Christophe Sabot
email: sabot@math.univ-lyon1.fr

Laurent Tournier
email: tournier@math.univ-lyon1.fr

Reversed Dirichlet environment and Directional transience of random walks in Dirichlet random environment

. In a first part of this paper, we also give a probabilistic proof of this property as an alternative to the change of variable computation used in [Sa09].

Introduction

Random walks in a multidimensional random environment have attracted considerable interest in the last few years. Unlike the one-dimensional setting, this model remains however rather little understood. Recent advances focused especially in two directions: small perturbations of a deterministic environment (cf. for instance [START_REF] Bolthausen | Multiscale analysis of exit distributions for random walks in random environments[END_REF]) and ballisticity (cf. [START_REF] Sznitman | Topics in random walk in random environment[END_REF] for a survey), but few conditions are completely explicit or known to be sharp.

The interest for random Dirichlet environment stems from the desire to take advantage of the features of a specific multidimensional environment distribution that make a few computations explicitly possible, and hopefully provide through them a better intuition for the general situation. The choice of Dirichlet distribution appears as a natural one considering its close relationship with linearly reinforced random walk on oriented edges (cf. [START_REF] Pemantle | Phase transition in reinforced random walk and RWRE on trees[END_REF] and [START_REF] Enriquez | Edge oriented reinforced random walks and RWRE[END_REF]). The opportunity of this choice was further confirmed by the derivation of a ballisticity criterion by N.Enriquez and C.Sabot [START_REF] Enriquez | Random walks in a Dirichlet environment[END_REF] (later improved by L.Tournier [START_REF] Tournier | Integrability of exit times and ballisticity for random walks in Dirichlet environment[END_REF]), and more especially by the recent proof by C.Sabot [START_REF] Sabot | Random walks in random Dirichlet environment are transient in dimension d ≥ 3[END_REF] that random walks in Dirichlet environment on Z d are transient when d ≥ 3.

The proof of the ballisticity criterion in [START_REF] Enriquez | Edge oriented reinforced random walks and RWRE[END_REF] relies on an integration by part formula for Dirichlet distribution. This formula provides algebraic relations involving the Green function that allow to show that Kalikow's criterion applies. As for the proof of [START_REF] Sabot | Random walks in random Dirichlet environment are transient in dimension d ≥ 3[END_REF], one of its key tools is the striking property (Lemma 1 of [START_REF] Sabot | Random walks in random Dirichlet environment are transient in dimension d ≥ 3[END_REF]) that, provided the edge weights have null divergence, a reversed Dirichlet environment (on a finite graph) still is a Dirichlet environment.

In this paper we prove with this same latter property that random walks in Dirichlet environment on Z d (d ≥ 1) are directionally transient as soon as the weights are nonsymmetric. More precisely, under this condition, our result (Theorem 1) states that directional transience happens with positive probability: for some direction ℓ, P o (X n • ℓ → +∞) > 0. Combined with S.Kalikow's 0-1 law from [START_REF] Kalikow | Generalized random walk in a random environment[END_REF], this proves that, almost surely, |X n • ℓ| → +∞. Furthermore, in dimension 2, the 0-1 law proved by M.Zerner and F.Merkl (cf. [START_REF] Zerner | A zero-one law for planar random walks in random environment[END_REF]) enables to conclude that, almost surely, X n • ℓ → +∞.

The above mentioned property of the reversed Dirichlet environment was derived in [START_REF] Sabot | Random walks in random Dirichlet environment are transient in dimension d ≥ 3[END_REF] by means of a complicated change of variable. In Section 2 of the present paper, we provide an alternative probabilistic proof of this important property.

Definitions and statement of the results

Let us precise the setting in this paper. Let G = (V, E) be a directed graph, i.e. V is the set of vertices, and E ⊂ V × V is the set of edges. If e = (x, y) is an edge, we respectively denote by e = x and e = y its tail and head. We suppose that each vertex has finite degree, i.e. that finitely many edges exit any vertex. G is said to be strongly connected if for any pair of vertices (x, y) there is a directed path from x to y in G. The set of environments on G is the set

∆ = {(p e ) e∈E ∈ (0, 1) E | for all x ∈ V, e∈E,e=x p e = 1}.
Let (p e ) e∈E be an environment. If e = (x, y) is an edge, we shall write p e = p(x, y). Note that p extends naturally to a measure on the set of paths:

if γ = (x 1 , x 2 , . . . , x n ) ∈ V n is a path in G, i.e. if (x i , x i+1 ) ∈ E for i = 1, . . . , n -1, then we let p({γ}) = p(γ) = p(x 1 , x 2 )p(x 2 , x 3 ) • • • p(x n-1 , x n ).
To any environment p = (p e ) e∈E and any vertex x 0 we associate the distribution P (p)

x 0 of the Markov chain (X n ) n≥0 on V starting at x 0 with transition probabilities given by p. For every path γ = (x 0 , x 1 , . . . , x n ) in G starting at x 0 , we have

P (p) x 0 (X 0 = x 0 , . . . , X n = x n ) = p(γ).
When necessary, we shall specify by a superscript P (p),G x 0 which graph we consider.

Let (α e ) e∈E be positive weights on the edges. For any vertex x, we let

α x = e∈E,e=x α e
be the sum of the weights of the edges exiting from x. The Dirichlet distribution with parameter (α e ) e∈E is the law P (α) of the random variable p = (p e ) e∈E in ∆ such that the transition probabilities (p e ) e∈E,e=x at each site x are independent Dirichlet random variables with parameter (α e ) e∈E,e=x . Thus,

dP (α) (p) = x∈V Γ(α x ) e∈E Γ(α e ) e∈E p αe-1 e e∈ e E dp e ,
where E is obtained from E by removing arbitrarily, for each vertex x, one edge with origin x (the distribution does not depend on this choice). The corresponding expectation will be denoted by E (α) . We may thus consider the probability measure

E (α) [P (p)
x 0 (•)] on random walks on G.

Time reversal Let us suppose that G is finite and strongly connected. Let Ǧ = (V, Ě) be the graph obtained from G by reversing all the edges, i.e. by swapping the head and tail of the edges. To any path γ = (x 1 , x 2 , . . . , x n ) in G we can associate the reversed path γ = (x n , . . . , x 2 , x 1 ) in Ǧ.

For any environment p = (p e ) e∈E , the Markov chain with transition probabilities given by p is irreducible, hence (G being finite) we may define its unique invariant probability (π x ) x∈V on V and the environment p = (p e ) e∈ Ě obtained by time reversal: for every e ∈ E,

pě = π e π e p e .
For any family of weights α = (α e ) e∈E , we define the reversed weights α = (α e ) e∈ Ě simply by αě = α e for any edge e ∈ E, and the divergence of α is given, for x ∈ V , by div(α)(x) = α xαx .

The following proposition is Lemma 1 from [START_REF] Sabot | Random walks in random Dirichlet environment are transient in dimension d ≥ 3[END_REF].

Proposition 1. -Suppose that the weights α = (α e ) e∈E have null divergence, i.e. ∀x ∈ V, div(α)(x) = 0 (or equivalently αx = α x ).

Then, under P (α) , p is a Dirichlet environment on Ǧ with parameters α = (α e ) e∈ Ě .

In section 2, we give a neat probabilistic proof of this proposition. It also lies at the core of the proof of the directional transience.

Directional transience on Z d We consider the case V = Z d (d ≥ 1) with edges to the nearest neighbours. An i.i.d. Dirichlet distribution on G is determined by a 2d-vector α = (α 1 , β 1 , . . . , α d , β d ) of positive weights. We define indeed the parameters α on E by α (x,x+e i ) = α i and α (x,x-e i ) = β i for any vertex x and index i ∈ {1, . . . , d}. In this context, we let

P o = E (α) [P (p) o (•)] be the so-called annealed law. Theorem 1. -Assume α 1 > β 1 . Then P o (X n • e 1 → n +∞) > 0.
The 0-1 law proved by S.Kalikow in Lemma 1.1 of [START_REF] Kalikow | Generalized random walk in a random environment[END_REF] and generalized to the non-uniformly elliptic case by M.Zerner and F.Merkl in Proposition 3 of [START_REF] Zerner | A zero-one law for planar random walks in random environment[END_REF], along with the 0-1 law proved by M.Zerner and F.Merkl in [START_REF] Zerner | A zero-one law for planar random walks in random environment[END_REF] (cf. also [START_REF] Zerner | The zero-one law for planar random walks in i.i.d. random environments revisited[END_REF]) for the two-dimensional random walk allow to turn this theorem into almost-sure results:

Corollary. -Assume α 1 > β 1 . (i) If d ≤ 2, then P o -almost surely, X n • e 1 → n +∞. (ii) If d ≥ 3, then P o -almost surely, |X n • e 1 | → n +∞.
Remark This theorem provides examples of non-ballistic random walks that are transient in a direction. Proposition 12 of [START_REF] Tournier | Integrability of exit times and ballisticity for random walks in Dirichlet environment[END_REF] shows indeed that the condition 2 j (α j + β j )α iβ i ≤ 1 for a given i implies P o -almost surely Xn n → n 0, because of the existence of "finite traps", so that any choice of small unbalanced weights (less than 1 4d for instance) is such an example. Another simple example where such a behaviour happens was communicated to us by A.Fribergh as well. It is however believed (cf. [START_REF] Sznitman | Topics in random walk in random environment[END_REF] p.227) that transience in a direction implies ballisticity if a uniform ellipticity assumption is made, i.e. if there exists κ > 0 such that, for every edge e, P-almost surely p e ≥ κ.

