0 . Whenever φ is zero on a non-trivial interval I ⊂ [0, t f ], we say that u is a singular control, and that the trajectory contains a singular arc. The sign of φ is fundamental in order to obtain the optimal synthesis. By taking the derivative of φ, we get:

φ = λ s x(s in -s)µ ′ (s) v . (19)
Moreover, we can show that λ s < 0 (see e.g. [START_REF] Dochain | Minimal time control of fed-batch processes for growth functions with several maxima[END_REF], [START_REF] Bayen | Optimal synthesis for the minimal time control problems of fed-batch processes for growth functions with two maxima[END_REF]). This implies that any extremal trajectory satisfies the property:

s(t) > s =⇒ φ(t) > 0 ; s(t) < s =⇒ φ(t) < 0.
Now, if an extremal trajectory contains a singular arc on some time interval I := [t 1 , t 2 ], then we have φ = φ = 0 on I, hence we have µ ′ (s) = 0 and s = s on I. Therefore, by solving ṡ = 0, we obtain easily the expression of the singular control given by ( 14), see e.g. [START_REF] Bayen | Minimal time control of fed-batch bioreactor with product inhibition[END_REF]. Moreover, we can estimate the time of a singular trajectory as follows (see e.g. [START_REF] Bayen | Optimal synthesis for the minimal time control problems of fed-batch processes for growth functions with two maxima[END_REF]):

t 2 -t 1 = 1 µ(s) ln M + v(t 2 )[s in -s] M + v(t 1 )[s in -s] , (20)

III. OPTIMAL SYNTHESIS WITHOUT CONTROLLABILITY

ASSUMPTION

In this part, we provide a description of optimal trajectories for problem (6) when ( 12) is not satisfied. To this end, we introduce solutions of the system backward in time that will allow us to describe where optimal trajectories have a switching point.

A. Study of the domain D

In view of ( 12), we introduce a mapping η : (0, s in ) → R by

η(s) := 1 µ(s) - M s in -s . ( 21
)
By definition of η, we have:

u s (v) = 1 ⇐⇒ v = η(s).
Now, we set v * := η(s) so that the singular arc is controllable provided that v * ≥ v m . Therefore, we assume throughout this section that v * satisfies:

v * < v m . (22)
It follows that the singular arc is controllable only provided that the volume is less than v * . Indeed, for v > v * equality (14) no longer defines an admissible control in [0, 1]. Next, we will consider the two following cases:

• Case 1: v * ≤ 0, • Case 2: 0 < v * < v m .
Remark 1: Case 1 means that the singular arc is never controllable over (0, v m ]. As the function η can take negative values, v * can be negative. We now define the curve Ĉ (resp. Č) whose graph is the mapping

v ∈ (0, v m ] -→ γ(v) (resp. v ∈ (0, v m ] -→ γ(v))
as follows. The mapping γ (resp. γ) is the unique solution of the equation:

ds dv = -µ(s) M v + s in -s + s in -s v , (23)
which passes through the point (s, v m ) (resp. (s, v * )). In other words, γ(v) and γ(v) are solution of ( 9) backward in time with a constant control u = 1. These curves will play a major role in our optimal synthesis contrary to the case where the singular arc is controllable (see Figure 2 and Table I for parameter values). In fact, they will indicate sub-domains where optimal trajectories have a switching point. Let v (resp. v) the first volume value such that γ(v) / ∈ (0, s in ) (resp. γ(v) / ∈ (0, s in )). The next proposition is concerned with monotonicity properties of γ and γ.

Proposition 1: (i) The curve γ is either decreasing on [v, v m ], either there exists a unique v 1 ∈ (v, v m ) such that γ(v 1 ) ∈ (0, s in ) and dγ dv (v 1 ) = 0. Moreover, in the latter case, γ is increasing on [v, v 1 ] and is decreasing on

[v 1 , v m ]. (ii)
The mapping γ is increasing on (v, v * ] and decreasing on [v * , v m ] and dγ dv (v * ) = 0. Proof: We omit the proof that can be found in [START_REF] Bayen | Minimal time problem for a fed-batch bioreactor with saturating singular control[END_REF]. In the second case of Proposition 1 (i), we can show that there exists values of M such that there exists a unique v * ∈ (0, v * ) such that:

γ(v * ) = s, ( 24
)
see Fig. 2.

B. Optimal synthesis when the singular arc is never admissible

In the case where v * ≤ 0 (case 1), we have the following optimality result (see also Fig. 2).

Proposition 2: Assume that v * ≤ 0. Then, optimal controls satisfy the following:

• If s 0 ≤ s, then, there exists t 0 > 0 such that u = 1 on [0, t 0 ], u = 0 on [t 0 , t f ] where t 0 is such that v(t 0 ) = v m and s(t f ) = s ref . • If s < s 0 < γ(v 0 ), then, there exists t 0 > 0 such that u = 0 on [0, t 0 ], u = 1 on [t 0 , t 1 ], u = 0 on [t 1 , t f ] where t 0 ≥ 0, s < s(t 0 ) < s 0 , v(t 1 ) = v m , and
s(t f ) = s ref . • If s 0 ≥ γ(v 0 ), then, there exists t 0 > 0 such that u = 0 on [0, t 0 ], u = 1 on [t 0 , t 1 ], u = 0 on [t 1 , t f ] with t 0 > 0, s < s(t 0 ) < γ(v 0 ), v(t 1 ) = v m , and s(t f ) = s ref .
Proof: We omit the proof that can be found in [START_REF] Bayen | Minimal time problem for a fed-batch bioreactor with saturating singular control[END_REF].

Remark 3: In the second case of proposition 2, the switching time t 0 from u = 0 to u = 1 may be zero and it can be found numerically, see section IV.

Next, we consider the second case where 0 < v * < v m . It is easy to see that for initial conditions such that v 0 > v * , optimal controls are given by proposition 2 (indeed, the singular arc is defined only for v 0 ≤ v * ). Therefore, we only deal with v 0 < v * . The next Proposition is illustrated on Fig.

2.

Proposition 3: Assume that v 0 < v * and that there exists 0 < v * < v * such that γ(v * ) = s. Then, optimal controls satisfy the following:

• If s 0 ≤ γ(v 0 ), then, there exists t 0 > 0 such that we have u = 1 on [0, t 0 ], u = 0 on [t 0 , t f ] where t 0 is such that v(t 0 ) = v m . • If γ(v 0 ) < s 0 < γ(v 0 ) and s 0 ≤ s, then, there exists t 0 > 0 such that we have u = 1 on [0, t 0 ], u = u s on [t 0 , t 1 ], u = 1 on [t 1 , t 2 ], u = 0 on [t 2 , t f ], where s(t 0 ) = s, t 1 -t 0 ≥ 0, v(t 1 ) < v * , and v(t 2 ) = v m . • If γ(v 0 ) ≤ s 0 < s, then, there exists 0 < t 0 < t 1 < t 2 such that we have u = 1 on [0, t 0 ], u = u s on [t 0 , t 1 ], u = 1 on [t 1 , t 2 ], u = 0 on [t 2 , t f ] where s(t 0 ) = s, v(t 1 ) ∈ (v * , v * ), v(t 2 ) = v m . • If s 0 ≥ s and v 0 ≤ v * , then, there exists 0 < t 0 < t 1 < t 2 such that we have u = 0 on [0, t 0 ], u = u s on [t 0 , t 1 ], u = 1 on [t 1 , t 2 ], u = 0 on [t 2 , t f ] where s(t 0 ) = s, v(t 1 ) ∈ (v * , v * ), and v(t 2 ) = v m . • If s 0 ≥ s,
and v 0 > v * , then, the optimal control is one of the following types:

-

either u = 0 on [0, t 0 ], u = u s on [t 0 , t 1 ], u = 1 on [t 1 , t 2 ], u = 0 on [t 2 , t f ] where s(t 0 ) = s and 0 < t 0 < t 1 , v(t 2 ) = v m , -either u = 0 on [0, t 0 ], u = 1 on [t 0 , t 1 ], u = 0 on [t 1 , t f ] where t 0 ≥ 0, s < s(t 0 ) < γ(v 0 ), v(t 1 ) = v m . Proof:
The proof that can be found in [START_REF] Bayen | Minimal time problem for a fed-batch bioreactor with saturating singular control[END_REF].

