Mikhail Y Laktionov

Ekaterina B Zhulina

Oleg V Borisov

Proteins and polyampholytes interacting with polyelectrolyte brushes and microgels: the charge reversal concept revised

Weak polyampholytes and globular proteins among them can be efficiently absorbed from solution by polyelectrolyte brushes or microgels even if the net charge of the polyampholyte is of the same sign as that of the brush/microgel. We use a mean-field approach for calculating free energy of insertion of a probe polyampholyte molecule into a polyelectrolyte brush/microgel. We anticipate that insertion of the polyampholyte into similarly charged brush/microgel may be thermodynamically favorable due to the gain in the cumulative reionization free energy of the pH-sensitive acidic and basic residues. Importantly, we demonstrate that the polyampholyte (protein) charge sign inversion upon transfer from the bulk of the solution into the brush/microgel does not provide sufficient condition to assure negative re-ionization free energy balance. Thus (in the absence of other driving or stopping mechanisms) charge sign inversion does not necessarily provoke spontaneous absorption of the polyampholyte into the brush/microgel.

Introduction

Interaction and electrostatically driven complexation of charged macromolecules or (bio)nanocolloids with polyelectrolyte brushes, branched polyelectrolytes and ionic microgels is actively investigated both experimentally and theoretically which is motivated primarily by biomedical applications. [START_REF] Kabanov | Macromolecular complexes in chemistry and biology[END_REF][START_REF] Kabanov | Interpolyelectrolyte and block ionomer complexes for gene delivery: physico-chemical aspects[END_REF][START_REF] Pergushov | Micelllar Interpolyelectrolyte Complexes[END_REF][START_REF] Schallon | Nanoparticulate Nonviral Agent for the Effective Delivery of pDNA and si RNA to Differentiated Cells and Primary Human T Lymphocytes[END_REF][START_REF] Malmsten | Biomacromolecules in Microgels -Opportunities and Challenges for Drug Delivery[END_REF][START_REF] Bysell | Microgels and Microcapsules in Peptide and Protein Drug Delivery[END_REF][START_REF] Xu | Distribution of Ionizable Groups in Polyampholyte Microgel Controls Interactions with Captured Proteins: From Blockade and "Levitation" to Accelerated Release[END_REF] Polyelectrolyte brushes are arrays of ionically charged polymer chains end-tethered to planar substrate ("planar polyelectrolyte brushes") or to the surface of colloidal particles ("colloidal polyelectrolyte brushes"). [START_REF] Birshtein | Polymer Brushes[END_REF][START_REF] Ballauff | Polyelectrolyte brushes[END_REF][START_REF] Minko | Responsive Polymer Materials: Design and Applications[END_REF][START_REF] Rühe | Polyelectrolyte Brushes[END_REF] The latter resemble micellar-like nanostructures formed upon self-assembly of amphiphilic ionic block copolymers in aqueous media. [START_REF] Borisov | Selfassembled structures of amphiphilic ionic block copolymers: theory, selfconsistent field modeling and experiment[END_REF] Strong local electrostatic field created by brush-forming polyions allows for entrapping of a major fraction of mobile counterions, which are inevitably present in the solution, in the inner volume of the polyelectrolyte brush. As a result, the brush swells due to excess osmotic pressure exerted by these coutnerions. [START_REF] Pincus | Colloid stabilization with grafted polyelectrolytes[END_REF][START_REF] Borisov | Collapse of grafted polyelectrolyte layer[END_REF] Similar effects of localization of couterions and osmotic swelling occur in strongly branched polyelectrolytes [START_REF] Borisov | Conformations and solution properties of star-branched Polyelectrolytes[END_REF] (stars, dendrimers, dendrigrafts, etc.) and in cross-linked polyelectrolyte microgels. [START_REF] Kramarenko | Collapse of Polyelectrolyte Macromolecules Revisited[END_REF] When the brush/microgel and the "guest" polyelectrolyte are charged oppositely, then Coulomb interactions lead to interpolyelectrolyte complexation with concomitant coacervation and collapse of the brush/microgel that may provoke the loss of colloidal stability.

Charged globular protein can be absorbed from the solution in the intramolecular volume of polyelectrolyte brush-like architectures. A striking feature discovered by M.Ballauff and coworkers is that absorption of the proteins may occur on the "wrong" side of the isoelectric point, that is, when the net charge of the protein globule has the same sign as the charge of the polyelectrolyte brush [START_REF] Wittemann | Adsorption of proteins on spherical polyelectrolyte brushes in aqueous solution[END_REF][START_REF] Wittemann | Interaction of proteins with linear polyelectrolytes and spherical polyelectrolyte brushes in aqueous solution[END_REF][START_REF] Becker | Proteins and polyelectrolytes: A charge relationship[END_REF] or microgel [START_REF] Welsch | Adsorption of proteins to functional polymeric nanoparticles[END_REF][START_REF] Yigit | Protein Sorption to Charged Microgels: Characterizing Binding Isotherms and Driving Forces[END_REF] (see also recent comprehensive reviews [START_REF] Achazi | Understanding the interactions of polyelectrolyte architectures with proteins and biosystems[END_REF][START_REF] Czeslik | Adsorption mechanism, secondary structure and local distribution of proteins at polyelectrolyte brushes[END_REF] ).

Although non-electrostatic forces may contribute to the overall proteinpolyelectrolyte interaction balance, it was convincingly demonstrated [START_REF] Wittemann | Adsorption of proteins on spherical polyelectrolyte brushes in aqueous solution[END_REF][START_REF] Wittemann | Interaction of proteins with linear polyelectrolytes and spherical polyelectrolyte brushes in aqueous solution[END_REF] that an increase in the ionic strength in the solution suppresses protein absorption into polyelectrolyte brushes. The latter observation suggest that driving force for the absorption is of electrostatic origin.

A number of hypotheses were proposed to explain this counterintuitive phenomenon. All these hypotheses were based on the fact, that globular proteins comprise amino acid residues with both positive and negative ionic charges. Moreover, these positively or negatively charged residues are weak basic (tertiary amine) or acidic (carboxyl) groups and can be either ionized (through protonation or deprotonation, respectively) or deionized depending on the pH in the surrounding medium. Hence, most of globular proteins can be classified as weak or pH-sensitive polyampholytes. The overall charge of the polyampholyte vanishes at some particular pH=pI corresponding to the isoelectric point (IEP), whereas the polyampholyte acquires net negative or positive charge at pH above or below IEP, respectively.

Importantly, acidic and basic residues on the globule surface are not randomly distributed, but rather form "patches" of pH-dependent negative and positive surface charge. Therefore, even above (below) IEP one fined patches of positive (negative) charge on the globule surface. As it was suggested by M.Ballauff and coworkers, [START_REF] Becker | Proteins and polyelectrolytes: A charge relationship[END_REF][START_REF] Leermakers | On the mechanisms of interaction of globular proteins with polyelectrolyte brushes[END_REF] this patchy charge distribution plays the key role in the protein absorption by similarly charged polyelectrolyte brushes. It was demonstrated theoretically 24 using self-consistent field approach that there is an essential asymmetry in the interaction of brush-forming polyelectrolyte chains with patches of charge with different signs on the globular surface: The free energy gain due to polyelectrolyte chains adsorption onto oppositely charged patch exceeds the free energy penalty for polyelectrolyte depletion from similarly charged patch. This asymmetry is related to localization of small mobile counterions near strongly charged patches on the globule surface and their release upon adsorption of polyelectrolyte chains on these patches, which was also confirmed by molecular dynamics simulations. [START_REF] Yigit | Like-charged proteinpolyelectrolyte complexation driven by charge patches[END_REF][START_REF] Yigit | Interaction of charged patchy protein models with like charged polyelectrolyte brushes[END_REF][START_REF] Xu | Interaction of Proteins with Polyelectrolytes: Comparison of Theory to Experiment[END_REF] As a result, free energy gain due to polyelectrolyte attraction to the oppositely charged patches may overcompensate the free energy losses due to depletion from similarly charged patches even if the latter occupies larger area on the globule surface and provide the same sign of the net charge of the globule as that of the polyelectrolyte brush.