Under the hypothesis of Theorem 1 we conjecture that, for any d ≥ 1,

P o (X n • e 1 → n +∞) = 1,
and that the following identity is true:

P o (D = +∞) = 1 - β 1 α 1 ,
where D = inf{n ≥ 0|X n • e 1 < 0}. In this paper, only the "≥" inequality is proved. o (•)] characterizes the distribution P of the environment, as soon as we assume that P is a distribution on Ω = x∈V P x such that P-almost every environment p is transitive recurrent. This results from considering the sample environment p (n) at time n:

for every e = (x, y) ∈ E, p (n) e = |{0 ≤ i < n|(X i , X i+1 ) = e}| |{0 ≤ i < n|X i = x}| .
From the recurrence assumption and the law of large numbers we deduce that p (n) makes sense for large n and that, as n → ∞, it converges almost-surely under P o (•)], to a random variable p with distribution P. This way we can recover the distribution of the environment from that of the annealed random walk.

We are thus reduced to proving that, for any finite path γ in G,

E (α) [p(γ)] = E (α) [p(γ)]. (1)
The only specific property of Dirichlet distribution to be used in the proof is the following "cycle reversal" property:

Lemma 2. -For any cycle σ = (x 1 , x 2 , . . . , x n (= x 1 )) in G, E (α) [p(σ)] = E (α) [p(σ)], i.e. E (α) [p(x 1 , x 2 ) • • • p(x n-1 , x n )] = E (α) [p(x n , x n-1 ) • • • p(x 2 , x 1 )].
Proof. Remembering that the annealed random walk in Dirichlet environment is an oriented edge linearly reinforced random walk where the initial weights on the edges are the parameters of the Dirichlet distribution (cf. [START_REF] Enriquez | Edge oriented reinforced random walks and RWRE[END_REF]), the left-hand side of the previous equality writes:

E (α) [p(σ)] = e∈E α e (α e + 1) • • • (α e + n e (σ) -1) x∈V α x (α x + 1) • • • (α x + n x (σ) -1)
,

where n e (σ) (resp. n x (σ)) is the number of crossings of the oriented edge e (resp. the number of visits of the vertex x) in the path σ. The cyclicity gives n e (σ) = n ě(σ) and n x (σ) = n x (σ) for all e ∈ E, x ∈ V . Furthermore, by assumption αx = α x for every vertex x, and by definition α e = αě for every edge e. This shows that the previous product matches the similar product with α and σ instead of α and σ, hence the lemma.

If σ = (x 1 , x 2 , . . . , x n ) is a cycle in G (thus x n = x 1
), the definition of the reversed environment gives:

p(σ) = p(x 1 , x 2 )p(x 2 , x 3 ) • • • p(x n-1 , x n ) = π x 2 π x 1 p(x 2 , x 1 ) π x 3 π x 2 p(x 3 , x 2 ) • • • π xn π x n-1 p(x n , x n-1 ) = p(x 2 , x 1 )p(x 3 , x 2 ) • • • p(x n , x n-1 ) = p(σ) (2)
Therefore, the previous lemma gives:

E (α) [p(σ)] = E (α) [p(σ)]
, which is Equation (1) for cycles.

Consider now a non-cycling path γ in G: γ = (x = x 1 , . . . , x n = y) with x = y. The same computation as above shows that p(γ) = π x π y p(γ).

It is a well-known property of Markov chains that πx πy = E (p)

y [N y x ]
where N y x is the number of visits to x before the next visit of y: N y x = Hy i=0 1 {X i =x} , with H y = inf{n ≥ 1 | X n = y}. It can therefore be written, in a hopefully self-explanatory schematical way:

π x π y = ∞ k=0 P (p) y (N y x > k) = ∞ k=0 p ( = x, y = x, y x y = x, y y ≥ k times ) = ∞ k=0 p ( k times = x, y = x, y x y ) ,
where the subscripts " = x, y" mean that the paths sketched by the dashed arrows avoid x and y, and "(≥)k times" refers to the number of loops. Using the notation { x y γ } for the event where the walk follows the path γ (which goes from x to y), the Markov property at the time of k-th visit of x gives:

p(γ) = ∞ k=0 p ( k times = x, y = x, y x y ) p(γ) = ∞ k=0 p ( k times = x, y = x, y x γ y y) .
The paths in the probability on the right-hand side are cycles. Taking the expectation under P (α) of both sides, we can use the Lemma to reverse them. We get:

E (α) [p(γ)] = ∞ k=0 E (α) [p ( k times = x, y = x, y x γ y y )].
It only remains to notice that the summation over k ∈ N allows to drop the condition on the path after it has followed γ:

E (α) [p(γ)] = E (α) [p(γ)].
This concludes the proof of the proposition.

Directional transience. Proof of Theorem 1

For any integer M , let us define the stopping times:

T M = inf{n ≥ 0|X n • e 1 ≥ M } T M = inf{n ≥ 0|X n • e 1 ≤ M },
and in particular D = T -1 . In addition, for any vertex x, we define

H x = inf{n ≥ 1 | X n = x}.
Let N, L ∈ N * . Consider the finite and infinite "cylinders"

C N,L = {0, . . . , L -1} × (Z N ) d-1 ⊂ C N = Z × (Z N ) d-1 ,
endowed with an i.i.d. Dirichlet environment of parameter α. We are interested in the probability that the random walk starting at o = (0, 0, . . . , 0) exits C N,L to the right. Specifically, we shall prove: To that aim, we introduce the finite weighted graph G = G L,N defined as follows (cf. also figure 1). The set of vertices of G is V = C N,L+1 ∪{δ}. Let us denote by L = {0}×(Z N ) d-1 and R = {L}×(Z N ) d-1 the left and right ends of the cylinder. The edges of G inside C N,L+1 go between nearest vertices for usual distance. In addition, "exiting" edges are introduced from the vertices in L and R into δ, and "entering" edges are introduced from δ into the vertices in L. The edges in C N,L+1 (including exiting edges) are endowed with weights naturally given by α except that the edges exiting from R toward δ have weight α 1β 1 (> 0) instead of α 1 . At last, the weight on the entering edges is α 1 . These weights, let us denote them by α ′ , are chosen in such a way that div α ′ (x) = 0 for every vertex x (including δ, necessarily). We are thus in position to apply Proposition 1.

E (α) P (p),C N o (T L < T -1 ) ≥ 1 - β 1 α 1 . (3)
Using Equation (2) to reverse all the cycles in G that start at δ and get back to δ from R (for the first time), we get the following equality:

P (p),G δ (X H δ -1 ∈ R) = P (p), Ǧ δ (X 1 ∈ R) .
In the reversed graph Ǧ, there are N edges that exit δ to L (with reversed weight β 1 ), and N edges that exit δ to R (with reversed weight α 1β 1 ). Combined with the Dirichlet distribution of p under P given by Proposition 1 (notice that in fact we only reverse cycles, hence Lemma 2 suffices), the previous equality gives:

E (α ′ ) P (p),G δ (X H δ -1 ∈ R) = E (α ′ )   e∈ Ě,e=δ p(e)   = N (α 1 -β 1 ) N (α 1 -β 1 ) + N α 1 = 1 - β 1 α 1 . (4)
In the second equality we used the fact that under P (α ′ ) the marginal variable p e follows a beta distribution with parameters (α ′ e , ( e ′ ∈ Ě,e ′ =δ α′ e ′ ) -α′ e ), and that the expected value of a Beta random variable with parameters α and β is α α+β . Alternatively, this equality follows also from the distribution of the first step of an oriented edge reinforced random walk with initial weights given by α ′ .