Substrate

Volume Fig. 2. Optimal trajectories (in solid red lines) for various initial conditions Top: case 1 (v * ≤ 0), see Proposition 2 for the optimal synthesis; bottom: case 2 (0 < v * < vm), see Proposition 3. In blue dashed lines, trajectories Ĉ and Č which pass through (s, v * ) and (s, vm). In green, the mapping

v 0 -→ s b (v 0 ) (see Section IV).
Remark 4: For the last item of the previous Proposition, the optimal trajectory is either B 0 -SA -B 1 -B 0 or B 0 -B 1 -B 0 where B 0 (resp. B 1 ) denotes an arc Bang u = 0 (resp. u = 1), and SA denotes a singular arc.

The next section will clarify this remark from a numerical point of view, and will provide an estimation of the switching time t 0 for items 2, 3, and 4 of Proposition 3.

IV. NUMERICAL SIMULATIONS

A. Determination of the maximal optimal volume

We will focus on the second case (0 < v * < v m ). Our aim in this part is to determine the optimal volume, denoted v a , above which it is not optimal to stay on the singular arc, as follows.

For t a (v 0 ) of this strategy is (recall (20), see also [START_REF] Bayen | Optimal synthesis for the minimal time control problems of fed-batch processes for growth functions with two maxima[END_REF]):

v 0 ∈ [v * , v * ],
t a (v 0 ) = ln M+v0(sin-s) M+v * (sin-s) µ(s) + v m -v 0 + s † (v0) s ref dσ µ(σ) M vm + s in -σ , (25)
where s † (v) is the substrate concentration when the trajectory reaches v = v m . We now show that τ admits a minimum v a ∈ [v * , v * ] that we will characterize hereafter. First, notice that s † (v 0 ) is obtained after integrating the ODE:

ds dv = -µ(s) M v + s in -s + s in -s v , s(v 0 ) = s. (26)
Moreover, (26) can be equivalently written as ds dv = g(v, s) where g is the right hand-side of (26). By the classical dependence of the solution of an ODE on parameters, we denote by s(v, s, v 0 ) the unique solution of (26). It is standard that

v 0 ∈ R * + -→ s(v, s, v 0 ) is of class C 1 for all v > 0. It follows by composition that v 0 -→ t a (v 0 ) is of class C 1 on [v * , v * ].
Consequently, it admits a minimum on this interval. By differentiating s(v, s, v 0 ) w.r.t. v 0 , we get:

∂s ∂v 0 (v, s, v 0 ) = -g(v 0 , s)e v v 0
∂g ∂s (s(w,v0,s),w)dw , hence ds † dv0 (v 0 ) = ∂s ∂v0 (v m , s, v 0 ). We have used a classical resul in ODE on the dependance of the trajectory w.r.t. initial conditions. Now, we know from the PMP that v 0 = v * and v 0 = v * are not admissible (see also Proposition 3), hence v a necessarily satisfies dta dv0 (v a ) = 0. So, if we put θ(v 0 ) := vm v0 ∂g ∂s (w, s(w, s, v 0 ))dw, we obtain by taking the derivative of (25) w.r.t. v 0 : v0) (27) This equation allows to obtain numerically the volume v a ∈ (v * , v * ) above which extremal trajectories stop to be singular. As an example, we find v a ≃ 1.67 (see Fig. 3 and Table I for the values of the parameters).

dt a dv 0 (v 0 ) = v * -v 0 M sin-s + v 0 1 - µ(s)( M v0 + s in -s) µ(s † )( M vm + s in -s) e θ(

B. Determination of the switching curve

Now, we determine the optimal switching time for the trajectories B 0 B 1 B 0 starting with s 0 > s.

For each v 0 ∈ (v * , v m ), we search numerically s b (v 0 ) ∈ [s, γ(v 0 )] which minimizes the time t b (s b ) to reach the target starting from (γ(v 0 ), v 0 ) with the strategy: u = 0 until s b , u = 1 until v m , u = 0 until s ref . Assuming that for each v 0 there is a unique minimum, this allows us to define the curve C b whose graph is the mapping v 0 -→ s b (v 0 ), represented in green on Fig. 2. We find that for v 0 < v a , we have s b (v 0 ) = s, while s b (v 0 ) > s for v 0 ∈ (v a , v m ). This result allows to conclude numerically which structure is optimal whenever s 0 > s and v 0 > v * (see the last case of Proposition 3):

• if v 0 ≤ v a , the optimal strategy is u = 0 on [0, t 0 ], u = u s on [t 0 , t 1 ], u = 1 on [t 1 , t 2 ], u = 0 on [t 2 , t f ] where s(t 0 ) = s, v(t 1 ) = v a , v(t 2 ) = v m . • if v 0 > v a , the optimal strategy is u = 0 on [0, t 0 ], u = 1 on [t 0 , t 1 ], u = 0 on [t 1 , t f ] where t 0 ≥ 0, s(t 0 ) = min(s(0), s b (v 0 )), v(t 1 ) = v m .
To conclude, based on the optimal synthesis (Proposition 3) and the computations of the mapping v 0 -→ s b (v 0 ) and v a , optimal trajectories for various initial conditions (s 0 , v 0 ) are represented on Fig. 2. We have shown that the optimal synthesis is quite different as the one of Theorem 2. In particular, we have pointed out that it is not optimal for a trajectory to stay as long as possible on the singular arc. Moreover, the curve v 0 -→ s b (v 0 ) is the switching curve for optimal trajectories and it will allow to give an optimal feedback control.

We thus obtain the following optimal feedback control. Proposition 4: The optimal feedback steering any initial state in D to the target is given by: V. CONCLUSION We have studied an optimal control problem for a fedbatch bioprocess in presence of a non-controllable singular arc. This situation can arise for example when the initial biomass concentration is high. Thanks to Pontryagin Maximum Principle and a careful study of the switching function, we have obtained a characterization of optimal trajectories of the problem. Our main result is that it is not optimal for a trajectory to stay as long as possible on the singular arc. Moreover, we have obtained switching curves that can be used to determine an optimal feedback control of the problem. A more detailed insight into the determination of these switching curves (for instance using the theory of conjugate points [START_REF] Bonnard | Second order optimality conditions in the smooth case and applications in optimal control[END_REF]) will deserve more investigation. The analysis of the problem when the singular arc is nonadmissible everywhere is more delicate as in the standard case. We believe that this kind of study can be also the starting point to study optimal control problems in a more general setting in presence of non-controllable singular arcs.

u 2 [s, v] :=                    0 if v = v m , 0 if s > s b (v), v ≥ v a , 1 if s ≤ s b (v), v m > v ≥ v a , 0 if s > s, v < v a , 1 if s < s, v < v a , u s (v) if s = s, v < v a ,
The implementation of the optimal strategy will also deserve future works. Given that model parameters are generally poorly known, and that all the state variables cannot be measured, it will be a key challenge to determine on-line the switching points.

  ) In the rest of the paper, we write u(•) a control in open loop and u[•] a feedback control depending on the state (s, v).

Remark 2 :

 2 The previous proposition implies that the backward curve Č leaves the domain D through the linesegment {0} × [0, s in ] whereas Ĉ leaves the domain D either through the line-segment {0} × [0, s in ] or through the linesegment {s in } × [0, s in ], see Figure1.

Fig. 1 .

 1 Fig. 1. The curve Ĉ leaves the domain through {0}×[0, s in ] when M = 1, and through {s in } × [0, s in ] when M = 25.

  consider the strategy u = u s from v * to v 0 , u = 1 until v m and then u = 0 until s ref . The time

Fig. 3 .

 3 Fig. 3. Time ta(v) to reach T from (s, v * ) with the strategy: singular arc until the switching volume v, u = 1 until vm, u = 0 until s ref . We find that ta(v) has a unique minimum for v = va (see Section IV)

TABLE I PARAMETER

 I VALUES (ARBITRARY UNITS) OF SIMULATIONS FOR THE OPTIMAL SYNTHESIS (SEE FIG. 2 AND 3)

	vm	s in	s ref	M	μ	k	g
	7	10	0.1	25 (case 1)	0.5	1	0.11
				1 (case 2)

ACKNOWLEDGEMENTS

We would like to thank Professor C. Lobry and A. Rapaport for helpful discussions on the subject.