An alternative hypothesis ("charge reversal") proposed in refs [START_REF] Biesheuvel | A modified box model including charged polyelectrolyte brush: self-consistent field theory[END_REF][START_REF] De Vos | Field theoretical analysis of driving forces for the uptake of proteins by like-charge polyelectrolyte brushes: effects of charge regulation and patchiness[END_REF] is based on the fact that both degree of dissociation of carboxylic groups and degree of protonation of the amine groups depend on the local pH, that is protein is a weak (pH-sensitive) polyampholyte. Because of the counterion entrapping, local pH in polyelectrolyte brush (or in polyelectrolyte microgel) differs from that in the bulk of the solution. That is, pH in the polycationic brush is higher and pH in the polyanionic brush is lower than that in the bulk of the solution, pH b . (Here and below subscript "b" refers to the bulk of the solution (the buffer)). Therefore, if pH≤pI≤pH b the protein is charged negatively in the solution, but acquires positive charge inside polyanionic brush. Similarly, if pH b ≤ pI≤ pH the protein is charged positively in the solution but acquires negative charge inside polycationic brush. Hence, the net charge of the protein globule may change the sign upon transfer of the globule from the solution into the similarly charged brush. It is anticipated, that the resulting Coulomb attraction between the brush and the protein drives absorption of the protein by the brush.

Although the charge reversal hypothesis provides a plausible explanation of the protein absorption by charged brush/microgel on the "wrong" side of the IEP, as we demonstrate below, the condition pH≤ pI≤ pH b (or pH b ≤ pI≤ pH) is not sufficient for providing thermodynamic driving force for the absorption of negatively (positively) charged protein by polyanionic (polycationic) brush. A proper account of the free energy of re-ionization of acidic and basic residues leads to stronger requirement rather than simple reversal of charge of the protein upon transfer into the brush for assuring a decrease in the free energy of ionic interactions in the system which provides driving force for such absorption and can make it a spontaneous process.

Re-ionization free energy of a polyampholyte

Consider a polyampholyte molecule (e.g., the protein globule) comprising N + monomer units capable of acquiring positive (elementary) charge upon protonation, and N -monomer units capable of acquiring negative (elementary) charge upon dissociation of a proton. When the molecule is immersed into solution with buffered local pH, the fractions of positively and negatively charged ionizable monomer units are, respectively

α + = (1 + K + /[H + ]) -1 ≡ (1 + 10 pH-pK + ) -1 (1)
and

α -= (1 + [H + ]/K -) -1 ≡ (1 + 10 pK --pH ) -1 (2)
where [H + ] is local concentration of hydrogen ions, K + and K -are respective acidic ionization constants of basic and acidic monomer groups.

The overall net charge of the polyampholyte (measured in the elementary charge units) equals

Q = α + N + -α -N - (3)
A polyelectrolyte brush (or polyelectrolyte microgel) which is in equilibrium with the bulk of the solution with fixed pH b and salt concentration (i.e. with fixed chemical potentials of all mobile ionic species) is characterized by an excess electrostatic potential Ψ (which is also referred to as Donnan potential) with respect to its reference (zero) value in the bulk of the solution. Below we consider negatively charged (polyanionic) brush or microgel, where the excess potential is negative, Ψ ≤ 0. Therefore, local concentration of hydrogen ions inside the brush (or in the microgel) can be expressed as

[H + ] = [H + ] b exp(-eΨ/k B T ) ( 4
)
where concentration of hydrogen ions in the bulk of the solution is by defi-

nition [H + ] b = 10 -pH b .
The free energy of the polyampholyte molecule inserted into the brush can be presented as

F/k B T = N -α -ln α -+(1-α -) ln(1-α -)-α -ln K --α -ψ+α -ln[H + ] b + N + α + ln α + + (1 -α + ) ln(1 -α + ) + α + ln K + + α + ψ -α + ln[H + ] b (5)
which comprises (i) mixing entropies of ionized and non-ionized monomers (logarithmic terms); (ii) electrostatic free energies of positively and negatively charged monomers (the reduced Coulomb potential inside the brush is defined as ψ ≡ eΨ/k B T ); (iii) ionization free energies through respective ionization constants K -and K + = K w /K base,+ where K w = [H + ][OH -] = 10 -14 ; and (iv) accounts for equilibrium with the reservoir with fixed chemical potential of hydrogen ions,

k B T ln[H + ] b .
Minimization of the free energy, eq 5, with respect to α + and α -eq 4 leads to eqs 1 and 2 for equilibrium degrees of ionization of basic and acidic monomer units, respectively. Upon substitution of α + and α -into eq 5 one finds that the free energy contribution due to ionization of positively and negatively charged monomer units of polyampholyte inside the brush can be presented as

F ion /k B T = N + ln(1 -α + ) + N -ln(1 -α -) (6)
and the reference state F ion = 0 corresponds to absence of charges, α + = α -= 0. Transfer of a polyampholyte molecule from the bulk of the solution into the brush/microgel leads to change in ionization degree of both basic and acidic monomer units. The concomitant change in the free energy can be expressed, with the account of eq 6 as

∆F ion /k B T = N + ln 1 -α + 1 -α b+ + N -ln 1 -α - 1 -α b- (7)
where α b+ , α + , α b-, α -and the respective degrees of ionization of basic and acidic monomer units in the bulk of the solution and in the brush/microgel. Since all mobile ions are distributed between the brush and the bulk of the solution according to Boltzmann law, eq 4, lower electrostatic potential inside the brush/microgel implies that local concentration of H + ions inside the brush/microgel is larger than in the bulk of the solution. That is, pH in the brush/microgel is smaller than pH in the bulk of the solution which we denote as pH b . Consequently, as follows from eqs 1 and 2, degree of ionization of acidic monomer units inside polyanionic brush is lower and degree of ionization of basic monomer units inside the polyanionic brush is higher than in the bulk of the solution.

Because basic monomer units are stronger ionized inside the polyanionic brush than in the bulk of the solution, α + ≥ α b+ , the first term in eq 7 is negative. On the contrary, acidic monomer units are weaker ionized inside the polyanionic brush than in the bulk of the solution, α -≤ α b-, and the second term in eq 7 is positive. Hence, the sign of ∆F ion is determined by the balance between re-ionization free energies of basic and acidic monomer units and depends on polyampholyte composition (numbers of acidic and basic monomer units), pH b in the reservoir, ionization constants of basic and acidic monomers and on the shift of the electrostatic potential between the solution and interior of the brush/microgel.

Polyampholyte absorption threshold vs isoelectric points

At this point it is convenient to introduce a set of dimensionless variables:

x = K -/[H + ] b ≡ 10 pH b -pK - (8)
controlled by the bulk value of pH, the asymmetry of ionization constants characterized by parameter

u = K - K + ≡ 10 pK + -pK - (9)
and fraction of positively charged ionizable monomer units

f + = N + N -+ N + ≡ N + N Σ (10)
were N + + N -= N Σ is total number of ionizable (basic and acidic) monomer units in the polyampholyte. Finally,

λ = exp(eΨ/k B T ) (11)
serves as externally controlled (e.g. by salt concentration) parameter. We remark that λ ≤ 1 since Ψ ≤ 0 for an anionic brush/microgel. Disregarding gradients in the charge distribution on grafted (brush-forming) or crosslinked polyelectrolyte chains we can present the excess potential (the Donnan potential) as (see e.g. ref [START_REF] Borisov | Conformations and solution properties of star-branched Polyelectrolytes[END_REF] )

λ ≡ exp( eΨ k B T ) = - α p c p c s + 1 + ( α p c p c s ) 2 ≤ 1 ( 12
)
where α p is fraction of (negatively) permanently charged monomer units and c p is polymer concentration inside the brush/microgel and c s is concentration of all types of monovalent mobile ions in the bulk of the solution. Here we remark that in brushes/microgels formed by weak (pH-sensitive) polyelectrolytes the degree of ionization α p depends itself on pH and salt concentration in the buffer and increases as a function of c s that was predicted theoretically [START_REF] Zhulina | Theory of ionizable polymer brushes[END_REF][START_REF] Klein Wolterink | Screening in Solutions of Star-Branched Polyelectrolytes[END_REF] and confirmed in experiments [START_REF] Konradi | Interaction of Poly(methacrylic acid) Brushes with Metal Ions: Swelling Properties[END_REF][START_REF] Del Castillo | Large Changes in Protonation of Weak Polyelectrolyte Brushes with Salt Concentration -Implications for Protein Immobilization[END_REF] and computer simulations. [START_REF] Landsgesell | Grand-Reaction Method for Simulations of Ionization Equilibria Coupled to Ion Partitioning[END_REF] Using eqs 1, 2 and 4 the degree of ionization of basic and acidic monomer units in the bulk of the solution and inside the brush/microgel can be expressed as