Let us show how this equality implies the lower bound (3). First, because of the symmetry of the weighted graph, the distribution of X 1 under E

(α ′ ) [P (p),G δ (•)] is uniform on L, hence, since o ∈ L: E (α ′ ) P (p),G δ (X H δ -1 ∈ R) = E (α ′ ) P (p),G o (X H δ -1 ∈ R) .
(5)

Then we relate the exit probability out of C N,L endowed with parameters α with the exit probability out of C N,L+1 endowed with modified parameters α ′ . These weights α and α ′ coincide in C N,L ; furthermore, a random walk on the cylinder C N , from o, reaches the abcissa L before the abcissa L + 1, so that:

E (α) P (p),C N o (T L < T -1 ) ≥ E (α ′ ) P (p),C N o (T L+1 < T -1 ) = E (α ′ ) P (p),G o (X H δ -1 ∈ R) . (6)
The equalities (4) and (5), along with (6), finally give (3).

To conclude the proof of the theorem, let us first take the limit as N → ∞. Due to the ellipticity of Dirichlet environments, Lemma 4 of [START_REF] Zerner | A zero-one law for planar random walks in random environment[END_REF] shows that the random walk can't stay forever inside an infinite slab:

0 = E (α) [P (p) o (T L = T -1 = ∞)] = lim N →∞ E (α) [P (p) o (T L ∧ T -1 > T ⊥ N )]
where T ⊥ N = inf{n ≥ 0| |X n -(X n • e 1 )e 1 | > N } is the first time when the random walk is at distance greater than N from the axis Re 1 . Using this with the stopping time T ⊥ N/2 , we can switch from the cylinder C N,L to an infinite slab:

β 1 α 1 > 0.
Moreover, Lemma 4 from [START_REF] Zerner | A zero-one law for planar random walks in random environment[END_REF] shows more precisely that if a slab is visited infinitely often, then both half-spaces next to it are visited infinitely often as well. The event {D = ∞} ∩ {lim inf n X n • e 1 < M } has therefore null P o -probability for any M > 0, hence finally:

P o (X n • e 1 → n +∞) ≥ lim M →∞ P o (D = ∞, lim inf n X n • e 1 ≥ M ) = P o (D = ∞) > 0.
This concludes the proof of the theorem.

2

  Reversed Dirichlet environment. Proof of Proposition 1 It suffices to prove that, for a given starting vertex o, the annealed distributions E (α) [P (p) o (•)] and E (α) [P (p) o (•)] on walks on Ǧ are the same. Indeed, the annealed distribution E[P (p)

Figure 1 :

 1 Figure 1: Definition of the weighted graph G. The top and bottom rows are identified (the graph is based on a cylinder) and the weights are the same on each line. Both vertices called δ are the same.

E- β 1 α 1 .

 1 (α) [P (p),C N o (T L < T -1 )] = E (α) [P (p),C N o (T L < T -1 , T ⊥ N/2 > T L )] + E (α) [P (p),C N o (T L < T -1 , T ⊥ N/2 ≤ T L )] = E (α) [P (p) o (T L < T -1 , T ⊥ N/2 > T L )] + o N (1) = E (α) [P (p) o (T L < T -1 )] + o N (1) ′ ,hence, with (3),E (α) [P (p) o (T L < T -1 )] ≥ 1The limit when L → ∞ simply involves decreasing events:P o (D = +∞) = E (α) [P (p) o (D = +∞)] = lim L→∞ E (α) [P (p) o (T L < T -1 )] ≥ 1 -

This research was supported by the french ANR Project MEMEMO


Esme Candish

Aminreza Khodabandeh

Marianne Gaborieau

Thomas Rodemann

Robert A Shellie

Andrew A Gooley

Emily F Hilder
email: emily.hilder@unisa.edu.au

Poly(

Keywords: sample preparation, porous polymer monolith, solid phase extraction, grafting, biocompatible, mass spectrometry

come

Introduction

The highly selective and sensitive analysis of drugs and metabolites by mass spectrometry (MS) has emerged as an essential tool in many fields including clinical chemistry, forensic toxicology and pharmaceutical research. Blood, plasma and serum are among the most difficult samples to analyze as they are poorly compatible with MS [1]. Matrix components, including cellular material, proteins and non-volatile lower molecular weight solutes can severely reduce assay sensitivity by suppressing the electrospray ionization (ESI) of the analyte [2]. Furthermore, matrix components will complicate data and foul the instrumentation. The key to achieving a highly sensitive and accurate analysis is to present a purified sample to the MS.

Preparing a sample for analysis is labor intensive and time consuming, often taking over 80% of the total analysis time [3]. Fortunately, rapid workflows can be achieved using miniaturized solid phase extraction (SPE). Hydrophobic and ion-exchange adsorbents are extremely efficient for removal of salts and non-adsorbed matrix compounds but protein rich matrices remain problematic. Proteins exhibit both hydrophobic and ionic interactions resulting in their non-specific adsorption and precipitation on solid surfaces [4]. Non-specific protein adsorption will foul the adsorbent bed, leading to a diminished extraction performance and a reduced assay sensitivity [4,5]. In addition, protein fouling often leads to the blockage of the SPE cartridge, which can result in sample wastage. Therefore, additional labor intensive and time consuming sample pretreatment procedures including centrifugation, protein precipitation and filtration are routinely introduced into the workflow prior to SPE.

Alternatively, the morphology and chemistry of the SPE adsorbent can be engineered to improve biocompatibility by reducing the non-specific adsorption of proteins.

Restricted access materials (RAM) can facilitate the rapid and efficient purification of biological samples; both non-adsorbed low molecular weight compounds and the protein rich matrix can be eliminated using a single adsorbent [6][7][8][9][10]. A diffusion barrier (physical or chemical) limits the accessibility of proteins to the adsorbent's surface. We have recently demonstrated that large surface area polymer monoliths adsorbents based on poly(divinyl benzene) (PDVB) are highly promising for the size selective sample clean-up of small molecules [11]. Inherent micro-(0.2-2 nm) and mesopores (2-50 nm) present a physical barrier, which dictates that larger matrix components pass through the adsorbent bed unretained, thus producing more purified extracts. Unfortunately, the hydrophobic nature of the PDVB remains problematic for protein-rich samples. Scope exists to reduce the non-specific surface adhesion of proteins by introducing surface functionality that act as a chemical barrier [12][13][14].

To achieve the unique pore structure of the PDVB adsorbent it is necessary to fabricate from a high concentration of crosslinking monomer. Therefore, approaches to introduce a chemical barrier are limited to a post-polymerization reaction whereby a layer is grafted to the adsorbent surface [15,16]. It is well established that grafting the monomer poly(ethylene glycol) methacrylate (PEGMA) onto a hydrophobic surface results in a reduction of non-specific proteins binding through a combination of the hydrophilicity and the steric hindrance imparted by the long PEG chains [17,18]. Grafting reactions undertaken thermally often exploit the residual reactive initiators, employed for living/controlled polymerizations, as attachment points for further functionalization [10,19,20]. Unfortunately, this approach often necessitates specialized initiators and the conditions of polymerization can be limited. A simpler protocol involves utilizing the residual vinyl groups (RVGs) of the PDVB that remain unreacted for further functionalization [21][START_REF] Tripp | Reactive & Functional Polymers[END_REF][23]. Gaborieau et al. determined that PDVB particulate can possess 39-43% accessible RVGs that are available for further functionalization [23]. The purpose of the current study was to develop an approach for the PEGMA grafting of PDVB adsorbents using the RVGs. Grafted adsorbents were optimized to reduce non-specific protein interactions while preserving hydrophobic capacity and analyte interactions. Grafting was characterized using scanning electron microscopy (SEM), non-specific protein binding, solid-state nuclear magnetic resonance (NMR) spectroscopy and frontal analysis. The suitability of the resulting adsorbent was demonstrated for the improved miniaturized SPE purification of highly complex, protein rich, biological samples prior to MS.