Franc ¸ois De Vleeschouwer

Heleen Vanneste

Dmitri Mauquoy

Natalia Piotrowska

Fernando Torrejo

Thomas Roland

Ariel Stein

Gae ¨l Le Roux

Emissions from Pre-Hispanic Metallurgy in the South American Atmosphere

Metallurgical activities have been undertaken in northern South America (NSA) for millennia. However, it is still unknown how far atmospheric emissions from these activities have been transported. Since the timing of metallurgical activities is currently estimated from scarce archaeological discoveries, the availability of reliable and continuous records to refine the timing of past metal deposition in South America is essential, as it provides an alternative to discontinuous archives, as well as evidence for global trace metal transport. We show in a peat record from Tierra del Fuego that anthropogenic metals likely have been emitted into the atmosphere and transported from NSA to southern South America (SSA) over the last 4200 yrs. These findings are supported by modern time back-trajectories from NSA to SSA. We further show that apparent anthropogenic Cu and Sb emissions predate any archaeological evidence for metallurgical activities. Lead and Sn were also emitted into the atmosphere as by-products of Inca and Spanish metallurgy, whereas local coal-gold rushes and the industrial revolution contributed to local contamination. We suggest that the onset of pre-Hispanic metallurgical activities is earlier than previously reported from archaeological records and that atmospheric emissions of metals were transported from NSA to SSA.

. Results of the AMS radiocarbon dating of 8 selected samples from Karukinka.

Introduction

Background

Population expansion and territorial colonisation increased over the course of the last 5000 years in South America. Pre-Colombian civilizations flourished first in the Northern Andes and populations progressively migrated to the South. South American animal domestication and agriculture was followed by several periods of population expansion and metallurgical activities. In particular, copper extraction and smelting started in northwestern South America as well as in Argentina around 1400 BC and spread with the various South American civilizations (Chavin, Nasca, Tiwanaku, Inca). Incas also increasingly used silver, mainly around Peru, the Titicaca basin and Potosi (Bolivia). These changes in land use as well as the extraction and processing of metals up to the present day have released detectable amounts of trace elements into the atmosphere during the second half of the Holocene (ca. 4 kyrs cal BP to present).

Whereas South American metallurgy has been documented by archaeological finds, there is no record of how the resulting trace metals dispersed into the South American atmosphere through time and how far they have been transported away from production sites. To estimate the extent of anthropogenic trace metal emissions in South America, continuous and well-dated records of long-term metal deposition away from production centres are needed to highlight ancient metallurgical activities, given the absence/lack of archaeological evidence.

Our study of an ombrotrophic peat profile from Tierra del Fuego provides a continuous history of metallurgical activities in South America, and is therefore a valuable alternative to discontinuous archaeological findings. Andean trace metals of anthropogenic origin are found in our peat record from Tierra del Fuego, which suggests that they have been dispersed widely from their source areas. The data suggest for the first time that trace metals from pre-Hispanic and Hispanic Andean metallurgical activities were transported from North to South America, due to occasional North to South wind trajectories.

Scope

Continental archives, in particular raised/ombrotrophic peatlands or bogs have proven to be very useful in reconstructing past environmental changes and human activities [START_REF] Hong | Atmospheric heavy metals in tropical South America during the past 22 000 years recorded in a high altitude ice core from Sajama, Bolivia[END_REF][START_REF] Renberg I Persson | Preindustrial atmospheric lead contamination detected in Swedish lake sediments[END_REF][START_REF] Shotyk | History of atmospheric lead deposition since 12,370 14 C yr BP from a peat bog, Jura Mountains, Switzerland[END_REF]. Peat bogs are exclusively fed by atmospheric inputs and can therefore provide key archives of atmospheric metal deposition through time [START_REF] Shotyk | History of atmospheric lead deposition since 12,370 14 C yr BP from a peat bog, Jura Mountains, Switzerland[END_REF]. Long distance transport of trace metals is possible and hence suitable peat archive records can be used to identify the history of metallurgy even in remote areas [START_REF] Vleeschouwer | Peat as an archive of atmospheric pollution and environmental change: A case study of lead in Europe[END_REF][START_REF] Shotyk | Biogeochemistry and cyclcing of lead[END_REF][START_REF] Rosman Kjr Chisholm | Lead from Carthaginian and Roman Spanish mines isotopically identified in Greenland ice dated from 600 B.C. to 300 A.D[END_REF]. In this study, we reconstruct the timing of metal deposition in SSA during the last 4200 years using a high-resolution record of metal/La ratios as well as Pb isotopes in a radiocarbon-dated peat core from Karukinka bog, located in the central western part of Isla Grande de Tierra del Fuego (Fig. 1). The ombrotrophic section of the core provides a reliable record of trace metallurgical activities throughout the Andes because 1/peat is mostly (.95%) composed of organic matter, hence trace to ultra-trace amounts of metals are detectable, 2/Modern-time particle back-trajectories supported by studies showing analogies between past and present climate [START_REF] Fagel | Climate oscillations evidenced by spectral analysis of Southern Chilean lacustrine sediments: the assessment of ENSO over the last 600 years[END_REF][START_REF] Haug | Southward migration of the Intertropical Convergence Zone through the Holocene[END_REF][START_REF] Markgraf | Paleoclimate reconstruction along the Pole-Equator-Pole transect of the Americas (PEP 1)[END_REF][START_REF] Killian | A review of Glacial and Holocene paleoclimate records from southernmost Patagonia (49-55uS)[END_REF] prove the feasibility of atmospheric transport from NSA to SSA and 3/the low level of local environmental and atmospheric contamination (no metallurgical activities before the 20 th century) provides ideal background conditions to record long-range trace metal deposition.

Materials and Methods

Location

Karukinka Park is a protected area managed by the Chilean branch of the Wildlife Conservation Society (WCS) and located at the southwestern edge of Isla Grande de Tierra del Fuego. (Fig. 1). Access and coring permits were obtained by contacting Ricardo Muza at the WCS office in Punta Arenas (http://www. karukinkanatural.cl/). The mean annual rainfall is 400 mm.yr 21 and the mean annual temperature is 5uC. The reserve contains numerous peat bogs dominated by Sphagnum magellanicum with a sporadic cover of Marsippospermum grandiflorum and Empetrum rubrum. Occasional pools on the peatlands contain Sphagnum falcatulum and are fringed by Tetroncium magellanicum. Dense stands of deciduous forests surround the peat bog ecosystems and are dominated by Nothofagus pumilio. Peat deposits have accumulated continuously in this zone since the early Holocene (ca. 10kyrs cal BP) [START_REF] Auer | The Pleistocene of Fuego-Patagonia. Part 4: Bog profiles[END_REF][START_REF] Rabassa | The peatlands of Argentine Tierra del Fuego as a source for paleoclimatic and paleoenvironmental information[END_REF] Several bogs were identified as ideal dust traps because they are located on high altitude points in open areas solely dominated by Sphagnum magellanicum mosses. The site we sampled is a small 1-km diameter peat bog (S 53.86002u, N 69.57639u, 245 m a.s.l.) located at a relatively high altitude, 200 m aside from the main valley and 20 m above the main river. Given this, significant minerogenic input of river sediments is highly unlikely.

Coring

Peat profiles were recovered from Karukinka bog in February 2012 using a stainless steel Russian corer of 10 cm internal diameter and 50 cm length [START_REF] Belokopytov | Giktorf's peat borers Torfa ˆnaa promyslennost[END_REF]. The barrel of the corer was cleaned between each section using deionized water. The cores were photographed, described, wrapped in plastic film and packed into PVC tubes, which were subsequently stored in wooden boxes and shipped to France. An overlapping core were collected in order to avoid any possible disturbance at the top and bottom of each master core section, which can occur because the tip of the corer can slightly disturb the directly underlying peat layer. However, this core was not analyzed because we did not detect any contamination in the master core. The maximum depth (bottom of the peat bog) was reached at 4.5 m. In this study, we have only analysed the ombrotrophic part of the section (0-344 cm, see section 4.1).