α b+ = u x + u ; α + = u u + λx ( 13
)
α b-= x 1 + x ; α -= λx 1 + λx (14)
The insertion free energy, eq 7, can be then expressed as

∆F ion N Σ k B T = ln x + 1 λx + 1) 1-f + λ(x + u) λx + u f + (15)
The condition ∆F ion ≤ 0 is fulfilled when

x + 1 λx + 1 1-f + f + λ(x + u) λx + u ≤ 1 ( 16
)
that is, at x ≤ x abs where x abs is found from the equation

x abs + 1 λx abs + 1 1-f + f + λ(x abs + u) λx abs + u = 1 (17)
The net charge of the polyampholyte molecule in the bulk of the solution and inside the brush (gel) can be expressed as

Q b = α b+ N + -α b-N -= N Σ f + x 2 + 2ux + u (x + 1)(x + u) - x x + 1 ( 18
)
and

Q in = α + N + -α -N -= N Σ f + (λx) 2 + 2uλx + u (λx + 1)(λx + u) - λx λx + 1 (19)
respectively. As follows from eqs 18 and 19,

Q b (x = 0) = Q in (x = 0) = N Σ f + Q b (x = ∞) = Q in (x = ∞) = -N Σ (1 -f + )
The net charge of the polyampholyte in the bulk of the solution vanishes, Q b = 0, in the isoelectric point x = x IEP specified as

x IEP = u 2 • 2f + -1 1 -f + + u 2 4 2f + -1 1 -f + 2 + uf + 1 -f + (20)
As follows from eqs 18, 20 the net charge of the polyampholyte in the bulk of the solution

Q b ≤ 0 at x ≥ x IEP and Q b ≥ 0 at x ≤ x IEP .
Inside the brush the net charge of the polyampholyte vanishes at x = x IEP where

x

IEP = λ -1 x IEP (21)
Hence, the net charge of the polyampholyte inside the brush vanished at x = x IEP ≥ x IEP , i.e., at pH b ≥ pI. In the range of x IEP ≤ x ≤ λ -1 x IEP the polyampholyte is charged negatively in the bulk of the solution, but positively inside the brush/microgel, i.e., charge sign inversion occurs upon insertion of the polyampholyte into the brush/microgel. It is easy to check that in the particular case of u = 1, eq 20 has a simple form:

x

IEP = f + 1 -f + ( 22
)
and solution of eq 17 can be presented as

x abs = 1 -λ f + λ f + -λ (23)
The analysis of eqs 17, 20, 21 proves that the inequality

x IEP ≤ x abs ≤ x IEP ≡ λ -1 x IEP (24)
holds at arbitrary u as long as λ ≤ 1 (negatively charged brush), which is also illustrated in Figures 1 and2.

In Figure 1 the dependences of the polyampholyte molecule net charge outside and inside the brush, Q b and Q in , respectively, and their difference, ∆Q = Q in -Q b , are plotted as a function of x = 10 pH b -pK -. As one can see from Figure 1, in the limit of x → ∞ (extreme alkaline pH b range) the polyampholyte is negatively charged both in the buffer and inside the brush. A decrease in x (in pH b ) leads to a decrease in the magnitude of the polyampholyte negative charge both in the buffer and inside the brush. Inside the brush the polyampholyte charge vanishes at x = x IEP while the polyampholyte in the buffer remains negatively charged. Further decrease in pH b results in increasing positive charge of the polyampholyte inside the brush (and simultaneous decreasing magnitude of the negative charge outside the brush), but the insertion free energy penalty remains positive, i.e., there is no electrostatic driving force for the polyampholyte uptake in the brush. Only when pH b drops down to the value corresponding to x abs , the insertion free energy becomes negative.

Hence, ionic interactions provide driving force for spontaneous absorption of polyampholyte by anionic brush/microgel in the range of the pH b corresponding to the conditions x ≤ x abs , when the net charge of the polyampholyte is still negative in the bulk of the solution, but positive inside the brush. However, the inversion of the polyampholyte charge sign inside the brush does not necessarily imply negative insertion free energy ,i.e.,, there is no driving force for the absorption in the x(pH b ) range corresponding to x abs ≤ x ≤ λ -1 x IEP and the polyampholyte is repelled from the polyelectrolyte brush.

As it follows from eq 20, the value of x IEP corresponding to vanishing net polyampholyte charge in the solution depends only on the polyampholyte composition (fraction of positive charged residues f + ) and ratio of ionization constants u = K -/K + . The values of x IEP and x abs depend additionally on λ, that is, on the magnitude of excess electrostatic potential inside the brush/microgel, see eqs 17, 21.

The dependences of the threshold values of x IEP , x IEP , and x ads on f + (at λ, u = const), on u (at λ, f + = const and on λ (at f + , u = const) are presented in Figures 2a,b,c.

Figure 2a and 2b illustrate that x IEP , x IEP and x abs increase (corresponding pH b values grow) when the fraction f + of basic monomer units in the polyampholyte or the value of pK + increase. As one can see from Figure 2c, x abs and x IEP grow upon a decrease in λ (increasing magnitude of negative electrostatic potential inside the brush), whereas x IEP (the isoelectric point in the solution) is obviously independent of λ. Remarkably, both x abs and xIEP /x abs , do increase upon a decrease in λ. This is easy to check in the particular case of u = 1 when

x IEP /x IEP = λ -1 ; x abs /x IEP ∼ λ -f + ; x IEP /x abs ∼ λ f + -1
Hence, for smaller λ (stronger anionic charge of the brush or lower ionic strength of the solution), both the polyampholyte charge sign inversion and the insertion free energy sign inversion occur at larger x, i.e. at higher pH b . Therefore, at given pH b ≥ pI the uptake of polyampholyte in the brush/microgel can be provoked by a decrease in λ caused, e.g, by a decrease in the ionic strength of the solution which is consistent with experimental observation. [START_REF] Wittemann | Interaction of proteins with linear polyelectrolytes and spherical polyelectrolyte brushes in aqueous solution[END_REF] In Figure 3 we present 3D diagram of states for the polyampholyte molecule as a function of composition, f + , excess electrostatic potential inside the brush/microgel, λ = exp(eΨ/k B T ), and pH in the solution, for a typical set of ionization constants K -= 5.5 and K + = 4.5. The lower (blue) surface corresponds to the pI of the polyampholyte in the solution, above it the polyampholyte in the buffer is negatively charged. The upper (green) surface corresponds to the polyampholyte charge inversion threshold: below this pH value in the buffer the polyampholyte gets positively charged (the charge sign inversion occurs) upon insertion from the buffer into the brush/microgel. The brownish surface sandwiched between the blue and the green ones corresponds to absorption threshold: the re-ionization free energy upon insertion of the polyampholyte into the brush/microgel is positive at higher pH b and negative at lower pH b .

Crossing of the boundary pH b (x IEP ) upon a decrease in λ or in pH b , as well as upon an increase in f + leads to the charge sign inversion for the polyampholyte inserted into the brush, but does not imply appearance of the driving force for the absorption. Further decrease in λ or in pH b (an increase In Figures 4a andb we show the cross-sections of the 3D diagram presented in Figure 3 corresponding to variation of fraction f + of cationic groups at constant λ (Figure 4a) and to variation in the excess electrostatic potential inside the brush/microgel, that is, variation in λ at constant composition of the polyampholyte (constant f + ). Evidently the larger the fraction f + of cationic groups, the higher are pH thresholds corresponding to charge sign inversion and to the onset of absorption. However, the difference between these two characteristic pH b values does not noticeably change upon variation in f + . The pH-range at which re-ionization favours absorption widens and pH IEP and pH abs increase when λ decreases that corresponds to larger difference of the electrostatic potential between the brush/microgel and the buffer. Such decrease in λ can be caused, e.g., by increasing fraction of negatively charged monomer units in the brush or by a decrease in the salt concentration in the buffer (see eq 12). Obviously at λ → 1 all the three surfaces separating different charge inversion/absorption regimes intersect at the line corresponding to IEP of the polyampholyte in the solution, given by eq 20.