Experimental section

Chemicals and materials

DVB (containing 80% 1,3-DVB + 1,4-DVB and 20% 1-ethyl-3-vinylbenzene + 1ethyl-4-vinylbenzene), poly(ethylene glycol) methyl ether methacrylate average M n 950 (PEGMA 950 ), poly(ethylene glycol) methacrylate average M n 360 (PEGMA 360 ), poly(ethylene glycol) diacrylate average M n 258 (PEGDA 258 ), anisole, 1-dodecanol, benzophenone (99%), ibuprofen, human serum albumin (HSA) lyophilized powder ≥97%, fluorescamine, ammonium hydroxide and sodium tetraborate were all purchased from Sigma Aldrich (Castle Hill, Australia). The monomer inhibitors were removed using a packed bed of basic alumina. The initiator, 2,2'-azo-bisisobutyronitrile (AIBN), was obtained from MP Biomedicals (Santa Ana, CA, USA) and purified by recrystallization with methanol. The high performance liquid chromatography (HPLC) grade solvents; acetone, acetonitrile, methanol and toluene were all purchased from Sigma Aldrich. The water was purified with a Milli-Q system (Millipore, Bedford, MA, USA). Polyethylene (PE) tubing (1.57 mm i.d.) was obtained from SDR Scientific (Chatswood, Australia). The MEPS cartridge assembly included frits, shank and needle hub components (SGE Analytical Science, Ringwood, Australia).

Sample collection

Blood samples were obtained by finger lancet (Acc-Chek Softclix, Roche Diagostics, Castle Hill, Australia) from a healthy female volunteer and stored in EDTA miniCollect tubes (ThermoScientific, Scoresby, Australia). All blood samples were centrifuged to obtain the plasma fraction. The plasma was diluted with water (20% v/v) and spiked with 50 ng mL -1 ibuprofen.

Instrumentation

An inverted fluorescence microscope (Nikon, Eclipse Ti-U, Japan) was employed to determine protein adsorption with the violet pass excitation (lex at 380-420 nm) and emission (lem at 450 nm) filters (Semrock, Rochester, NY, USA). The microscope was operated with NIS-Elements BR 3.10 software (Melville, NY, USA).The Brunauer-Emmett-Teller (BET) surface area and microporosity were assessed by argon adsorption/desorption at 77 K using a Tristar II analyzer (Particle and Surface Science, Gosford, Australia). Microporous surface area was determined using t-plots and the pore size was assessed using the non-localized density functional theory (NLDFT). The macroporous properties of the adsorbent materials were measured using an Autopore IV mercury intrusion porosimeter. The surface morphologies of the adsorbents were analyzed using a Hitachi SU-70 field emission SEM, the polymer monoliths were sputter-coated with platinum.

The chemical functionality of the adsorbents was determined using solid-state 13 C cross-polarization magic angle spinning (CP-MAS) NMR spectroscopy, 1 H MAS NMR spectroscopy and attenuated total reflectance fourier transform infrared (ATR-FTIR) spectroscopy. 13 C CP-MAS NMR was carried out to determine the presence of vinyl groups on the PDVB using a Bruker DPX 200 spectrometer (Bruker, Alexandria, Australia) operating at 50 MHz Larmor frequency for 13 C, using a 4 µs 90° pulse, a 4 ms contact time, a 3 s repetition delay, and 15 360 transients. 13 C CP-MAS NMR carried out to determine the presence of PEGMA in the PDVB utilized a Bruker DRX 300 spectrometer operating at 75 MHz Larmor frequency for 13 C, using a 5 µs 90° pulse, a 4 ms contact time, a 3 s repetition delay, and 13 112 to 16 472 transients. Quantitative 1 H MAS NMR spectra of the grafted samples were recorded on a Bruker DRX 300 spectrometer at 300 MHz Larmor frequency for 1 H, using a 10 µs 90° pulse, a 10 s repetition delay, and 64 transients. In order to determine true relative signal integrals it was checked that the magnetization was fully recovered between pulses for 1 H NMR signals of both PDVB and PEGMA (Section S5) [START_REF] Claridge | High-Resolution NMR Techniques in Organic Chemistry[END_REF].

All experiments were carried out with 4 mm solid-state MAS NMR probeheads at a MAS rotational frequency of 10 kHz. 1 H and 13 C pulses were calibrated with adamantane and a mixture of 3 singly 13 C labeled alanines. The 13 C chemical shift scale was externally referenced to tetramethylsilane at 0.0 ppm using adamantane by setting the CH resonance to 38.5 ppm [START_REF] Morcombe | [END_REF]. All data was collected using Topspin software (Bruker, Alexandria, Australia). ATR-FTIR spectroscopy was used to determine the functional groups of the adsorbents. The spectra were recorded on a Vertex 70 spectrometer (Bruker Optic, Ettlingen, Germany) in the 500-4000 cm -1 region, 32 scans were signal-averaged with a spectral resolution of 4 cm -1 using a single reflection diamond ATR attachment, Platinum ATR (Bruker Ettlingen, Germany).

Miniaturized SPE adsorbent performance was assessed using an ICS3000 system (ThermoScientific, Scoresby, Australia) consisting of two quaternary solvent pumps, an autosampler and an ultraviolet (UV) detector. Analyte recovery and protein adsorption from the offline SPE extraction was analyzed using a ProteCol C8 ( MS experiments were performed using a micrOTOF-Q MS (Bruker, Preston, Australia) equipped with an ESI source. The instrument was operated as follows: +3.5

kV capillary potential (negative ion mode), nitrogen nebulizer gas at 1.7 bar, nitrogen dry gas at 220°C and a flow rate of 4 L min -1 . Calibration of the TOF-MS was carried out prior to each analysis by direct infusion of the low concentration multimode tune mix (Agilent Technologies, Mulgrave, Australia) at 5 µL min -1 . Data was collected using micrOTOF control 2.2 with a mass range of 100 -600 m/z at an acquisition rate of 0.5 Hz. Ibuprofen was monitored using the extracted ion 205.12 ±0.01 m/z. The analyte identity was confirmed using standard addition. All ion traces were processed using Compass Data Analysis 4.0 (Bruker, Preston, Australia) and smoothed using the Gauss smoothing algorithm at 2.003 s.

Preparation of the monolithic SPE adsorbents

The fabrication of PDVB adsorbents was based on an approach described by Sýkora et al. [26]. Briefly, the composition included 40% w/w DVB, 8% w/w toluene, 52% The adsorbents were then dried overnight under vacuum at 25 °C.

The post-polymerization thermal grafting reaction was developed from an approach described by Tripp et al. [START_REF] Tripp | Reactive & Functional Polymers[END_REF]. The grafting mixture was prepared in toluene containing 5-20% w/w functional monomer (PEGMA 360 or PEGMA 950 ), 1% w/w of the crosslinking monomer, PEGDA 258 , and 1% w/w AIBN. The bulk monolith was submerged in the grafting mixture, while the PE tubes containing monolith were manually flushed with grafting mixture then submerged in the grafting solution. The mixture was polymerized at 65°C for 20 h. The products were Soxhlet extracted with methanol for 24 h and dried under vacuum overnight.

For testing the MEPS format was explored, MEPS cartridges were filled with monolith weighing 2 ± 0.2 mg. Once assembled the cartridges were flushed with 20 mL of 95:5 methanol:water (v/v).

Fluorescence assay of protein adsorption

To determine HSA adsorption, a stock solution of the protein was prepared at 1 mg mL -1 in 10 mM sodium tetraborate buffer. A 3 mg mL -1 solution of fluorescamine in acetone was prepared. A final protein concentration of 50 µg mL -1 was achieved by combining 5% HSA solution and 15% fluorescamine solution in a 10 mM sodium tetraborate buffer. Adsorbents were first conditioned with 250 µL of methanol and equilibrated with 250 µL of water at 50 µL min -1 . A 400 µL aliquot of the protein solution (50 mg L -1 HSA) was passed through the adsorbent and any unretained protein was flushed out with 250 µL of water, a flow rate of 50 µL min -1 was used.

The fluorescence intensity of the cross-section of the adsorbent was analyzed. The signal of each adsorbent was compared with the corresponding blank (adsorbents imaged prior to the protein-binding assay Section S4).

Adsorbent performance for SPE

Breakthrough times and adsorption behavior for the SPE adsorbents were determined using frontal analysis. The MEPS cartridge was inserted between the dual quaternary pump and the UV detector. Pump 1 delivered the solvent solutions while Pump 2 delivered the aqueous analyte solution. The adsorbent was first conditioned with 2.5 mL of methanol:water (95:5 %v/v) using a continuous flow of 1 mL min -1 then equilibrated with 2.5 mL water:methanol (95:5 %v/v) at the same flow rate. The probe analytes, anisole (100 mg L -1 ) and ibuprofen (10 mg L -1 ), were employed. A lower concentration of ibuprofen was employed due the lower aqueous solubility.

The analyte in aqueous solution was continuously pumped over the adsorbent bed and the cartridge effluent was monitored at 254 nm. Uracil was employed to determine the void volume of the system (including cartridge). Each analysis was undertaken in duplicate. Backpressure of SPE cartridges was monitored to determined permeability.