Subsampling, depth correction and density calculation

A clean slicing and sub-sampling procedure was applied following published guidelines [START_REF] Givelet | Suggested protocol for collecting, handling and preparing peat cores and peat samples for physical, chemical, mineralogical and isotopic analyses[END_REF][START_REF] Vleeschouwer | Coring and subsampling of peatlands for palaeoenvironmental research[END_REF] to ensure minimal contamination and disturbance, whilst maintaining the highest sampling resolution. Samples were handled with powder free latex gloves. The cores were frozen at 220uC, unpacked and sliced contiguously at 1 cm resolution using a stainless steel band saw in a room specifically designed to process pristine samples. Each slice was cleaned with mQ (18 mega Ohm clean) water. The edges of each semi-cylindrical slice were then removed and the slices were subsampled using ceramic knives and plastic cutting boards. The dimensions of the still-frozen samples dedicated to geochemistry were measured using a vernier caliper in order to know the exact thickness of the sample and the loss of material due to each cut. The thickness of each sample ranges from 0.8 to 1.3 cm with 72% of the samples being 1.060.1 cm thick. The average loss on each cut (i.e. [total core length -cumulative sample thickness]/Nr of samples) is 0.18 cm. We have taken this loss due to cutting into account and reassessed the exact mid-point depth of each sample. All the subsamples were finally stored in a freezer at 220uC.

According to literature recommendations [START_REF] Vleeschouwer | Coring and subsampling of peatlands for palaeoenvironmental research[END_REF], the frozen samples dedicated for geochemistry were also measured for their lateral dimensions in order to accurately calculate their volume. These frozen samples were then freeze-dried to remove their water. The resulting dry samples were subsequently weighed and their densities calculated by dividing their dry weight by their respective volume.

Macrofossil analysis and selection for radiocarbon measurement

Plant macrofossil samples (5 cm 3 ) were boiled with 5% NaOH and sieved (mesh size 180 mm). Macrofossils were scanned using a binocular microscope (610-650), and identified using an extensive reference collection of type material collected during fieldwork in the study region in 2012. Plant macrofossils were assessed using the Quadrat and Leaf Count (QLC) method [START_REF] Barber | Climatic change and human impact in north Cumbria: peat stratigraphic and pollen evidence from Bolton Fell Moss and Walton Moss[END_REF]. Volume percentages were estimated for all components with the exception of seeds, macroscopic charcoal particles, fungal fruit bodies and minerogenic grains, which were counted and expressed as the number (n) present in each sub-sample. The selection of plant macrofossils in 8 samples for AMS radiocarbon dating was made following established protocols [START_REF] Mauquoy | Late Holocene climatic changes in Tierra del Fuego based on multiproxy analyses of peat deposits[END_REF]. The selection of aboveground plant macrofossils ensures that accurate dates are assigned (i.e. free of rootlets and therefore of contamination).

Radiocarbon sample preparation and measurement

The macrofossils from the 8 samples selected according to literature recommendations [START_REF] Vleeschouwer | Coring and subsampling of peatlands for palaeoenvironmental research[END_REF][START_REF] Piotrowska | Constructing deposition chronologies in peat deposits using radiocarbon dating[END_REF] for radiocarbon dating mainly consisted of Sphagnum magellanicum leaves and stems except the basal sample, which consisted of unidentified graminoids. The deepest dated sample is at 440.6 cm depth (below this depth and down to the base of the core at 450 cm, no datable macrofossils could be retrieved). Enough macrofossils (15-20 Sphagnum stems or graminoid fragments) were picked to ensure a minimum carbon mass of ca. 1 mg. All samples were prepared at the GADAM Centre (Gliwice, Poland) using acid-alkali-acid washing (to remove carbonate, bacterial CO 2 and humic/fulvic acid), drying, combustion and graphitisation [START_REF] Piotrowska | Status report of AMS sample preparation laboratory at GADAM Centre, Gliwice, Poland[END_REF]. Radiocarbon concentrations were measured at the Rafter Radiocarbon Laboratory (Lower Hutt, New Zealand) and at DirectAMS Laboratory (Bothell, USA) following established protocols [START_REF] Donahue | Isotope-Ratio and Background Corrections for Accelerator Mass Spectrometry Radiocarbon Measurements[END_REF]. Results are reported in Table 1.

Acid digestion in clean lab and dilutions

Dried samples were manually crushed using a clean agate pestle and mortar, which was rinsed between each sample using mQ water and pro analysis grade ethanol. They were then stored in 20ml plastic vials. Approximately 100 mg of dried and homogenized peat from each interval was weighed out and digested in Teflon beakers in a clean room with class 100 flow benches. All acids used were of ultrapure (purchased Optima grade) quality. The digestion procedure consisted of three steps each succeeded by a slow evaporation at 55uC [START_REF] Roux | Preparation of peat samples for inorganic geochemistry used as palaeoenvironmental proxies[END_REF]: (1) a mixture of 0.5 ml HF and 2 ml 16M HNO 3 was added and left on the hotplate at 110uC for 2 days; (2) 1 ml of H 2 0 2 was added to react for 6 h at room temperature; (3) 2 ml of 16M HNO 3 was added and left at 90uC for 2 days to finalize the digestion. Although peat is mainly composed of organic matter, a strong acid digestion is needed to dissolve any silica-based mineral residue that forms part of the remaining inorganic content (for peat bogs this is likely to be atmospheric dust) [START_REF] Roux | Preparation of peat samples for inorganic geochemistry used as palaeoenvironmental proxies[END_REF][START_REF] Roux | Weathering of inorganic matter in bogs[END_REF]. The samples were subsequently dissolved in 2 ml of 35% HNO 3 , transferred into 15 ml polypropylene tubes (FalconH), and further diluted with Milli-Q water up 14 ml. All samples were stored in a fridge for future analyses.

Q-ICP-MS analyses

After proper dilution Cu, Pb and La concentrations were measured on a Quadrupole ICP-MS (Agilent Technologie 7500 ce) at the Observatoire Midi Pyre ´ne ´e, Toulouse. The ICP-MS was calibrated using a synthetic multi-element standard, which was run every 8 samples, while an In-Re solution was used as an internal standard. The accuracy of the analyses was assessed by analysis of 3 international certified reference materials (SRM1947-peach leaves; SRM1515-apple leaves and GBW-07063-bush branches and leaves) and is reported in Table 2. The reproducibility, determined by repeat analyses (n = 3) of 1 CRM and 2 peat samples, is better ,5% for Cu and Pb, and , 12% for Sb, Sn and La. The blanks for all elements considered here, are ,0.01%. Final dilution factors were approximately 700.

HR-ICP-MS

After the measurements of the Pb concentration by Q-ICP-MS, mother solutions of selected samples were sub-sampled and diluted to adjust the Pb concentration to 500 mg.kg 21 prior to analyses by HR-ICP-MS at the Observatoire Midi-Pyre ´ne ´es, Toulouse [START_REF] Krachler | Optimising accuracy and precision of lead isotope measurement ( 206 Pb, 207 Pb, 208 Pb) in acid digests of peat with ICP-SMS using individual mass discrimination correction[END_REF]. A 500-mg.kg 21 SRM 981 standard solution was used to control and correct mass bias by standard bracketing of every sample. 206 Pb/ 207 Pb ratios in the NIMT peat standard were 1.17560.001 (n = 4) compared to the reported value of 1.176360.0004 [START_REF] Yafa | Development of an ombrotrophic peat bog (low ash) reference material for the determination of elemental concentrations[END_REF].

Sample preparation for tephra analysis

Based on the visual examination of the core as well as the lead isotope profile, 3 samples (237.5 cm, 34-301 yrs cal BC; 347 cm, 2354-2130 yrs cal BC; and 428 cm, 5745-5327 yrs cal BC) were processed to determine their potential volcanic origin by identifying glass shards. Dry samples were burned in a muffle furnace (550uC, 4 h) and ash residues were fine-sieved with the 10-125 mm fractions mounted in Canada Balsam on glass slides. Plane-polarised light, at 6100-400 magnification, was used to differentiate between tephra and other minerogenic material and 300 shards/grains were counted to determine their respective proportion. Fresh glass shards were identified and photographed.

Generation of the age-depth model

The 8 radiocarbon dates as well as two eruptions from Mt. Hudson (428 cm depth, 5745-5327 yrs cal BC) and Mt. Burney (347 cm depth, 2354-2130 yrs cal BC) were used as chronological time markers with the top of the core set to year 2012 AD (when it was recovered). CLAM software [START_REF] Blaauw | Methods and code for classical' age-modelling of radiocarbon sequences[END_REF] was used to construct a cubic smooth spline age-depth model using the SHCal04 calibration curve [START_REF] Mccormac | SHCal04 Southern Hemisphere Calibration, 0-11.0 cal kyr BP[END_REF]. The age-depth model is presented in Fig. 2. Based on 10000 iterations minimum and maximum ages for 2 sigma confidence interval were determined for each sample, as well as maximum likelihood age point estimates based on the weighted average of all generated age-depth curves. Maximum likelihood ages are given with the geochemical data in Table 3.