Comparison of theory to experiment

Our theory describes the electrostatically driven uptake of protein from buffer solution by polyelectrolyte brushes/microgels. We model polyelectrolyte brush/microgel as a reservoir comprising polyelectrolyte chains with the average concentration c p of monomer units, and mobile monovalent ions distributed via Donnan equilibrium with buffer solution. The gradients in polymer concentration and chain ionization are neglected. The electrostatic potential difference Ψ between the interior of polyelectrolyte brush/microgel and buffer solution is accounted for via parameter λ (eq 12) in terms of the average concentration c p of the monomer units, their degree of ionization, α p , and the concentration c s of mobile ions in buffer solution. The brush geometry and chemical nature of the tethered polyelectrolytes are accounted via the dependence c p (α p , c s ). For strong polyelectrolytes, e.g., poly(styrenesulfonic acid), α p is constant specified by the degree of polystyrene sulfonation, while for weak polyelectrolytes, e.g., poly(acrylic acid), α p is pH-sensitive, and is governed by the brush geometry, grafting density of polyelectrolyte chains, pH b and the ionic strength in buffer solution. [START_REF] Borisov | Conformations and solution properties of star-branched Polyelectrolytes[END_REF] To confront the predictions of our model to experiment, we focus on uptake of globular proteins by colloidal polyelectrolyte brushes. [START_REF] Wittemann | Adsorption of proteins on spherical polyelectrolyte brushes in aqueous solution[END_REF][START_REF] Guo | Spatial Dimensions of Colloidal Polyelectrolyte Brushes As Determined by Dynamic Light Scattering[END_REF] Absorption of bovine serum albumin (BSA) in poly(acrylic acid) (PAA) spherical brushes decorating poly(styrene) latex particles was thoroughly investigated in ref. [START_REF] Wittemann | Adsorption of proteins on spherical polyelectrolyte brushes in aqueous solution[END_REF] A detailed characterization of the experimental system (L16) was performed in ref. [START_REF] Guo | Spatial Dimensions of Colloidal Polyelectrolyte Brushes As Determined by Dynamic Light Scattering[END_REF] It was concluded by the authors that BSA uptake by PAA spherical brushes occurred via two, time separated stages. During the first (quick and reversible) stage, BSA progressively accumulates inside PAA brushes. During the second (later) stage, protein molecules bind to polyelectrolyte chains exhibiting minor changes in the secondary structure. At this stage, an additional, non-electrostatic protein -chain interactions are plausible. Notably, our current model addresses the first stage of the electrostatically driven protein uptake, in which the structure of the polyelectrolyte brush/microgel remains essentially unperturbed by interaction with proteins.

The pH-sensitivity of PAA brush was experimentally confirmed by 3fold increase in the brush thickness L upon an increase in pH b from pH b = 2 up to neutral conditions, pH b = 7, at low salt, c s = 10mM (Figure 2 in ref [START_REF] Wittemann | Adsorption of proteins on spherical polyelectrolyte brushes in aqueous solution[END_REF] ). The increase in L of the spherical brush formed by polyelectrolytes with contour length L c = 209 nm and N p acidic groups per chain (tethered with grafting density σ to the surface of core particle with radius R c ≈ 50 nm), from L ≈ 100 nm (pH b = 5) to L ≈ 120 nm (pH b = 7) leads to the decrease in the average concentration of monomer units,

c p = N p σL -1 /[1 + L/R c + (L/R c ) 2 /3]
, approximately twice. At low salt (i.e., in the osmotic polyelectrolyte brush regime with α p c p /c s 1), the degree of ionization α p of monomer units with dissociation constant K p varies as The IEP of BSA was experimentally estimated as pI = 5.1, indicating that charging inversion of the protein occurs in the examined range of pH b . That is, overall protein charge Q b changes its sign from positive to negative at pH b > 5.1. By modelling BSA molecule as a compact globule containing N Σ available ionizable groups with dissociation constants K -= K + (i.e., u = 1), the free energy of protein insertion (eq 15) reduces to

α p ∼ (K p /c p ) 1/2 . 15 As a result, parameter λ ∼ c s /(α p c p ) ∼ c
∆F ion N Σ k B T = ln x + 1 λx + 1 -ln x abs + 1 λx abs + 1 (25)
As it follows from Figure 4, the choice of f + and λ affects the value of pH abs , corresponding to the protein absorption threshold (shown by dashed lines in Figures 4a and4b). Recall that ∆F ion < 0 (and the protein absorption is promoted) if x < x abs , or, equivalently pH b < pH abs . By choosing, e.g., f + = 0.5 and λ = 0.05, one finds ∆F ion < 0 for experimentally examined [START_REF] Wittemann | Adsorption of proteins on spherical polyelectrolyte brushes in aqueous solution[END_REF] values of pH b = 5.1 and 6.2, and ∆F ion > 0 for pH = 7.2. The reduction in absolute value of ∆F ion < 0 is also expected upon addition of salt in buffer solution via the increase in λ ∼ c s .

In the electrostatically driven protein uptake, the slope of the equilibrium absorption isotherm at low concentrations of BSA is governed by exp(-∆F ion /k B T ). By introducing the difference between x abs and x, ∆x = (x abs -x) with 0 < ∆x x abs , the free energy of protein absorption reduces to

∆F ion N Σ k B T = ln x abs + 1 -∆x λx abs + 1 -λ∆x -ln x abs + 1 λx abs + 1 ≈ ∆x (1 -λ -1 ) (1 + x abs )(λ -1 + x abs ) ∼ -∆x (26)
since λ = exp(eΨ/k B T ) < 1. Therefore a larger deviation from the absorption threshold, ∆x = (x abs -x) > 0, should lead to the larger slope of absorption isotherm. These expectations are in accord with the data in Figures 6 and8 in ref. [START_REF] Wittemann | Adsorption of proteins on spherical polyelectrolyte brushes in aqueous solution[END_REF] A decrease in the initial slope of the absorption isotherm is clearly seen in Figure 6 upon increasing c s . As for the pH-dependence of the slope in Figure 8, it is difficult to distinguish the difference in slopes for for pH b = 5.1 and pH b = 6.2 due to extremely sharp increase in the absorbed amount at small concentrations of BSA. However, at pH b =7.2, the dramatic decrease in slope is clearly seen, indicating reduction/elimination of the electrostatic driving force. Overall, the conclusion about the electrostatic origin of BSA absorption by PAA colloidal spherical brushes [START_REF] Wittemann | Adsorption of proteins on spherical polyelectrolyte brushes in aqueous solution[END_REF] is in reasonable agreement with our theory.

Discussion and conclusions

To summarize, our theory proves that transfer of a polyampholyte molecule (e.g., globular protein) containing pH-sensitive acidic and basic monomeric groups from the buffered solution into a similarly charged polyelectrolyte brush (or a microgel) can lead to a decrease in the ionic part of the free energy of the system thus making uptake of the polyampholyte molecule thermodynamically favorable. The negative balance in the free energy is assured by overcompensation of the free energy losses due to suppression of ionization of similarly (with respect to the brush) charged monomeric groups by gain in the free energy due to promoted ionization of the oppositely charged ones.

One of the predictions of our theory is separation of the protein absorption threshold by a pH-window from IEP in buffer solution. For both weak and strong polyelectrolyte brushes/microgels, λ = exp(eΨ/k B T ) ∼ c s /(α p c p ) decreases upon decreasing c s in buffer solution and increasing charge concentration (α p c p ) in the polyelectrolyte brush/microgel, widening the window between x abs and x IEP as x abs /x IEP ∼ λ -f + (see also Figure 2c).