Analyte recovery and protein adsorption for the SPE adsorbents was determined by offline SPE. The processed extracts were analyzed by HPLC-UV to determine peak area. The SPE cartridge was attached to a 100 µL CDF-MEPS syringe (a modified MEPS needle [27]), sample and solvent were aspirated into the syringe barrel with the valve in Position 2 and dispensed with the valve in Position 1. In all cases the flow rate to aspirate was 3.5 mL min -1 while dispensing was achieved at 1.0 mL min -1 . To determine the protein adsorption of the SPE adsorbents the extraction workflow involved preconditioning the adsorbent with 100 µL of methanol and then 100 µL of water. Next, a 100 µL aliquot of HSA (1000 mg L -1 aqueous solution) was passed through the adsorbent bed. To assess the depletion of the HSA the effluent was collected and compared with a non-extracted HSA solution (n=3). To determine analyte recovery the adsorbent was conditioned and equilibrated as above with methanol and water respectively. Next, a 100 µL aliquot of the 10 mg L -1 ibuprofen in aqueous solution was applied to the adsorbent, then any unretained compounds were removed with 100 µL of water. The analyte was eluted with 100 µL of methanol and extracts were analysed by HPLC-UV (n=3). The recovery was determined by comparing the peak area for an extracted ibuprofen solution with a non-extracted ibuprofen sample. For the MS analysis the 50 µL CDF syringe was employed. The sample was processed as above. The elution solvent employed was acetonitrile with 0.3% v/v ammonium hydroxide. The processed sample was delivered directly to the ESI source at a flow rate of 20 µL min -1 (n=3). Following the first elution a second 50 µL elution was performed using 1% formic acid:isopropanol (60:40 %v/v) to clean the bed.

Results and discussion

Characterization of the PDVB adsorbent

The desirable attributes of an adsorbent for the rapid and efficient miniaturized SPE of protein rich biological samples are: high permeability, large surface area and biocompatibility. For fast SPE workflows, large macropores are favorable to ensure maximum flow rates can be employed. The macroporous properties of the PDVB were investigated using mercury intrusion porosimetry. The PDVB adsorbent presents a large macropore size of 6.7 µm, this infers a highly permeable morphology. The permeability was measured using an HPLC pump and the pressure-drop across the device was determined. The permeability of the PDVB adsorbent was estimated to be 6.7 x 10 -15 m 2 (23% propagated error).

For a sensitive assay a large surface area is desirable, as extraction capacity is strongly related to the interactable surface. SEM provides a visual assessment of the clustered interconnected globules of the PDVB adsorbent (Figure 1A). Agglomerates of 5-10 nm microspheres roughened the surface of the interconnected globules. The specific surface area of PDVB was determined to be 497 mg g -1 by argon adsorption/desorption. The PDVB isotherm presents a combination of a Type I isotherm and a Type IV isotherm, indicative of micro-and mesopores respectively (Figure 1B). The Type H2 hysteresis loop suggests the mesopores are small and exist over a broad distribution in sizes and shape. NLDFT confirms these pores range from 1 to 6 nm, but the vast majority are micropores of less than 2 nm (Figure 1C). The slope of the t-plot suggests that 93% of the total pore volume (0.30 cm 3 g -1 ) is attributed to the micropores (0.28 cm 3 g -1 ). Only a small percentage of the surface area (16 mg g -1 ) can be attributed to the external surface. The innate process of fabricating PDVB monoliths creates a physical size exclusion barrier that inhibits proteins from gaining access to the large internal surface. However the external hydrophobic surface remains problematic for protein fouling of the PDVB.

Figure 2 shows the solid-state 13 of the monomer units [23]. Some of these RVGs will be present in the internalized structure and unavailable for modification; the others are available for the grafting of a chemical barrier to restrict non-specific protein interactions.

PEG functionalization of PDVB adsorbents

The hydrophilic monomer, PEGMA, was selected for grafting as it can be covalently attached to the adsorbent's surface via a free radical polymerization [4,17]. The mechanism of protein resistance is believed to be a combination of the hydrophilic glycol groups and the steric hindrance imparted by the long PEG chains. We chose to explore two different PEGMA monomers with average molecular weights of M n 360 and 950 g mol -1 . To achieve grafting, the preformed PDVB was submerged in a solution containing 5-20% of the PEGMA monomer and 1% of the crosslinking monomer, PEGDA 258 , in toluene. Polymerizations containing 10 and 20% PEGMA 360 in the graft solution yielded a sticky transparent gel. For this reason higher concentrations of PEGMA were not explored for either of the monomers. Following polymerization, the graft solution of PEGMA 950 (all concentrations) remained liquid.

Characterization of the PEGMA grafted PDVB adsorbents

To achieve successful miniaturized SPE it is critical that the analyte is able to quickly and efficiently interact with the adsorbent. Therefore, the hydrophilic layer must be carefully optimized, as it should be appropriately dense to prevent proteins from accessing the hydrophobic surface but it also must remain sufficiently thin and permeable to ensure that the analyte can efficiently interact with the hydrophobic inner surface [28][29][30]. The adsorbents were visually inspected using SEM to determine the presence of the grafted PEGMA (Figure 3). The grafted layer was clearly evident for adsorbent prepared using 20% and 10% PEGMA 360 in the graft solution. The PDVB-g-PEGMA 360 prepared with 20% PEGMA 360 displayed a thick coating of the grafted layer which masked the globular features that are characteristic of the PDVB scaffold (Figure 1A). The thickness of the PEGMA 360 layer raised concern over the suitability of this adsorbent for miniaturized SPE, thus PDVB-g-PEGMA 360 (20%) was not assessed further. The lower concentration of 10%

PEGMA 360 in the graft solution yielded a thinner sheet like layer which draped the polymer globules of the PDVB. On the other hand, the adsorbent prepared from 5%

PEGMA 360 in the graft solution showed no evidence of a PEGMA graft layer, appearing identical to the PDVB scaffold (Figure 1A). Additional SEM images over wider range of magnifications can be seen in Figure S1. In contrast, all adsorbents prepared from the longer chain monomer, PEGMA 950 , appeared identical to the PDVB precursor (Figure 3 and1A). This cast doubt over the success of the PEGMA 950 grafting.

To determine the success of grafting, PDVB-g-PEGMA 950 (20% PEGMA) was probed using a crude assay to assess surface hydrophilicity. A 10 µL droplet of an aqueous solution of green food dye was pipetted onto the surface of the bulk adsorbents. The PDVB was employed as negative control. As successful grafting of PDVB-g-PEGMA 360 (10% PEGMA) was evident from the SEM assessment, this adsorbent was employed as a positive control. The aqueous dye solution was repelled from the hydrophobic PDVB, beading on the surface. In contrast, the aqueous solution penetrated the PDVB-g-PEGMA 360 wetting the adsorbent (Figure S2). As with PDVB-g-PEGMA 360 , the aqueous dye solution immediately penetrated the PDVB-g-PEGMA 950 , suggesting the successful the grafting of PEGMA 950 .

To evaluate the extent of protein interactions with each of the SPE adsorbents a qualitative fluorescent assay was employed. As HSA is the most abundant human plasma protein it was selected as the probe. HSA is a 67 kDa globular protein with The PDVB presents a substantial fluorescent signal, despite the small pores preventing HSA from accessing the large internal surface area it appears a considerable amount of HSA has been adsorbed on the external hydrophobic surface (Figure 4). A high fluorescent signal was seen for both PDVB-g-PEGMA 360 and PDVB-g-PEGMA 950 that was prepared with 5% PEGMA in the graft solution. This result suggests 5% PEGMA is not sufficient to prevent protein interactions, therefore these adsorbents were not analyzed further. In contrast, increasing the PEGMA concentration to 10% in the graft solution decreased the fluorescent signal. The fluorescent signal observed for PDVB-g-PEGMA 360 was reduced substantially. The signal for PDVB-g-PEGMA 950 was also lessened but to a lower extent. PDVB-g-PEGMA 950 prepared using 20% PEGMA 950 in the graft solution yielded a much lower fluorescent response compared to PDVB-g-PEGMA 950 prepared using 10% PEGMA 950 . As the grafting of PEGMA 950 (20%) and PEGMA 360 (10%) was sufficient to substantially reduce protein binding these adsorbents were further investigated.