Back trajectory calculation

The Hybrid Single Particle Lagrangian Integrated Trajectory (HYSPLIT) model [START_REF] Draxler | Description of the HYSPLIT_4 modeling system[END_REF] has been used to run a series of backtrajectories from January 1, 1948 until December 31, 2012. The model is driven by meteorological data fields from the NCEP/ NCAR Reanalysis Project (http://ready.arl.noaa.gov/ gbl_reanalysis.php and ftp://arlftp.arlhq.noaa.gov/archives/ reanalysis/) [START_REF] Kalnay | The NCEP/NCAR 40-Year Reanalysis Project[END_REF]. The tools (http://www.meteozone.com/home/ tutorial/html/traj_freq.html) used to generate the trajectories are publicly available for the scientific community to reproduce the results. Previous observational and modelling palaoclimate studies have shown strong analogies between present and past climate and wind conditions [START_REF] Fagel | Climate oscillations evidenced by spectral analysis of Southern Chilean lacustrine sediments: the assessment of ENSO over the last 600 years[END_REF][START_REF] Haug | Southward migration of the Intertropical Convergence Zone through the Holocene[END_REF][START_REF] Markgraf | Paleoclimate reconstruction along the Pole-Equator-Pole transect of the Americas (PEP 1)[END_REF][START_REF] Killian | A review of Glacial and Holocene paleoclimate records from southernmost Patagonia (49-55uS)[END_REF], and that climate variability is partly controlled by modes or short-term pluriannual to decadal-scale oscillations [START_REF] Fagel | Climate oscillations evidenced by spectral analysis of Southern Chilean lacustrine sediments: the assessment of ENSO over the last 600 years[END_REF]. Therefore, our calculations, which cover more than 60 years, encompass these climatological drivers. The calculation includes back-trajectories initialized from the measurement site every six hours and the duration of each trajectory is set to 15 days. The trajectory starting heights are set to 500 m above ground level to represent air parcels within the planetary boundary layer reaching the measurement site. A total of 93440 trajectories were computed for this study. Once all the trajectories were calculated, the model counts the number of trajectories that fall within each grid cell from an arbitrarily set horizontal grid that covers the Southern Hemisphere. The grid resolution is 1.0 degree. The trajectory frequency (F) is just the sum of the number of trajectories (T) that passed through each (i, j) grid cell divided by the total number (N) of trajectories analyzed:

F i,j ~100 S T i,j =N
Note that all trajectories are counted once in the source location grid cell and once per intersecting grid cell.

Out of the 64 years investigated in this model (from 1948 to 2012, figure S1 for more information), air parcels coming partially from NSA and the pre-Hispanic metallurgical territories 7 times (Fig. 3). This represents more than 10% of the total period. As each of the peat slice from Karukinka represents several decades of peat accumulation, and assuming that the 64-year climatology used to drive the back-trajectories is representative of the atmospheric circulation of the past 4500 years, each slice from Karukinka is likely to capture several years where small but significant amounts of metals coming from Pre-Hispanic metallurgical territories are incorporated in the air parcels travelling to Tierra del Fuego. stage (Fig. 4 is followed by a transition to a bog (zone KAR12-PB01B-B) at 344 cm depth (2269-2057 yrs cal BC). This transition stage is however largely dominated by Sphagnum magellanicum. The ombrotrophic stage at 344 cm persists to the present, with a dominance of Sphagnum magellanicum (up to 95% of the total macrofossil assemblage). Moreover, from 344 cm to the surface, the peat bulk density is low (,0.2 g.cm 22 ), which also suggests ombrotrophic conditions spanning this depth interval. These low bulk density values indicate that minimal inwash of mineral material from lateral streams has occurred and therefore the peat deposits are likely to record predominantly low atmospheric dust inputs. Under these conditions, previous studies have shown that the elements analysed here (Pb, Cu, Sn, Sb) are not prone to post-depositional mobility and should therefore effectively record the history of metallurgy and other human activities [START_REF] Shotyk | Biogeochemistry and cyclcing of lead[END_REF][START_REF] Rausch | Comparison of atmospheric deposition of copper, nickel, cobalt, zinc, and cadmium recorded by Finnish peat cores with monitoring data and emission eecords[END_REF][START_REF] Shotyk | Antimony in recent peat from Switzerland and Scotland: comparison with natural background values (5,320 to 8,020 14C yr BP), correlation with Pb, and implications for the global atmospheric Sb cycle[END_REF][START_REF] Allan | High-resolution reconstruction of atmospheric deposition of trace metals and metalloids since AD 1400 recorded by ombrotrophic peat cores in Hautes-Fagnes, Belgium[END_REF][START_REF] Meharg | First comprehensive peat depositional records for tin, lead and copper associated with the antiquity of Europe's largest cassiterite deposits[END_REF].

Tephra identification

Of the 3 samples submitted for petrographic analysis, tephra horizons were identifiable in two (347 cm depth, 2354-2130 yrs cal BC and 428 cm depth, 5745-5327 yrs cal BC). The remaining sample contained small numbers of tephra shards but did not exceed baseline levels, which are typically present in such close proximity to numerous volcanic sources.

The tephrochronology of SSA is relatively well known [START_REF] Stern | Holocene tephrochronology record of large explosive eruptions in the southernmost Patagonian Andes[END_REF]. Subsequently, the two tephra horizons identified in the Karukinka peat core could be confidently attributed to known eruptions based on the characteristics of the shards present and their respective ages, as calculated from the age-depth model. The Mt. Hudson eruption was identified at 428 cm depth (5745-5327 yrs cal BC), with shard colour and morphology consistent with an eruption from the predominantly andesitic southern Volcanic Zone in which Mt. Hudson is located (Fig. 5). Conversely, the transparent, rhyolitic shards found at 347 cm were consistent with the Mt. Burney eruption (2354-2130 yrs cal BC) and the Austral Volcanic Zone in which the volcano is situated [START_REF] Stern | Holocene tephrochronology record of large explosive eruptions in the southernmost Patagonian Andes[END_REF]. These tephra horizons have been identified in various peat cores from Southern South America [START_REF] Stern | Holocene tephrochronology record of large explosive eruptions in the southernmost Patagonian Andes[END_REF][START_REF] Stern | The Holocene tephrochronology of southernmost Patagonia and Tierra del Fuego[END_REF][START_REF] Heusser | Deglacial palaeoclimate at Puerto del Hambre, subantarctic Patagonia, Chile[END_REF][START_REF] Mcculloch | ate-glacial and Holocene palaeoenvironmental change in the central Strait of Magellan, southern Patagonia. Palaeogeogr[END_REF]. The presence of a younger eruption from Mt. Burney reported in the literature at 358 cal BC -201 cal AD in Southern Patagonia [START_REF] Stern | Holocene tephrochronology record of large explosive eruptions in the southernmost Patagonian Andes[END_REF] is not clear in the Karukinka peat sequence, since no significant glass shard concentrations were found in the layer corresponding to this time range.

3. Brief description of the raw geochemical results (Cu, Pb, La, 206 Pb/ 207 Pb) and calculation of Metal/La UCC ratio

Copper and Sb concentrations remain below 7 mg.kg 21 from the base of the core to approximately 250 cm depth (Table 3). From 250 cm to 150 cm depth, the Cu concentration increases periodically, reaching more than 24 mg.kg 21 to finally decrease rapidly and remain low from 150 cm depth to the surface. Lead and Sn concentrations remain mostly below 1 mg.kg 21 except around 250 cm-150 cm where concentrations double and peak twice. Lanthanum concentration are stable between ca. 1 mg.kg 21 and 6.5 mg.kg 21 from 344 cm to 250 cm depth, and then increase twice from 3 mg.kg 21 to 15 mg.kg 21 between 250 cm to 150 cm, to then decrease below 1 mg.kg 21 . These increases in metals and La between 150 and 250 cm are not interpreted here since the concentration profiles are prone to changes in the accumulation rates or in the influx of dust to the bog. In order to accurately interpret changes in metal inputs to the bog, we will be using Me/ La ratios in the rest of the manuscript. Upper Continental Crust (UCC)-normalized metal/La ratios were calculated by dividing the metal/La ratio by the same ratio in the UCC, with 25 mg.kg 21 , 17 mg.kg 21 , 0.2 mg.kg 21 , 5.5 mg.kg 21 and 30 mg.kg 21 as the UCC concentrations for Cu, Pb, Sb, Sn and La, respectively [START_REF] Taylor | The geochemical evolution of the continental crust[END_REF].