Hence, in full accordance with experimental observations, [START_REF] Wittemann | Adsorption of proteins on spherical polyelectrolyte brushes in aqueous solution[END_REF][START_REF] Wittemann | Interaction of proteins with linear polyelectrolytes and spherical polyelectrolyte brushes in aqueous solution[END_REF] our theory correctly predicts that an increase in salt concentration at constant buffer pH b leading to an increase in λ = exp(eΨ/k B T ) reduces the driving force and may ultimately suppress uptake of a polyampholyte (protein) by the similarly charged polyelectrolyte brush or microgel at sufficiently high ionic strength of the solution.

When the net re-ionization free energy balance is negative, the polyampholyte net charge changes its sign upon transfer from the bulk of the solution in the brush (charge inversion). However, the opposite is not true, i.e., in spite of earlier claims, 28 the charge inversion is not sufficient for making the absorption thermodynamically favorable. In other words, even if the charge inversion of polyampholyte inside the brush/microgel occurs, the reionization free energy upon insertion may be still positive. A closer approach to the IEP in terms of pH b is required to make insertion free energy negative.

Upon calculating the contribution to the free energy of the system related to the presence of ionically charged (ionizable) monomeric groups as a function of the polyampholyte parameters (f + , K -, K + ) and environmental controlled parameters (pH b , λ = exp(eΨ/k B T )) we used a number of simplifying approximations.

At fixed pH b and c s , uptake of protein molecules from buffer solution by polyelectrolyte brush/microgel is presumed to occur without changing the state of polyelectrolyte brush (i.e., at a fixed value of λ). This assumption is justified for small protein concentrations in buffer solution and no binding of protein molecules to polyelectrolyte chains. Under these conditions, the protein uptake is electrostatically driven, and is accompanied by re-ionization of protein amino acids. For naturally disordered proteins, re-ionization could incorporate all N Σ ionizable amino acids in the primary sequence. How-ever, for globular proteins with compact ternary structure, the number N Σ of re-ionizable amino acids could be much less and associated merely with protein/water interface. Notably, the initial slope of the protein absorption isotherm is governed by exp(-∆F ion /k B T ) and is mediated, in particular, by buffer parameters c s and pH b .

Another essential assumption used is neglecting (anti)cooperativity of reionization process, that is, dependence of the effective ionization constants of the ionogenic groups on their position inside polyampholyte molecule and on the actual charge of the polyampholyte. This approximation implies that ionization of each acidic or basic monomer group is controlled solely by average electrostatic potential inside the brush/microgel and is not affected by charge state of other ionogenic groups in the polyampholyte. Evaluation of the effective ionization constants requires specifying the spacial distribution of ionogenic groups in the polyampholyte molecule (on the surface of the protein globule) that is far beyond the scope of mean-field approach applied here. We also neglected spacial gradients in the charge density and related electrostatic potential distribution inside the brush/microgel. Furthermore, we assumed pH and λ = exp(eΨ/k B T ) to be independently controlled parameters that is the case for strong polyelectrolyte brushes/microgels but not for weak (pH-sensitive) ones, where electrostatic potential Ψ is a function of pH in the buffer as well. The dependence of λ on pH b introduces new feature in the pH-controlled polyampholyte absorption by weak polyelectrolyte brush/microgel and will be studied in our forthcoming publication. The interplay between pH-dependent ionization of weak polyelectrolyte gel and absorbed by the gel proteins was recently investigated by Longo and Szleifer, 37 who pointed that re-ionization of the protein upon transfer into the gel significantly enhances absorption.

Finally, we have to stress that we have evaluated only the contributions to the free energy change related to re-ionizaton of ionogenic monomer groups upon insertion of a polyampholyte (or a protein globule) into polyelectrolyte brush/microgel . There are other contributions to the free energy which have to be accounted for to evaluate the net free energy balance. The insertion of a globule into the brush requires the work to be done against excess osmotic pressure acting in the brush. This leads to a positive contribution ∆F vol to the full insertion free energy which has to be overcompensated by negative ionic contribution ∆F ion evaluated above. The re-ionization may provide a dominant driving force for uptake of weakly charged polyampholytes, intrinsically disordered proteins or, in some cases, of globular proteins with quasi-uniform surface distribution of basic and acidic residues. For most of globular proteins with pronounced patchiness in distribution of positive and negative surface charge the asymmetry of interactions of both with brush-or gel-forming polyelectrolyte chains and counterion release can be equally or even more relevant in the overall balance of interactions leading to the protein uptake on the "wrong" side of the IEP.

In spite of all these limitations and reservation, we believe that our theory provides a valuable insight into generic physical mechanisms of spontaneous uptake of weak polyampholytes (e.g. globular proteins) by similarly charged polyelectrolyte brushes (or microgels).

Figure 1 :

 1 Figure 1: Dependences of of the polyampholyte molecule net charge outside and inside the brush, Q b and Q in , respectively, on x = 10 pH b -pK -at f + = 0.5; u = 1; λ = 0.2

Figure 2 :Figure 3 :

 23 Figure2: The dependences of x IEP , x IEP , and x ads on f + for λ = 0.1; 0.5; 0.9; u = 1 (a), on u for λ = 0.1; 0.5; 0.9; f + = 0.5 (b) and on λ for u = 0.1; 1.0; 10; f + = 0.5(c).

Figure 4 :

 4 Figure 4: The crossections of the 3D diagram of states (shown in inserts) at constant values of λ (indicated in the figure) and varied fraction of cationic monomer units f + (a) and at constant values of f + (indicated in the figure) and varied λ (b).

  approximately 1.3 times) upon an increase in pH b from 5 to 7.

Acknowledgments

This work was financially supported by the Government of the Russian Federation (Grant 08-08), RFBR grant 20-53-12036 and by the European Union's Horizon 2020 research and innovation program under the Marie Sklodowska-Curie programme (grant agreement No 823883).

Conflict of interests

The authors declare no conflict of interest.


Saheli Mitra

Susana Marín-Aguilar

Srikanth Sastry

Frank Smallenburg

Giuseppe Foffi

Correlation between plastic rearrangements and local structure in a cyclically driven glass

The correlation between local structure and the propensity for structural rearrangements has been widely investigated in glass forming liquids and glasses. In this paper we use the excess two-body entropy S 2 and tetrahedrality n tet as the per-particle local structural order parameters to explore such correlations in a three-dimensional model glass subjected to cyclic shear deformation. We first show that for both liquid configurations and the corresponding inherent structures, local ordering increases upon lowering temperature, signaled by a decrease in the two-body entropy and an increase in tetrahedrality. When the inherent structures, or glasses, are periodically sheared athermally, they eventually reach absorbing states for small shear amplitudes, which do not change from one cycle to the next. Large strain amplitudes result in the the formation of shear bands, within which particle motion is diffusive. We show that in the steady state, there is a clear difference in the local structural environment of particles that will be part of plastic rearrangements during the next shear cycle and that of particles that are immobile. In particular, particles with higher S 2 and lower n tet are more likely to go through rearrangements irrespective of the average energies of the configurations and strain amplitude. For high shear, we find very distinctive local order outside the mobile shear band region, where almost 30% of the particles are involved in icosahedral clusters, contrasting strongly with the fraction of < 5% found inside the shear band.