To characterize grafting both PDVB-g-PEGMA adsorbents were compared to the precursor PDVB using ATR-FTIR. The overlaid FTIR spectra revealed distinct changes in the chemical functionality (Figure 5). The spectrum corresponding to PDVB-g-PEGMA 360 exposed a distinct band at 1114 cm -1 which can be attributed to ether functionality. The bands at 1724 cm -1 and 1247 cm -1 can be credited to the ester functionality of the PEGMA monomer. Lastly, the wide band at 3449 cm -1 is indicative of the terminal hydroxyl group on the PEGMA 360 chain. The spectral changes are subtler for the PDVB-g-PEGMA 950 , the band at 1114 cm -1 (ether functional group) is the dominant change. The relative intensity of this band is expected due to the large number of the repeating glycol units. In contrast, only a very small band is seen at 1724 cm -1 relating to the single ester group on each of the attached monomer chains. Figure 5 infers a higher density of grafting was achieved for the shorter chain length PEGMA 360 .

The results above are corroborated by solid-state NMR, for 13 C CP-MAS NMR, the appearance of the signal at 71 ppm is attributed to the PEG side chain (Figure 2) [31].

Intrinsically, the 13 C CP-MAS NMR analysis is not quantitative as the signal enhancement is biased towards the more rigid components of a sample. Therefore, 1 H NMR was employed to quantitate PEGMA grafting, details of the experimental setup can be seen in the supplementary information Section S5. The 1 H NMR spectrum of PDVB exhibits a signal at 6.7 ppm, a small shoulder at 3.2 ppm is also seen (Figure 6) [23]. Following grafting, both PDVB-g-PEGMA adsorbents display a narrower signal for PDVB at 6.7 ppm (insert Figure 6), this suggests that the ungrafted PDVB is more rigid than the PDVB-g-PEGMA adsorbents (Figure 6) [31]. The difficulty to completely dry the PDVB-g-PEGMA adsorbents due to the introduction of the hydrophilic coating is the most likely cause for the increased mobility of the PDVB signal in the PDVB-g-PEGMA spectra. The strong signal at 3.5 ppm is attributed to the PEG chain, the PEG signal is much narrower than the PDVB signal as the chains have a higher mobility than the crosslinked PDVB monolith (Figure 6). The peak areas of the 1 H NMR spectra were employed to quantify the PEGMA content of the grafted adsorbents (Table S3 andS4). The PEGMA content was quantified in the grafted samples as the mass (m) ratio of PEGMA to PDVB using Equations (1).

! "#$%& ! "'() = + "#$ + "'() • -'() -"#$%& • . "#$%& . '() (1)
where I is the peak area of the monomer units in the NMR spectrum, N is the number of protons from which this signal originates and M is the molar mass of a monomer unit. Details of the equation derivation and the data processing are given in supporting information (Section S6). From this analysis the PEGMA content (% w/w) in the grafted samples was estimated to be 29.9 ± 1.6% for PDVB-g-PEGMA 360 and 7.7 ± 1.2% for PDVB-g-PEGMA 950 . This confirms that a higher degree of grafting was seen for the shorter chain length PEGMA 360 . The reason for the lower observed grafting density of PEGMA 950 was not investigated further but possible reasons may be a lower reactivity of the PEGMA 950 monomer or a steric hindrance due to the longer glycol chain limiting access to the surface.

The permeability was determined, as outlined above for PDVB, to determine if the grafted PEGMA layer was detrimental to the operation of the miniaturized SPE cartridge. The permeability was estimated to be 1.7 x 10 -15 m 2 (13% propagated error) and 3.7 x 10 -15 m 2 (12% propagated error) for PDVB-g-PEGMA 360 and PDVB-g-PEGMA 950 respectively. Unsurprisingly, the permeabilities of the miniaturized SPE cartridges were reduced following the grafting reaction. The PDVB-g-PEGMA 360 , with the most obvious grafted layer, demonstrated the greatest decrease in permeability. However, as the digital syringe drive was still able to successfully pass fluid through the miniaturized SPE cartridge at 1 mL min -1 the decreased permeability was not considered detrimental to the miniaturized SPE workflow.

Assessment of the adsorbent performance for SPE

Frontal analysis was employed to evaluate the performance of each SPE adsorbent as the shape of the breakthrough curve provides key insight into the adsorbents suitability for SPE. To achieve a complete SPE extraction a rapid an efficient equilibrium between the analyte and the adsorbent (a sharp breakthrough curve) is critical. The presence of the hydrophilic layer must not dramatically impede the interactions between the analyte and the inner hydrophobic surface. If the analyte is unable to achieve a rapid equilibrium with the adsorbent reduced extraction recoveries and high sample carryovers will result. To determine SPE suitability of the adsorbents, PDVB, PDVB-g-PEGMA 950 and PDVB-g-PEGMA 360 were prepared in MEPS cartridges, this format was explored as it could be seamlessly interfaced with HPLC-UV. Anisole (0.32 -0.42 nm 2 ) and ibuprofen (0.35 -0.65 nm 2 ) in aqueous solutions were employed as probe molecules. The capacity factor (k'), volume of analyte breakthrough (V B ), volume of analyte retention (V R ), the standard deviation of the derivative curve (2σ v ) and the amount of analyte adsorbed (q c ) were selected as metrics for the evaluation. The void volume (V 0 ) of the system, including the MEPS cartridge filled with each of the polymer monoliths, was determined using uracil. For all cartridges the V 0 was 80 µL.

The breakthrough curves of both anisole and ibuprofen can be seen in Figure 7. The V B is defined as the volume of sample that has passed through the adsorbent until 1% of the maximum analyte signal is measured at the outlet. V B can be calculated from Equation 2,

/ 0 = / 1 -24 5 (2)
where V R is the retention volume determined from the inflection point of the curve and 2σ v is the standard deviation of the derivative curve [32]. Results of anisole breakthrough for PDVB-g-PEGMA 950 mirrored the shape of the PDVB curve (Figure 7A). A small reduction in the V B was revealed which is consistent with the reduced k'

(Table 1). Further, a small decrease in the sharpness (2 σ v ) of the PDVB-g-PEGMA 950 curve was observed. In contrast, the anisole breakthrough curve for PDVB-g-PEGMA 360 shows a premature V B , the anisole is detected at the cartridge outlet almost immediately. The shallow breakthrough curve (large 2σ v ) suggests there is insufficient time for the analyte to penetrate the PEGMA 360 coating when a flow rate of 500 µL min -1 is employed.

Equation 3 was employed to determine the amount of analyte adsorbed (q c ) for each SPE adsorbent.

6 7 = 89 1 -9 : ;<= (3)
where t R , is the retention time, t 0 is the void time, f is the flow rate and C is the analyte concentration. Table 1 lists the q c for all adsorbents. The amount of analyte adsorbed was reduced slightly following the grafting of the PEGMA 950 layer. Interestingly, despite the reduced V B of PDVB-g-PEGMA 360 the q c of the adsorbent has largely been preserved. However, due the poor analyte diffusion through the PEGMA layer it was determined that this adsorbent is unsuitable for rapid miniaturized SPE workflows and PDVB-g-PEGMA 360 was not investigated further. To broaden the validation, ibuprofen breakthrough curves were compared for PDVB and PDVB-g-PEGMA 950 (Figure 7B). As with anisole, a small reduction in the k', V B , V R , σ v and q c of ibuprofen was observed following the grafting of PEGMA 950 (Table 1).

However, the overall shape of the curve was maintained relative to PDVB. The results indicate that the grafted layer of PEGMA 950 does not dramatically diminish analyte/adsorbent interactions. To support this observation the bulk PDVB-g-PEGMA 950 was analyzed using argon adsorption/desorption at 77 K (Figure S5). The isotherm of the PDVB-g-PEGMA 950 reveals a similar pore structure to PDVB. In addition, the surface area, 495 m 2 g -1 , was equivalent to the surface area of PDVB confirming that PEGMA 950 does not drastically alter analyte/adsorbent interactions.

To broaden the validation of biocompatibility the degree of protein binding was quantitated chromatographically. The amount of HSA adsorbed was measured for both the PDVB and the PDVB-g-PEGMA 950 using offline SPE. An aliquot of HSA was passed through the adsorbent bed and any reduction in peak area of the effluent corresponded to non-specific protein binding. The PDVB adsorbed 31 ± 2.41% of the HSA sample. Given that only a very small external surface area is available for interaction the amount of HSA adsorbed was substantial. While not completely eliminated, the amount of HSA adsorbed to the PDVB-g-PEGMA 950 was reduced to 12 ± 0.49%. In future studies, a further increase in the amount of PEGMA 950 in the graft solution could be investigated to aid the complete elimination of protein binding.