Lead isotopes fluctuate within a range typical for the Upper Continental Crust (1.18, 206 Pb/ 207 Pb ,1.21) with three slight shifts towards more radiogenic values around 425 cm, 351 cm and 195 cm depth. In shallower samples (above 60 cm depth), the 206 Pb/ 207 Pb shifts towards more radiogenic values, which are typical for a mix between anthropogenic and natural sources.

History of metallurgy as recorded in Karukinka bog

Whereas metallurgical activities occurred throughout South America since ca 2000 BC, there is no evidence that pre-Hispanic metal objects or artifacts reached Tierra del Fuego [START_REF] Palacios | Metalurgia Prehispa ´nica de Sudame ´rica[END_REF]. Hunters and gatherers as well as canoe-borne nomadic people from Tierra del Fuego were seemingly unaware of purified metals before the Argentinean and Chilean colonization of these territories occurred in the 19 th century [START_REF] Martinic | La Tierra de los Fuegos[END_REF][START_REF] Gusinde | Fueginos. Hombres primitivos en la Tierra del Fuego[END_REF]. Hence, compared to the Northern Hemisphere and mainly Europe, Tierra del Fuego remained almost pristine and free of any local metal contamination until the last two centuries.

The upper part of the Karukinka peat bog (4200 years) is ombrotrophic as demonstrated by the plant macrofossil succession (see Figure 4) and therefore is a suitable archive of atmospheric metal deposition. By accurately analysing Pb, Cu, Sb, Sn, La, and Pb isotopes in peat samples using Q-ICP-MS and HR-ICP-MS, supported by a 14 C age-depth model, we provide a unique history of anthropogenic activities and metallurgical developments in South America since the mid-Holocene (from ca. 4 kyrs cal BP) (Fig. 6). Modern-time back trajectory calculations further demonstrate that air parcels can be transported from the main metallurgical areas in NSA to SSA. The trajectory frequency plot confirms that this is the case for 7 times (1981,1997,1999,2000,2006,2008,2010) during the 64 investigated years. Accordingly, NSA, and especially northern Chile, southern Bolivia and southern Peru are potential source regions of metals found in Karukinka (Fig. 1). Assuming that the 64-year climatology is representative of the atmospheric circulation of the past 4200 years and as each peat slice from Karukinka represents several decades, each single peat slice is likely to capture several years where small but significant amounts of metals coming from pre-Hispanic metallurgical territories are incorporated in the air parcels travelling to Tierra del Fuego.

The lower part of the peat profile (6500 BC to 2200 BC) is not ombrotrophic and may have been exposed to local in-washed mineral inputs and changes in redox conditions. As we want to investigate atmospheric metal deposition, we therefore do not interpret this part of the profile in detail but focus our interpretations on the ombrotrophic part of the sequence (2200 BC to present day).

The oldest archaeological evidence for copper metallurgy dates back to ca. 1400 BC in Southern Peru and the following centuries saw its development mainly in the Bolivian Altiplano, Northern Chile but also in the Antofagasta and northeast Argentina regions. This is based on archaeological evidence for metal ore extraction and smelting, various ritual objects, artifacts and weapons [START_REF] Eerkens | Pre-Inca mining in the Southern Nasca Region, Peru[END_REF][START_REF] Lleras | Una revisio ´n crı ´tica de las evidencias sobre metalurgia temprana en Surame ´rica[END_REF][START_REF] Nu ´n ˜ez | Emergencia de comunidades pastoralistas formativas en el sureste de la Puna de Atacama[END_REF][START_REF] Salazar | Minerı ´a y Metalurgia en la Costa Arreica de la Regio ´n de Antofagasta, Norte de Chile[END_REF][START_REF] Salazar | Metalurgia prehispa ´nica en las sociedades costeras del norte de Chile (quebrada Mamilla, Tocopilla)[END_REF][START_REF] Salazar | Metalurgia en San Pedro de Atacama durante el Perı ´odo Medio: Nuevos Datos, Nuevas Preguntas[END_REF][START_REF] Scattolin | Una ma ´scara de cobre de tres mil an ˜os. Estudios arqueometalu ´rgicos y comparaciones regionales[END_REF]. In the Karukinka core, the Upper Continental Crust (UCC)normalized Cu/La UCC ratio increases starting at 2000 BC (ca. 600 yrs before the first archaeological evidence), suggesting that small but detectable amounts of Cu would be transported to SSA and that Andean Cu metallurgy may be older than previously reported. Antimony, associated with Cu ores such as tetrahedrite, also displays an increase in the Sb/La UCC ratio. A shift in Pb isotopes can also be observed around the BC-AD transition. Pb metallurgy was not yet developed in South America, therefore this shift may be an early manifestation of metallurgy of unknown origin or a shift in Pb isotopes from a natural origin. An eruption from Mt. Burney with a large associated age range (358 BC-201 AD) has been reported in various Patagonian sites [START_REF] Stern | Holocene tephrochronology record of large explosive eruptions in the southernmost Patagonian Andes[END_REF][START_REF] Stern | The Holocene tephrochronology of southernmost Patagonia and Tierra del Fuego[END_REF] and could explain the shift observed in Karukinka. While volcanic eruptions can shift isotopic ratios [START_REF] Hinkley | Lead (Pb) in old Antarctic ice: some from dust, some from other sources[END_REF], we found no volcanic material in this sample. Therefore, we interpret this shift as of an unknown anthropogenic origin, or caused by trace levels of Pb contained in the Cu ores that were processed by the Nazca populations at that time.

Civilizations such as the Chavin and Nazca are mainly known to have worked textiles and ceramics but they also processed copper [START_REF] Palacios | Metalurgia Prehispa ´nica de Sudame ´rica[END_REF][START_REF] Lleras | Una revisio ´n crı ´tica de las evidencias sobre metalurgia temprana en Surame ´rica[END_REF]. At the BC-AD transition, the Atacama Desert was the cradle of a new Cu metallurgical technique in the Ramaditas area, northern Chile, with wind being directed into partly opened furnaces to increase the temperature and melt the ores [START_REF] Graffam | Ancient Metallurgy in the Atacama: Evidence for Copper Smelting during Chile's Early Ceramic Period[END_REF]. The subsequent increase in metal production, as well as to a certain extent Chavin and Nazca metallurgical activities, also emitted metals into the atmosphere. Increases in Cu/La UCC and Sb/ La UCC between 80 BC and 200 AD in Karukinka and the modern-time back-trajectories indicate that some of these metals may have been transported south as far as Tierra del Fuego. Between 500 and 1000 AD, the Tiwanaku civilization spread through and dominated the South-Central Andes, the Titicaca Basin and northern Chile [START_REF] Augustyniak | Dating the Tiwanaku State[END_REF] and most likely increased the metallurgical activities in these areas [START_REF] Abbott | Intensive Pre-Incan Metallurgy Recorder by Lake Sediments from the Bolivian Andes[END_REF]. This constitutes a turning point in South American metallurgy, with the development of smelting techniques (wax moulds and tin-copper alloys) as recorded by a significant increase in metal objects and artefacts at this time in northern Chile, particularly in San Pedro de Atacama [START_REF] Hinkley | Lead (Pb) in old Antarctic ice: some from dust, some from other sources[END_REF]. Between 650 and 850 AD, metallurgical activities also developed in northwest Argentina [START_REF] Lleras | Una revisio ´n crı ´tica de las evidencias sobre metalurgia temprana en Surame ´rica[END_REF][START_REF] Scattolin | Actividades minero metalu ´rgicas prehispa ´nicas en el noroeste argentino.nuevas evidencias y su significacio ´n[END_REF]. This boom in Cu metallurgy is shown by a drastic increase in the Cu/La UCC and Sb/La UCC ratios in Karukinka, starting around 550 AD and culminating at 1100 AD. Interestingly, an increase in Cu/La UCC during this period was reported in an ice core collected in Bolivia [START_REF] Hong | Atmospheric heavy metals in tropical South America during the past 22 000 years recorded in a high altitude ice core from Sajama, Bolivia[END_REF], but the resolution was too low to make a conclusive interpretation.