I. INTRODUCTION

The mechanism involved in how a liquid loses its fluidity upon decreasing temperature or increasing density has been the subject of intense research investigations for several decades now 1 . One interesting feature that emerges in liquids close to the glass transition is dynamical heterogeneity, where over time scales corresponding to the α-relaxation time, some regions of the systems have significantly higher mobility than others. A growing body of literature now supports the existence of a correlation between this heterogeneity and the local structure [2][3][4][5][6] . One of the pioneering ideas in analysing the local structure on glasses was proposed by Charles Frank in 1952 indicating the possible prevalence of icosahedral clusters in the glassy regime 7 . Later, many studies corroborated the presence of locally favored structures in the glassy regime related to the slowdown of dynamics, with icosahedral clusters being the focus of attention in several studies [8][9][10][11][12][13][14] . Along similar lines, there have been proposals of simpler tetrahedronbased order parameters capable of capturing the changes in local structure and dynamics 11,[15][16][17][18][19] . In particular, a recent development was the introduction of the concept of tetrahedrality of the local structure n tet , which measures the number of tetrahedral clusters each particle is involved in 19 , based on the notion that, at least for some glass formers, most of the locally favoured structures (including icosahedral) can be dea) Electronic mail: s.marinaguilar@uu.nl b) Electronic mail: frank.smallenburg@universite-paris-saclay.fr c) Electronic mail: giuseppe.foffi@universite-paris-saclay.fr composed into tetrahedra. It was found that the particles with higher values of n tet are strongly correlated with slower dynamics. These studies are based strictly on local structures whose ability to fully predict glassy dynamics has been questioned 20 . In general, while a full understanding of the dynamics may require information beyond the static local environment, local structures represent an important starting point as testified by the success of machine learning based approaches that can be trained to predict the local mobility of different regions in a glassy fluid 3,21,22 . Moreover, the relevance of icosahedral and polytetrahedral local structure has been recently highlighted by novel unsupervised machine learning approaches, such as community inference 6 and neuralnetwork-based auto-encoders 5 . Apart from the heterogeneity in supercooled liquids, local structure can play an important role in the plastic rearrangements occurring in a sheared solid. When shearing a crystalline solid, rearrangement events are normally expected to occur in the vicinity of crystal defects. This idea can be extended to sheared glasses, where certain spots in the system may be "weaker" than others, and hence more prone to plastic rearrangement. These "soft spots" [23][24][25] or "shear transformation zones" 26,27 can be seen as analogues of crystal defects in glassy materials. In recent years, a variety of approaches have been explored to predict such "soft spots", including e.g. machine learning 28 , local vibrational modes 25 , local yield stress 29 . A summary of the efficiency of the different methods is reported in a recent paper by Richard et al. 30 , in which the authors compare the performance of a variety of methods to predict plasticity due to uniform shear deformation. For a 2d glass former, they demonstrated that rearrangements are indeed deeply encoded in the structure 30,31 . At the experimental level, soft colloidal glasses under thermal cycling display high correlation between local rearrangements and local twobody entropy 32 . In the aforementioned works, the role of local structure in amorphous systems undergoing cyclic mechanical deformation has not been studied in detail. Cyclic shear deformation is a commonly used technique to test mechanical properties of materials [33][34][35] , memory effects 36,37 , self organization 38 , and annealing of glass 39 . When a glass is deformed in the quasi-static limit at zero temperature 40 , compared to uniform shearing, the cyclic shear displays a sharp yielding transition from an absorbing to a diffusive state at a critical strain amplitude γ y 41 . Below this critical threshold γ max < γ y , after several training cycles of deformation, the system reaches a steady state. In this state, the plastic rearrangements during a deformation cycle are fully reversible 37,42,43 and localized. Above yielding, rearrangements are connected to large displacements and are irreversible. As a consequence plastic events become spatially organized, leading to the appearance of a shear band 44,45 . In this paper we use a binary repulsive 3d glass former to explore the structural properties of particles undergoing plastic rearrangements due to cyclic shear deformation. We analyze how the local structural order of the system is correlated with these rearrangements. In particular, we compute the perparticle two-body excess entropy S 2 and tetrahedrality n tet , the latter not having been explored yet in the context of plastic rearrangements. Our analysis focuses on the steady state and we look for structural differences between the local environments for subsets of particles that exhibit the largest and the smallest displacements in successive cycles. The paper is organized as follows: In section II, we discuss the glass former model in detail and we describe the simulation methods. In section III we introduce the structural descriptors. Thereafter, in section IV, we present our results in the following order. We first report (i) the variation of mean local ordering in liquid and inherent structures (IS) using the n tet and S 2 . (ii) We correlate the liquid structures and the corresponding IS at different temperatures. We then sample a high and a low temperature glass and shear them athermally for many deformation cycles with strain amplitudes γ max . (iii) In the steady state we compute the mean values of our structural order parameters as γ max varies in a range across the yielding amplitude γ y . (iv) From one cycle to the next we classify mobile and immobile particles with larger and smaller rearrangements respectively by computing the non-affine displacements D 2 min 26 . (v) In terms of n tet and S 2 we examine whether there is a difference between the mean local order of the mobile and immobile particles. (vi) Later, we use the topological cluster classification algorithm (TCC) 46 to point out the different cluster association between the two classes. (vii) Finally, we explore the structure inside and outside the shear band for the sheared glasses above the yielding transition. Our results show that there is a clear structural difference in local arrangements of the particles involved in the largest and smallest plastic rearrangements. Similarly, above yielding, we find that particles inside and outside the shear band have strongly different local environments.

II. METHODS

A. Glass-former model and Simulation

As a model glass system, we explore the behavior of purely repulsive Wahnström (WH) model 47 . It consists of a 50 : 50 mixture of particles with additive diameters interacting through a Lennard-Jones (LJ) potential,

V αβ (r) =    4ε αβ σ αβ r 12 - σ αβ r 6 -V r c , r < r c 0, r ≥ r c (1
) where α and β denote the type of particle (A or B), the cut-off distance is defined as r c = 2 1 6 σ αβ and V r c is the value of the LJ potential evaluated at r c . With this choice of the cut-off, only the repulsive part of the potential is retained. The potential parameters are defined with respect to type A:

σ BB /σ AA = 1.2, σ AB /σ AA = 1.1 and ε AA = ε BB = ε AB . We fix the total packing fraction to φ = π 6V (N A σ 3 AA + N B σ 3 BB ) = 0.
58, where V is the simulation box volume and the total number of particles N = N A + N B = 64000.

We perform molecular dynamics simulations using LAMMPS [START_REF] Plimpton | Fast parallel algorithms for short-range molecular dynamics[END_REF] with a fixed time step size of dt = 0.005τ, where

τ = mσ 2
AA /ε AA is the time unit, with m the particle mass. We use reduced LJ units where σ AA is the unit for distance and ε AA the energy unit. We prepare a set of equilibrated configurations at temperature T by running NV T molecular dynamics for 2 × 10 5 (high T ) -2 × 10 6 (low T ) time steps followed by constant energy (NV E) relaxation. From these equilibrated samples, we obtain the corresponding inherent structures (IS) by minimizing the potential energy, using conjugate gradient minimization, with a tolerance 10 -16 . The IS configurations are the "glass" configurations which we subject to shear deformation, which we associate with the temperature of the liquid simulations from which they are generated. The range of T ∈ [0.7, 2.0] for this WH system is sufficient to show the dynamical slowdown from high to low temperature as expected in a typical glass-former.

Next, to study the plastic re-arrangements in glasses, we focus on one high temperature (T = 1.5) and one low temperature (T = 0.7) glass, both quenched to the IS and sheared using the athermal quasi static shear (AQS) protocol 40,[START_REF] Maloney | [END_REF] . In AQS, the xz plane of a chosen IS configuration is sheared in small strain steps dγ = 0.0002 in one direction and each deformation step is followed by an energy minimization. A schematic representation of an AQS step is presented in Fig. 1. This is continued until the strain values reach a specified amplitude γ max in either direction, at which the strain direction is reversed. Therefore, one complete cycle of deformation has FIG. 1. Schematic representation of an Athermal Quasi Static (AQS) deformation move. In the top row we present a cartoon of the configuration and in the bottom one a simplified potential energy landscape. An initial configuration in its energy minima (left column) is deformed with small strain steps (central column). In this specific case, the system is not at a minimum anymore and the following minimization leads it into a new minima that corresponds to a new configuration (right column).

the following sequence:

γ = 0 → γ max → 0 → -γ max → 0. (2)
Hence, each cycle consists of 4 * γ max /dγ steps.

For each value of the parameter γ max , we deform the system for 100 -600 cycles until a steady state is reached. The energy of stroboscopic configurations, sampled at the end of each cycle with γ = 0, will depend on γ max . At γ max below the yielding transition, γ max < γ y where γ y is the yielding amplitud of the system, the energies attain a constant value in the steady state. For (γ max > γ y ), the energies fluctuate around a mean value in the steady state. In our WH system, γ y ≈ 0.06. For higher amplitudes than γ y the system yields, becoming diffusive.