The final performance qualifier of these adsorbents is analyte recovery; which is a critically important performance characteristic of miniaturized SPE. A comparative study of recovery was undertaken to benchmark the performance of the PDVB-g-PEGMA 950 against the PDVB adsorbent. Recovery of 10 mg L -1 ibuprofen was determined for the two adsorbents. The PDVB offered an analyte recovery of 92 ± 0.30% compared to 78 ± 0.93% using PDVB-g-PEGMA 950 . Reduced analyte recovery was expected given reduction the k' and V B seen in Table 1. Regardless of this reduction, 78% is still considered a high value for analyte recovery. This qualifies the PDVB-g-PEGMA 950 as a highly suitable biocompatible adsorbent for miniaturized SPE.

Application to a real sample

At-line ESI-MS was employed to finalize the assessment of PDVB-g-PEGMA 950 for a miniaturized SPE clean up of human plasma. By infusing the processed sample directly from the adsorbent bed into the ESI-MS using the at-line approach we can gain an impression of the effectiveness of the sample clean up [27]. A diluted (20% v/v) plasma sample spiked with ibuprofen was processed using both the PDVB and the PDVB-g-PEGMA 950 adsorbents. The extracted ion chromatogram of the elution profiles were compared between the PDVB and the PDVB-g-PEGMA 950 (Figure 8).

To this end, the peak area of the ibuprofen elution profile from the PDVB-g-PEGMA 950 was 30 % greater than the profile obtained using hydrophobic PDVB.

Here we see that the PDVB assay was crippled by the hydrophobic surface chemistry, the adsorbed protein matrix contaminants led to a dramatic reduction in assay sensitivity. The biocompatibility of the PDVB-g-PEGMA 950 adsorbent has been clearly demonstrated for the fast and efficient purification of analytes in highly complex biological samples.

Conclusions

The inherent pore structure of PDVB makes it highly appealing for a size selective miniaturized SPE. Despite the desirable pore structure of PDVB, use with highly complex biological samples rich in proteins can be problematic due to the nonspecific protein adsorption to the hydrophobic surface. PEGMA 360 and PEGMA 950 were grafted to the RVGs of the PDVB to provide biocompatible adsorbents. The grafted layer was carefully characterized for both PDVB-g-PEGMA 360 and PDVB-g-PEGMA 950 . The thick coating of PDVB-g-PEGMA 360 was a double-edged sword, while the coating prevented proteins from accessing the hydrophobic surface, substantial time was required for the analyte to interact with the adsorbent thus diminishing the adsorbents suitability for miniaturized SPE. The grafted PEGMA 950 did not dramatically affect the SPE adsorptive properties. The PDVB-g-PEGMA 950

revealed only a small reduction in the amount of analyte loaded on the adsorbent bed.

Furthermore, protein adsorption was substantially restricted by the grafted PEGMA 950 . The PDVB-g-PEGMA 950 adsorbent was demonstrated to produce cleaner extracts for a more sensitive ESI-MS assay. The superiority of the PDVB-g-PEGMA 950 adsorbent is extremely beneficial for the rapid and efficient miniaturized SPE purification of analytes in protein rich biological samples. Care was taken to ensure that the magnetization was fully recovered between pulses for 1 H NMR signals of both PDVB and PEGMA, in order to determine the true relative signal integrals. To ensure full magnetization recovery between pulses, 1 H NMR spectra must be recorded with a delay between pulses longer than 5 times the longitudinal relaxation time (T 1 ) for each signal of interest. One-dimensional inversion-recovery T 1 relaxation measurements (1D T 1 ) were conducted, each with a single delay in the "indirect dimension", for delays ranging from 0.1389 to 1.389 s, and corresponding delays between pulses of 5 × 1.44 times that value (0.2 to 2 s).

Section S5 ATR-FTIR experiments

Phasing the spectra was completed using the same approach as a conventional 1 H spectrum. Each signal is negative for short delay values and positive for long delay values, with a zero crossing occurring at T 1 /1.44. The PEG signals at 3.5 ppm exhibited T 1 values between 0.3 and 0.4 s (Fig. S4A), while the PDVB signals exhibited T 1 values between 1 and 2 s, very close to 1.5 s (Fig. S4B). Recording all subsequent 1 H NMR spectra experiments with a delay between pulses longer than 5 × 2 = 10 s ensured that full magnetization recovery took place and the 1 H NMR spectra were quantitative.

Section S7 Approach to calculate proportionality of PEGMA content in the grafted

PDVB samples

The PEGMA content could be quantified in the grafted samples in three ways: first, as the mass ratio of PEGMA to PDVB, second as the molar ratio of PEGMA monomer units to DVB monomer units, and finally as the molar ratio of ethylene glycol monomer units to DVB monomer units. Equations ( 1) to (3) were employed respectively:

= • • (1) = • (2) = • • / ( 3
)
where I is the peak area of the signal of a monomer unit on the NMR spectrum, N is the number of protons from which this signal originates in the monomer unit, N is the molar mass of a monomer unit, and X EG/PEGMA is the number of EG monomer units per PEGMA monomer unit.

Derivation of PEGMA to PVB ratios

The peak area (I) of a signal of a monomer unit on the NMR spectrum is proportional to the number of protons (N) from which this signal originates and to the amount (n in mol) of these monomer units. Let k be the proportionality constant in a given spectrum:

= • • (4) = • • (5)
! = + (6)
Using the relationship between the n, the molar mass (M) and the mass (m):

= • • (7) = • • (8)
The mass ratio of PEGMA and DVB monomer units can be obtained by combination of equations ( 7) and ( 8):

= • • (9) = $ %&%'( ) * • • (10)
The molar ratio of PEGMA and DVB monomer units can be obtained by combination of equations ( 4) and ( 5):

= • (11)
= $ %&%'( ) * •

This ratio can be converted to a molar ratio of ethylene glycol (EG) and DVB monomer units through the number X EG/PEGMA of EG monomer unit per PEGMA monomer unit: ) and a multiple of the ethylene glycol units (44.05 g•mol -1 ). The average number of ethylene glycol units is thus (360 g mol -1 -86.09 g mol -1 )/44.05 g mol -1 = 6.22.

= • • / (13)
b PEGMA 950 has a -CH 3 end group therefore its molar mass is one methyl methacrylate (100.12 g•mol -1 ) and a multiple of the ethylene glycol units (44.05 g•mol -1 ). The average number of ethylene glycol units is thus (950 g•mol -1 -100.12 g•mol -1 )/44.05 g•mol -1 = 19.3.

The peak area of the PEG signal was estimated by integrating the PEG signal at 3.5

ppm. The integration of the PEG signal is complicated by two situations, first a small component of the PDVB backbone overlaps with the base of the PEG signal. Second the PDVB signal for the ungrafted material is lower than the PDVB for the grafted adsorbents. As an approximate, the peak area of the PEG signal could either be underestimated by setting the baseline above the maximum of the PDVB signal or overestimated by setting the baseline at the level of the valley between this PDVB signal and the PEG signal. The peak area of the PDVB signal was determined as the difference between the total signal integral and the PEG signal integrals.

dimensions of 3 .

 3 8 nm × 15 nm, therefore any interactions are likely limited to the external surface of the adsorbent. A typical SPE workflow was employed for this assay. The adsorbent bed was conditioned with methanol then equilibrated with water, a 400 µL aliquot of 50 mg L -1 HSA labeled with fluorescamine was passed through the adsorbent bed. The adsorbent was flushed with water to remove any residual protein. The intensity of the fluorescent signal of each adsorbent was observed using an optical microscope. The intensity of the fluorescent signal provides a qualitative insight into the extent of protein interactions with the adsorbent; the desired low level of protein interaction is implied by low fluorescence intensity. The background fluorescent signal for each material can be seen in the supplementary information (FigureS3).

Figure 1 .

 1 Figure 1. The dry state characterization of PDVB. A) SEM images, top 2000× magnification and bottom 20 000× magnification. B) The argon adsorption (!) and desorption (ο) isotherms at 77 K and C) corresponding pore size data determined by NLDFT.

Figure 2 .

 2 Figure 2. Solid-state 13 C CP-MAS NMR spectra of all adsorbents. Bottom) PDVB, Middle) The PDVB-g-PEGMA 950 , Top) PDVB-g-PEGMA 360 . The chemical shifts are assigned in Scheme 1.

Figure 3 .

 3 Figure 3. SEM images of the surface of PDVB-g-PEGMA 360 and PDVB-g-PEGMA 950 prepared from 5, 10 and 20% PEGMA in the graft solution. The magnification of all images is 10 000× and the scale bar corresponds to 5 µm.