Copper production intensified under the control of the Incas (1480-1532 AD) [START_REF] Zori | Copper production in late prehispanic northern Chile[END_REF]. During the Inca domination and during the Hispanic colonization, northwest Argentina and northern Chile remained important copper production areas [START_REF] Scattolin | Actividades minero metalu ´rgicas prehispa ´nicas en el noroeste argentino.nuevas evidencias y su significacio ´n[END_REF][START_REF] Gonza ´lez | A sangre y fuego. Nuevos datos sobre la metalurgia Aguada[END_REF]. The Karukinka peat record highlights the period of Inca copper metallurgy with an increase of Cu/La UCC and Sb/La UCC starting at 1400 AD and persisting until ca. 1600 AD. At the same time, the mining of silver in the Andes triggered a substitution of copper to silver smelting from ca. 1450-1533 AD (i.e. during the Inca Period) around Peru, the Titicaca basin and Potosi, Bolivia [START_REF] Abbott | Intensive Pre-Incan Metallurgy Recorder by Lake Sediments from the Bolivian Andes[END_REF]. In order to reconstruct the Ag mining history, we report Pb and Sn data as these metals have been shown to be associated with Ag ores, for example in Potosi [START_REF] Abbott | Intensive Pre-Incan Metallurgy Recorder by Lake Sediments from the Bolivian Andes[END_REF]. An increase in Pb/La UCC and Sn/La UCC from ca. 1400 to ca. 1800 AD indicates that Pb and Sn were dispersed into the atmosphere and transported to SSA during the silver mining era. This is consistent with similar increases in Pb and Sn in NSA lake sediments [START_REF] Abbott | Intensive Pre-Incan Metallurgy Recorder by Lake Sediments from the Bolivian Andes[END_REF][START_REF] Gonza ´lez | A sangre y fuego. Nuevos datos sobre la metalurgia Aguada[END_REF] as well as the Inca and European Ag production, especially at the famous Potosi mine. The increase in Ag and Pb smelting is also clearly marked in Karukinka by a shift in the 206 Pb/ 207 Pb ratio towards values of ca.

1.19, corresponding to isotopic signatures found in Bolivian and Peruvian ores [START_REF] Cooke | A Millennium of Metallurgy Recorded by Lake Sediments from Morococha, Peruvian Andes[END_REF][START_REF] Desaulty | Isotopic Ag-Cu-Pb record of silver circulation through16 th -18 th century Spain[END_REF].

Although the most recent section of the Karukinka core is not as well chronologically constrained as the pre-Hispanic section, we can identify excursions in the metals/La UCC ratios and Pb isotopic profiles, which match historical data from the 18 th to 20 th century. We therefore present a tentative interpretation for this section of the profile. The early 18 th century sees a depletion in Ag resources and a transition to Sn mining in the Potosi area [START_REF] Abbott | Intensive Pre-Incan Metallurgy Recorder by Lake Sediments from the Bolivian Andes[END_REF], which could explain the increase in Sn/La UCC but also Cu/La UCC in Karukinka, reflecting the increase of bronze alloy production. The 19 th century is marked by a period of unrest associated with the independence of South American countries, especially Chile and Argentina and is associated with a decrease of Cu and Ag production. In Karukinka, the decrease of Cu/La UCC and Sn/ La UCC around 1750-1820 AD, the slight decrease in Pb/La UCC and Sb/La UCC around 1800 AD, and the shift in Pb isotopic values could reflected this period of unrest. Copper and Pb production remained low in South America until 1920-1930 AD [START_REF] Hong | Atmospheric heavy metals in tropical South America during the past 22 000 years recorded in a high altitude ice core from Sajama, Bolivia[END_REF]. The Fuegian gold rush in 1903-1908 AD [START_REF] Martinic | La mineria aurifera en la Region Austral Americana (1869-1950)[END_REF] and two phases of coal exploitation -during the first half of the 19 th century and from 1980 onwards [START_REF] Martinic | La mineria del carbon en Magallanes entre 1868-2003[END_REF] -lead to the first emissions of local anthropogenic metal particles in Tierra del Fuego. These events seem to be recorded in the Pb/La UCC profile, which increased drastically around 1900 AD. This regional 'industrial revolution' persisted into the 20 th century and saw the introduction of leaded gasoline as well as the intensification of lead metallurgy until 1980 [START_REF] Hong | Atmospheric heavy metals in tropical South America during the past 22 000 years recorded in a high altitude ice core from Sajama, Bolivia[END_REF]. This would be recorded by an increase in the Pb/La UCC at the beginning of the second half of the 20 th century to a maximum around 1970-1980 and by a shift in the Pb isotopic signatures towards less radiogenic values. The last period of atmospheric metal deposition recorded in Karukinka could be the introduction of unleaded gasoline in the late 1980's [START_REF] Onursal | Contaminacio ´n atmosfe ´rica por vehı ´culos automotores. Experiencias recogidas en siete centros urbanos de Ame ´rica latina[END_REF], which is seen by a drastic decrease in Pb/La UCC as well as a shift in Pb isotopes towards more radiogenic compositions, while the Cu/La UCC does not decrease, as Cu production is still flourishing in South America [START_REF] Hong | Atmospheric heavy metals in tropical South America during the past 22 000 years recorded in a high altitude ice core from Sajama, Bolivia[END_REF].

Conclusions

The Karukinka peat bog sequence records past metallurgical activities in South America, as the area was isolated from any local anthropogenic sources until the 19 th century. Based upon the excellent agreement between variations in UCC-normalized Cu, Pb, Sb and Sn to La ratios, modern-time back-trajectories and historical data, we conclude that Cu metallurgy predates archaeological evidence and that NSA atmospheric emissions from pre-Hispanic civilizations through to the Industrial Revolution are recorded in SSA.

Figure 1 .

 1 Figure 1. Upper Panel: A. Main polymetallic ores in South America (grey shaded area) and the extent of pre-Colombian civilisations and cultures discussed in the text (dark blue lines: 1-Chavin, 2-Nazca, 3-Tiwanaku; black square: 4-Ramaditas site black line: 5-Incas). B. Location of Karukinka bog in Tierra del Fuego. Lower Panel: back-trajectories for years when significant parcels of air masses travelled from NSA, specifically from the Inca, Tiwanaku and Nazca territories as well as from the Ramaditas site. Only the three most representative years are shown (see figure S1 for all years from 1948 to 2012). Other years(1997, 1999, 2000, 2008) are displayed in Fig.3. doi:10.1371/journal.pone.0111315.g001

Figure 2 .

 2 Figure 2. Age-depth model for the Karukinka peat core. Maximum likelihood ages are expressed in years Anno Domini (AD) and Before Christ (BC). The Hudson and Burney tephras are also used as chronological markers. doi:10.1371/journal.pone.0111315.g002

Figure 3 .

 3 Figure 3. Back trajectory frequency corresponding to calendar years where air parcels are partially coming from NSA. The year 1968 is given as a comparison to show a year where no air back trajectory is coming from NSA (see figure S1 for all years from 1948 to 2012). doi:10.1371/journal.pone.0111315.g003

Figure 4 .Figure 5 .

 45 Figure 4. Detailed macrofossil age and depth profile for the Karukinka peat profile. Mineral grains are from the Hudson and Burney tephras (red lines). doi:10.1371/journal.pone.0111315.g004

Figure 6 .

 6 Figure 6. UCC-normalized Cu, Sb, Pb and Sn vs. La ratios [37] and 206 Pb/ 207 Pb isotopic ratios in the Karukinka peat core during the last 4200 yrs (ombrotrophic part of the core). Maximum likelihood ages are expressed in years Anno Domini (AD) and Before Christ (BC). The age scale is cut at 1750 AD and the Sb/La UCC scale changes at 1750AD. 1. Onset of Cu metallurgy, 2. Chavin, 3. Nazca and Ramaditas, 4. Tiwanaku, 5. Inca (transition from Cu to Ag mining), 6. European Colonization, 7. Gold rush and bronze industry, 8. Leaded gasoline period, 9. Unleaded gasoline period (see text for details). doi:10.1371/journal.pone.0111315.g006

Table 2 .

 2 Cu, Pb and La determined by Q-ICP-MS compared to certified values of SRM1947, SRM1515 and GBW-07063 (mean 6 standard deviation, mg.kg 21 ).

	Measured	Certified

Table 3 .