B. Local structure observables

Local entropy

To characterize local structure at the particle level, it is useful to first consider a structural descriptor that is known to cor-relate with dynamical slowdown below the onset temperature. For this purpose, we consider the local entropy S 2 , defined in terms of pairwise structural correlations. In general, the entropy S can be expanded in terms of multi-particle correlation functions as S = S 1 + S 2 + S 3 + .., where S 1 is the ideal gas value, S 2 is the two-body excess entropy which can be calculated in terms of the pair distribution function, S 3 involves three-body correlations, etc. 50 . The two-body term S 2 [51][52][53] has been widely used in the context of transport coefficients and plasticity. In general, it measures the loss of entropy due to positional correlations with a lower value of S 2 corresponding to a more ordered structure. Even though S 2 has been introduced long ago 54,55 , it is a useful concept that continues to be widely used 17,[56][57][58] . As a matter of fact, recently, scaling relations have been reported between S 2 and diffusivity in glassy systems 59 and relaxation rates in cyclic sheared systems [START_REF] Galloway | Proc. Natl. Acad. Sci. U.S.A[END_REF] . In order to look at this quantity on a single-particle level, the two-body entropy of particle i can be defined as:

S i 2 = -2πρk B ∞ 0 [g i m (r) log g i m (r) -g i m (r) + 1]r 2 dr, (3)
where g i m (r) is the mollified radial distribution function 58,[START_REF] Piaggi | [END_REF] ,

g i m (r) = 1 4πNρr 2 ∑ i = j 1 2πσ 2 exp [-(r -r i j ) 2 /(2σ 2 )]. (4)
with r i j the distance between the i th and j th particle and σ a parameter setting the width of the Gaussian kernel. This perparticle radial distribution function is well behaved thanks to the broadening of the neighbor position from a delta peak to a narrow Gaussian. The parameter σ is chosen such that the global average of g i m (r) is close to the true radial distribution function g(r) and we have a smooth integral to compute S 2 . For our WH system, we choose σ = 0.09 and for each particle the integration in Eq. 3 is taken from zero to r max = 5.0. With this approach we obtain the per particle excess entropy that we call simply S 2 in the rest of the paper. When averaged over all particles in the system, and over different configurations, we obtain the average two-body entropy S 2 .

By its nature, S i 2 provides a measure for the local radial ordering of neighbors around particle i: many neighbors at the same absolute distance r, as one might find in highly symmetric local environments, will result in a low value of S i 2 . Hence, a more negative value of S 2 indicates higher local order.

Tetrahedrality

The second descriptor we characterize is the tetrahedrality of the local structure, n tet , which measures, for each particle, the number of tetrahedra it is involved in 19 . As some of the clusters that have been found in supercooled liquids, such as the icosahedral cluster, can be decomposed in a large number of tetrahedron, the tetrahedrality is a good descriptor of the local environment of a particle that correlates strongly with glassy dynamics in hard-sphere-like systems 5,19 .

In order to quantify the tetrahedrality, we first identify the neighbors of the particles through a modified Voronoi construction identical to the one done in the Topological Cluster Classification algorithm 46 . We define a tetrahedral cluster as a cluster formed by four particles which are all considered nearest neighbors of each other according to the Voronoi construction. For an individual particle, n tet is than simply the number of distinct tetrahedral clusters that include this particle. Finally, we can also characterize the global structure of a system through the average of the per-particle tetrahedrality n tet 19 . A higher value of n tet generally is related to more compact and long-lived clusters such as the icosahedral cluster.

III. RESULTS

A. Liquid and Inherent Structures

We begin our study by exploring the behavior of both S 2 and n tet for both liquid and inherent structures at different temperatures. In Fig. 2, we show the dependence of both structural quantities for temperatures between T = 0.7 -2.0.

In the liquid, upon lowering the temperature, the average local order increases as it can be deduced by the increase in the value of n tet and decrease of S 2 . As expected, for all temperatures, the inherent structure presents a higher degree of ordering that translates into a larger n tet and smaller S 2 in comparison to the liquid. However, for the IS, these quantities become essentially independent of T at sufficiently high temperatures. This trend is consistent with the previous observation of an increase of the number of locally favored structures (LFS) 58 when the dynamical slowdown start occurring and the aforementioned temperature can be identified as the onset temperature at which the dynamics start to be dominated by the potential energy landscape 62 . Above this temperature, the dynamics are typical of a high-temperature liquid and only marginally influenced by the energy landscape. As a consequence, in this regime, the system samples several independent minima of the IS and S 2 and n tet are insensitive to temperature variations. This means that the system explores a large range of energy basins with no specific bias toward a particular energetic level. Below the onset temperature, the system starts to explore preferentially basins with a low energy that decrease with temperature. start to be dominated by the landscape and the system tends to spend increasing time within individual basins corresponding to the IS. Consequently, an increased correlation may be expected between structural parameters for the liquid and the corresponding IS. To verify this, we inspect the relation between the values of our observables before and after quenching. In Fig. 3 a) and b) we show the 2d density plot of the S 2 and n tet values at temperature T = 0.7. We can observe a clear correlation between the observables in the liquid state and the IS. Note that in Fig. 3 b) there is an increase of correlation at a value of n tet = 20, corresponding to particles located in the center of icosahedral clusters. Hence, we can deduce that some specific structural motifs remain unchanged when liquid configurations are mapped to the IS. Clearly, this correlation might be temperature dependent. In order to have a broader picture of the correlation between the two sets of configurations (liquid and IS) at different temperatures, we calculate Spearman's rank order correlation 63 between the values of S 2 and n tet per particles in the liquid and the values in the corresponding IS. The results presented in Fig. 3 c) show that the correlation is stronger at lower temperatures, reaching a correlation of approximately 0.6 at T = 0.7 for both S 2 and n tet , demonstrating that the two-body entropy and tetrahedrality of the inherent structures are largely governed by the structure of the unquenched liquid as we observed in Fig. 3 a) and b). As expected, above the onset temperature, the correlation becomes negligible. This indicates that the local structure of a high temperature liquid is mostly insensitive to the underlying IS.

B. Deformed glassy structures

Now that we have characterized the static structure of our system using these descriptors, we can ask ourselves how these quantities evolve under cyclical shear deformation. To this end, we take the IS from the equilibrated configurations at a low and a high temperature of T = 0.7 and T = 1.5 and cyclically shear them with a range of strain amplitudes γ max . In particular, we distinguish two regimes of interest: I. Below yielding (γ max < γ y ): For small amplitudes of shear below yielding, the systems can anneal and, in the steady state, the system can visit lower energy minimum basins. In our system from γ max = 0.02 to γ max = 0.06, we have indeed more annealed absorbing states: the steady-state energy is lower than the one of the initial configuration, in agreement with what has been observed in the literature 41,64 . If the initial configuration corresponds to a high-temperature liquid, the annealing effect is more pronounced in comparison to low-temperature initial configurations 64 . In the steady state, the local plastic rearrangements during a cycle of deformation repeat from one cycle to the next 37 . Therefore, from one cycle to another, there is no appreciable net displacement when considering the configurations stroboscopically. II. Above yielding (γ max > γ y ): In this case, the system becomes diffusive. In the steady state, the plastic rearrangements during a cycle lead to a net displacement of particles. For a large enough system, like in our case, N = 64000, it is possible to observe the formation of shear bands of highly diffusive particles 44,65,66 . In the case of XZ shearing, the shear band consists of a slab of particles parallel either to the Y Z or the XY plane. These particles on average present net displacements, between two consecutive cycles, larger than the rest of the particles. Interestingly, the region inside the shear band is characterized by a lower local density and a higher local energy 44 . Now that we have established the macroscopic effects of shear on our system, we examine the impact of the shearing amplitude on the local structure. To this end, we first we compute S 2 and n tet in the steady state for strain amplitudes γ max ∈ [0.02, 0.09], i.e. both below and above γ y ∼ 0.06. In Fig. 4, we report the behaviour of the mean value of these local descriptors as a function of γ max , for temperatures T = 1.5 and T = 0.7. Note that γ max = 0.0 is actually the value for the initial IS configuration. The high temperature glass in the range γ max ∈ [0.0, 0.06] displays noticeable annealing and, as a consequence, an increase in local order reflected by a lower S 2 and a higher n tet . As we cross γ y , the diffusive state is reached, and the average local order decreases significantly. This behaviour implies that remarkably, at yielding, a maximum local order is attained. Moreover, the transition across γ y manifests in a cusp-like variation for both quantities. For T = 0.7, no strong annealing is observed below yielding and S 2 and n tet remain almost constant. Upon crossing γ y , the two observables display a discontinuous jump. Above yielding, the structural observables are the same for both temperatures, as has previously been observed for the energy 65 . The cusp for the high temperature and the jump for the low one is compatible with the existence of a critical temperature that divides two different regimes or yielding behaviour. 64 .