Figure 4 .

 4 Figure 4. Optical microscopy images of protein interactions for the developed adsorbents, fluorescently labeled HSA was flushed through the PDVB scaffold as well the grafted PDVB adsorbents with increasing percentages of PEGMA 950 and PEGMA 360.

Figure 5 .

 5 Figure 5. ATR-FTIR spectra of PDVB, PDVB-g-PEGMA 950 and PDVB-g-PEGMA 360 .

Figure 6 .

 6 Figure 6. Solid-state 1 H MAS NMR spectra of all adsorbents. A) PDVB, B) PDVB-g-PEGMA 950 , C) PDVB-g-PEGMA 360 .

Figure 7 .

 7 Figure 7. A) Anisole breakthrough for PDVB (---),PDVB-g-PEGMA 950 ( ) and PDVB-g-PEGMA 360 (-. -). B) Ibuprofen breakthrough for PDVB (---) and PDVB-g-PEGMA 950 ( ).

Figure 8 .Figure 1 Figure 2 Figure 3 Figure 4 Figure 5 Figure 6 Figure 7

 81234567 Figure 8. Comparison of sample clean up and at-line MS elution profiles of ibuprofen in a dilute plasma sample (20% v/v) using PDVB and PDVB-g-PEGMA 950 adsorbents

Figure 8 Analytical

 8 Figure 8

Fig. S2

 S2 Fig. S2 An aqueous solution of green food dye spotted onto PDVB, PDVB-g-PEGMA 360 and PDVB-g-PEGMA 950 as an estimate of surface grafting

Fig. S4

 S4 Fig.S4The one-dimensional inversion-recovery T 1 relaxation measurements of PDVB-g-PEGMA 360 to assess the relaxation of A) the PEG component at 3.5 ppm and B) the PDVB components at 6.7 ppm

  C CP-MAS NMR spectra of the PDVB, the chemical shift assignment is depicted in Scheme 1. Greater detail of the complete chemical shift assignments can be found in Table S1. The strong signals at 138 and 113 ppm (signals B and D) correspond to vinyl carbons adjacent and non-adjacent to the aromatic ring. Even though this NMR measurement is not quantitative, the strong B and D signals indicate the presence of unreacted vinyl groups on a significant fraction

Table 1 .

 1 Capacity factor, breakthrough volume, retention volume, standard deviation of the derivative curve and amount of analyte adsorbed for PDVB, PDVB-g-PEGMA950

	and PDVB-g-PEGMA360
	Adsorbent	k'	VB (mL)	VR (mL)	2σv (mL)	qc (mg g -1 )
	Anisole
	PDVB	39	2.8	3.1	0.7	129
	PDVB-g-PEGMA950	35	2.3	2.8	0.8	116
	PDVB-g-PEGMA360	34	0.5	2.6	1.6	106
	Ibuprofen
	PDVB	122	7.8	9.6	2.6	43
	PDVB-g-PEGMA360	118	7.2	9.8	2.7	38

Table S1

 S1 Complete13 C chemical shift assignment for13 C NMR spectra of PDVB, PDVB-g-PEGMA 950 and PDVB-g-PEGMA 360 (seen in Fig.2)

	Section S2 SEM analysis of the SEM analysis of the PDVB, PDVB-g-PEGMA 360 (10%) and PDVB
	PEGMA 950 at increasing magnification at increasing magnification
	Group	Chemical shift (ppm) Assignment
	A	145	aromatic ring (adjacent to ethyl or backbone)
	B	138	aromatic ring
	C	128	PDVB benzene ring
	D	113	C=C (C not adjacent to aromatic ring)
	E	71	CH 2 -CH 2 -O-of PEG side chain
		63	residual porogen 1-dodecanol
	F	41-50	CH back bone of PDVB and poly
			methacrylate backbone of PEGMA
			CH 2 back bone of PDVB and poly
			methacrylate backbone of PEGMA
		30	residual porogen 1-dodecanol
	G	30	CH 2 of ethyl on aromatic ring
		23	residual porogen 1-dodecanol
	H	16	CH 3 of ethyl on aromatic ring
	Fig. S1 SEM images of (A) PDVB, (B) PDVB SEM images of (A) PDVB, (B) PDVB-g-PEGMA 360 (10%) and (C) PDVB
	PEGMA 950 (20%). The magnification in (20%). The magnification in the top is 2000x, middle is 10
	bottom is 20 0000x

Note: There are three types of structural aromatics present in the PDVB monoliths (prepared from 80% DVB) 1) both the vinyl groups have reacted to form the crosslinked monolith, 2) the uncrosslinked ethylvinyl benzenes, 3) the DVB unit where only one of the vinyl groups has reacted leaving a residual vinyl groups (RVG).

Section S3 Crude analysis of surface hydrophilicity with aqueous solutions

Table S2

 S2 Spectral peak assignment for PDVB, PDVB-g-PEGMA 950 and PDVB-g-PEGMA (seen in Figure5)

	Wavelength (cm -1 )	Assignment
	3449	PEGMA terminal hydroxyl (-OH)
	2917,	methyl group (CH 3 )
	1724,	PEGMA ester (O=C-OR)
	1628, 1406, 1015, 987	residual vinyl groups (C=C)
	1600,	C-C stretching
	1509	para-disubstituted phenyl ring
	1484, 1016, 707	monosubstituted phenyl ring
	1444	C=CH 2 , and ring C-H
	1114	PEGMA ether
	905	mixed assigment: vinyl group, monosubstituted phenyl
		ring, disubstituted phenyl ringC
	832	mixed assignment: monosubstituted phenyl ring, para-
		disubstituted phenyl ring
	793	meta-disubstituted phenyl ring
	707	monosubstituted phenyl ring
	Hubbard, KL., Finch, JA, Darling, GD. (1998). React Funct Polym, 36, 1-16.

Section S6 1 H NMR experimental set up

  PEGMA for PEGMA 360 360 g•mol -1 (note: this is M n ) M PEGMA for PEGMA 950 950 g•mol -1 (note: this is M n ) X EG/PEGMA for PEGMA 360 6.22 a EG monomer units in a PEGMA unit X EG/PEGMA for PEGMA 950 19.3 b EG monomer units in a PEGMA unit a PEGMA 360 has an -OH end group therefore its molar mass is one methacrylic acid (86.09 g•mol -1

		= $ %&%'( )	* •	•	/	(14)
	Table S3 Constants needed for the calculations
	Name	Value	Unit
	N DVB N PEGMA for PEGMA 360 N PEGMA for PEGMA 950 M DVB	10 24.9 a 77.2 b 130.19	1 H nuclei per monomer unit 1 H nuclei per monomer unit 1 H nuclei per monomer unit g•mol -1
	M

Table S4

 S4 Integrals from NMR spectra

	Adsorbent	I total	I PEG	I PEG	I PEG
			(underestimate)	(overestimate)
	PDVB-g-PEGMA 360 2782.33	558.27	620.44	589 ± 31
	PDVB-g-PEGMA 950 1915.61	120.80	166.98

Acknowledgments

RAS is the recipient of an Australian Research Council Australian Research

Fellowship (DP110104923). EC is the recipient of an Australian Postgraduate Award. Support from the University of Tasmania Central Science Laboratory is gratefully acknowledged. We gratefully acknowledge Dr. Karsten Gömann and Dr. Sandrin Feig (University of Tasmania) for assistance with scanning electron microscopy and Patrice Castignolles (University of Western Sydney) for helpful discussions.

Table S5

The ratios of DVB to PEGMA for PDVB-g-PEGMA 950 and PDVB-g-PEGMA 360 determined as a mass ratio, a molar ratio of PEGMA to DVB as well as a ratio of EG units to DVB units Adsorbent + ,-./0 + ,123 (wt%) Section S10 Fabrication repeatability for PDVB and PDVB-g-PEGMA 950

To determine repeatability of the PDVB fabrication approach the dry state surface area and anisole retention volume (V B ) was measured for three different batches.

Table S6 The surface area of PDVB fabricated for three different batches

Batch

Surface area (m 2 g -1 )

1 497

2 487

3 491

Average 492

RSD 1%

Table S7 Anisole retention volume (V R ) for three different batches of PDVB Batch Amount adsorbed (mg g -1 )

1 129 2 127

3 120

Average 125 RSD

3.7%

ATR-FTIR was employed to assess the repeatability of the PEGMA 950 grafting. To account for variability in the mass of material being characterized all spectra were normalized (0,1). Two batches were assessed, for batch 1 three points across the material were measured (n=3) to assess intra batch grafting variability.