 3 Cu, Sn, Sb, Pb and La concentrations as well as206 Pb/ 207 Pb isotopic ratios in the Karukinka core.

	Depth	Density	Calendar Age	Cu	Sn	Sb	Pb	La	206 Pb/ 207 Pb	2s
	cm	(g.cm 23 )	(BC/AD)	(mg kg 21 )	(mg kg 21 )	(mg kg 21 )	(mg kg 21 )	(mg kg 21 )
	1.3	0.04	2010	1.59	0.04	0.02	0.14	0.14
	8.7	0.05	1998	0.99	0.02	0.02	0.14	0.07
	15.9	0.05	1985	0.84	0.03	0.02	0.28	0.25	1.181	0.001
	22.1	0.05	1972	0.84	0.02	0.02	0.27	0.08
	28.0	0.10	1957	0.55	0.04	0.01	0.31	0.08
	30.8	0.14	1949	0.84	0.02	0.01	0.48	0.55	1.182	0.003
	34.8	0.19	1936	0.89	0.02	0.01	0.20	0.20	1.179	0.0001
	36.0	0.15	1932	0.40	0.03	0.01	0.21	0.07
	42.0	0.15	1910	0.46	0.01	0.01	0.19	0.06
	50.4	0.07	1874	0.85	0.01	0.01	0.12	0.15	1.195	0.0003
	54.3	0.14	1853	1.14	0.06	0.02	0.23	0.30
	58.0	0.10	1831	0.81	0.01	0.01	0.15	0.15	1.185	0.004
	62.0	0.09	1805	1.20	0.02	0.01	0.17	0.16
	66.0	0.06	1776	1.91	0.01	0.01	0.11	0.11
	72.6	0.06	1722	0.69	0.13	0.04	0.10	0.10	1.191	0.001
	79.9	0.08	1653	0.99	0.01	0.01	0.15	0.20	1.193	0.001
	84.1	0.09	1607	1.39	0.03	0.01	0.16	0.20	1.188	0.003
	92.1	0.08	1509	2.24	0.02	0.01	0.17	0.33	1.190	0.002
	96.1	0.10	1454	2.38	0.02	0.01	0.13	0.46	1.198	0.004
	102.7	0.06	1355	2.12	0.03	0.02	0.14	0.63
	109.2	0.07	1244	2.48	0.03	0.02	0.22	0.63
	117.1	0.05	1093	3.63	0.02	0.02	0.12	0.54
	122.2	0.08	988	4.56	0.07	0.02	0.20	0.87	1.194	0.002
	127.7	0.09	871	3.43	0.05	0.02	0.19	0.64
	135.3	0.09	713	4.03	0.08	0.05	0.19	1.14
	142.7	0.08	571	3.62	0.04	0.01	0.37	3.13	1.193	0.001
	146.9	0.11	499	6.71	0.06	0.02	0.85	7.30	1.197	0.0002
	158.0	0.13	339	15.28	0.15	0.07	1.59	15.19	1.198	0.001
	161.8	0.12	293	21.53	0.11	0.10	2.00	14.76
	164.4	0.11	264	21.55	0.15	0.12	1.90	14.72
	168.0	0.10	228	17.10	0.12	0.08	1.56	11.37
	171.8	0.11	194	24.03	0.13	0.08	1.36	10.32	1.197	0.002
	175.4	0.08	165	22.19	0.06	0.07	1.28	9.35
	179.0	0.07	140	24.72	0.05	0.08	1.23	7.62

Table 3 .

 3 Cont.

	Depth	Density	Calendar Age	Cu	Sn	Sb	Pb	La	206 Pb/ 207 Pb	2s
	cm	(g.cm 23 )	(BC/AD)	(mg kg 21 )	(mg kg 21 )	(mg kg 21 )	(mg kg 21 )	(mg kg 21 )
	182.8	0.09	117	10.69	0.07	0.06	0.80	6.26
	190.0	0.08	83	18.05	0.05	0.06	0.91	5.37
	195.4	0.07	64	14.49	0.05	0.05	0.93	5.73	1.190	0.001
	201.3	0.06	46	12.46	0.05	0.07	0.68	4.81
	207.8	0.08	29	12.91	0.05	0.05	0.87	6.66
	215.2	0.07	6	9.1	0.1	0.1	1.4	5.4
	221.9	0.08	221	13.86	0.07	0.04	1.39	6.85
	227.4	0.06	250	21.63	0.26	0.07	1.63	7.68
	231.5	0.06	277	20.04	0.09	0.06	1.33	6.21	1.196	0.002
	237.5	0.16	2128	24.53	0.09	0.06	1.50	9.00	1.198	0.003
	244.3	0.13	2204	4.49	0.04	0.03	0.44	5.64
	250.2	0.09	2285	9.26	0.04	0.04	0.80	6.35
	253.1	0.14	2330	4.68	0.03	0.03	0.40	4.78	1.199	0.001
	258.7	0.12	2424	5.30	0.04	0.03	0.41	4.36
	264.8	0.10	2535	5.19	0.03	0.03	0.33	2.96
	270.2	0.10	2639	5.78	0.05	0.03	0.41	3.79	1.196	0.001
	274.0	0.09	2714	5.64	0.04	0.03	0.39	3.98
	279.7	0.10	2828	3.0	0.1	0.0	0.4	2.9
	285.2	0.09	2937	2.35	0.02	0.02	0.32	2.03	1.198	0.001
	289.3	0.09	21017	2.52	0.03	0.02	0.31	1.82
	293.4	0.14	21096	3.49	0.08	0.04	0.25	2.04
	300.9	0.10	21239	5.88	0.03	0.04	0.41	2.15	1.196	0.003
	304.9	0.09	21316	6.28	0.04	0.03	0.21	1.41
	308.6	0.08	21387	4.44	0.02	0.02	0.26	1.45
	312.2	0.13	21457	4.63	0.07	0.06	0.58	2.87	1.195	0.002
	320.5	0.10	21623	6.46	0.04	0.04	0.27	1.38
	325.6	0.13	21729	9.90	0.07	0.06	0.54	2.47
	330.8	0.11	21842	6.31	0.06	0.04	0.54	2.78	1.199	0.002
	334.8	0.12	21932	6.25	0.06	0.05	0.68	3.29
	339.8	0.12	22050	6.60	0.04	0.05	0.48	2.79
	343.9	0.12	22152	6.62	0.09	0.04	0.85	3.56

Sample depths and maximum likelihood calendar ages (positive values are ages Anno Domini, negative values are ages Before Christ) are also reported. doi:10.1371/journal.pone.0111315.t003

Acknowledgments

This research is supported by a Young Researcher Grant of the Agence Nationale de la Recherche (ANR) to F. De Vleeschouwer (Project ANR-2011-JS56-006-01 ''PARAD''). We thank Aure ´lie Lanzanova from Geoscience-Environnement-Toulouse (France), Maxime Enrico (EcoLab, Toulouse, France) and Adrien Claustres (EcoLab, Toulouse, France) for their help in the careful measurements of trace metals and lead isotopes. Barbara Saavedra, Susan Arismendi, Ricardo Muza and the Karukinka Park rangers from the Wildlife Conservation Society (WCS) are warmly thanked for facilitating access to the Karukinka Park. Se ´bastien Bertrand and Zakaria Ghazoui from RCMG (UGent, Belgium) and Jean-Yves De Vleeschouwer (ULg, Belgium) are thanked for their help in the field. Nelson Bahamonde (INIA, Punta Arenas, Chile) and Ernesto Teneb (UMag, Punta Arenas, Chile) are thanked for their logistical support in preparing the field mission. The authors gratefully acknowledge the NOAA Air Resources Laboratory (ARL) for the provision of the HYSPLIT transport and dispersion model and READY website (http://www.ready. noaa.gov) used in this publication. Supporting data are available in Figure S1. We thank one anonymous reviewers and Xabier Pontevedra Pombal for their valuable comments on an earlier version of this manuscript.

Results and Interpretation

Determination of the bog trophic status

Karukinka bog is located on a small hill beside the main Karukinka Valley. It consists of an open area where the surface vegetation is dominated by Sphagnum magellanicum mosses. The 5-cm resolution macrofossil diagram records a poor-fen stage (zone KAR12-PB01B-A from the base of the core (450 cm) to approximately 415 cm depth (4555-4215 yrs cal BC) and is composed of graminoid remains (Monocots undifferentiated). The presence of abundant macrofossil charcoal fragments indicates that local fires may have occurred during this period. The fen

Supporting Information