To examine the effects of these changes in local structure on the dynamics of individual particles, we need to quantify how much a given particle moves during a displacement cycle. When a system is sheared, the motion of the particles can be separated into affine and non-affine displacements 67 . One way of detecting and characterizing the local re-arrangements that the particles undergo during the cyclic deformations is through their non-affine displacement D 2 i as introduced in Ref. 26. We employ a modified version of D 2 i where we calculate the non-affine displacement as a function of the accumulated deformation γ instead of time: where r i (0) is the position of particle i at the beginning of a deformation cycle and r i (γ) is its position in the deformed box at a total accumulated deformation of γ. The sum in equation 5 is over the N i nearest neighbors of particle i, taken to be the neighbors within a cut off r cut = 1.4σ i j , which corresponds to the first minimum of g(r) of the full system. The matrix Γ is such that it minimizes the non-affine displacement giving a measure of local strain 26 . This definition best maps deviation from affine displacement at a local level 67 . In the case of a cyclic deformation below yielding, the system returns back to the same configuration each cycle once the steady state has been reached. Consequently, if we consider configurations at the beginning (γ = 0) and at the end (after 4γ max /dγ deformation steps) of the cycle, we would observe D 2 min (4γ max ) ≈ 0. However, we are interested in identifying which particles went through maximum re-arrangements during the deformation, regardless of reversibility. Therefore, we recorded the maximum value max(D 2 min ) for each particle in a cycle of deformation.

D 2 i (γ) = 1 N i N i ∑ j=0 [(r j (γ) -r i (γ)) -Γ(r j (0) -r i (0))] 2 , (5)
In Fig. 5, we plot the distribution of max(D 2 min ) for two strain amplitudes below and above yielding. It is evident that, above yielding, a long tail is observed in the distribution. This tail is due to the particles in the shear band that present the maximum degree of plastic rearrangement.

The calculation of max(D 2 min ) enables us to identify particles with high displacements, corresponding to those undergoing plastic rearrangements. At this stage it is natural to ask, is there any structural difference between the particles that undergo strong rearrangements and the particles that are immobile? In other words, at the end of a cycle, are particles with high and low max(D 2 min ) values characterized by a different structures at the beginning of a cycle? These are the questions that we will address in the remaining part of the paper. To answer these questions, we turn our attention again to the structural local descriptors S 2 and n tet . In the steady state, we compute both descriptors for each particle in the stroboscopic configurations at a given strain amplitude. Subsequently, we subject the system to a complete deformation cycle during which we compute the values of max(D 2 min ) for each particle. As in previous studies on dynamical heterogeneities 68 , we define a threshold based on the displacement of the individual particles that allows us to classify the particles as immobile or mobile if their max(D 2 min ) values are among the smallest or the largest respectively. In particular, we choose the top 5% as mobile particles and bottom 5% for the immobile ones. The resulting distributions are shown in Fig. 6. Here, we present the case of T = 0.7 and γ max = 0.06. However, the trend is similar in all the state points we investigated. From P(S 2 ), we see that immobile particles have significantly lower S 2 implying a stronger local order in comparison to the mobile particles. This trend is confirmed by the behaviour of P(n tet ) which shows that immobile particles are much more likely to be found in highly tetrahedral environments. Interestingly, for the immobile particles, we observe a strong peak at n tet = 18 and n tet = 20, the first corresponding to clusters formed by rings of 5 particles leading to high tetrahedrality and the second to particles at the center of an icosahedral cluster. This peak is essentially absent in the mobile particles, indicating that icosahedral caging has a strong effect on the local rearrangements in this system. The appearance of this cluster has been investigated on many occasions in supercooled liquid systems as they represent the fingerprint of geometrical frustration introduced by Frank 7 and have been found to be strongly linked to slow dynamics in a wide variety of systems [8][9][10][11][12][13]69,70 . We now turn our attention to the 0.02 0.04 0.06 0.08 - average structural characteristics of the mobile and immobile particles. In Fig. 7, we plot S 2 and n tet for immobile and mobile particles as a function of strain amplitude γ max and for both high (T = 1.5) and low (T = 0.7) temperatures. For the immobile particles, below yielding, n tet increases and S 2 decreases. These trends are more pronounced for T = 1.5 and can be ascribed to the stronger tendency to anneal and the consequent increase of local order, as we observed in Fig. 4. Above yielding, the immobile particles are present outside the shear band. We find that upon crossing the yielding amplitude (γ max > 0.06), both S 2 and n tet flattens out for the high temperature case. For the mobile particles, as expected, the values of S 2 and n tet are consistent with a less structured local structure. However, we notice that their structural features change very little between the two temperatures. In fact, both S 2 and n tet remain almost constant below and above yielding. Although the origin of this effect is not clear, it suggests that the most mobile particles are characterized by a similar local structure, which is independent of temperature, shear amplitude, and even the presence of a shear band. Equally interestingly, even though they do not participate in the shear bands in which strain is localized, the structure around immobile particles is significantly affected by shear band formation. So far, we have established that the particles characterised by different propensity to re-arrange are embedded, on average, in different local environment as reflected by the local descriptors considered here. However, we have not discussed the actual geometrical shape of the local cluster around a particle and how it is connected with plastic rearrangement. To this end, we classify the local arrangement of particles by using the Topological Cluster Classification algorithm (TCC) 46 . This algorithm detects predefined specific local clusters of different sizes. The detection is based on the connections between nearest neighbors, found via a modified Voronoi construction. As a first step, simple small clusters from 3 up to 7 particles are identified. Subsequently, more complex clusters are found as combinations of the simple ones. Note that each particle can simultaneously be part of multiple clusters. For the 5% most mobile and immobile particles we examine all possible clusters detected by TCC and we calculate the fraction of particles involved in a certain type cluster for each case. In Fig. 9 a) we show the results for mobile (red) and immobile (blue) particles for the case of T = 0.7 and γ max = 0.06. Here, we report only the cluster types that show significant differences between the two communities. In particular, we highlight the 13A cluster corresponding to an icosahedral cluster for which the largest difference between mobile and immobile particles is noticed. Similarly, another cluster that presents large differences, but less pronounced than the 13A case, is the 10B cluster which corresponds to a defective icosahedral cluster. It appears that these two clusters play an important role in determining whether a given particle will rearrange during a shear cycle. Note that the two of them in particular can be decomposed in several tetrahedral clusters characterised a particularly high value of n tet value. For example, for a particle located in the center of an icosahedral cluster, its n tet = 20 as each of the faces of the icosahedral form a tetrahedral cluster with the central particle. In the following, we will focus mainly on these two clusters. In particular, in Fig. 8(b-e), we show the fraction of particles involved in the 10B and 13A cluster for each of the communities as a function of γ max as they present the largest difference. For the immobile particles at T = 0.7, the population of 10B clusters is essentially constant, while the icosahedral cluster displays a noticeable maximum at the yielding transition. For T = 1.5, below yielding, there is a more pronounced effect as the fraction of immobile particles involved into these clusters grows with the amplitude γ max . Above yielding, this effect flattens out as no noticeable increase is observed. For mobile particles at both temperatures, it appears that the variation in the population of the two clusters is negligible. Similar results are found for other TCC clusters that differ in prevalence between the immobile and mobile particles, such as the ones shown in Fig. 9 a). Essentially, there are no appreciable changes in the structure of the mobile particles with different amplitudes. In contrast, the immobile ones show significantly different behavior across the yielding transition. These trends are consistent with the variation of S 2 and n tet in Fig. 7 and seems to confirm that the local environment in which the particles are embedded plays an important role. In particular, it is interesting to note that despite the immobile particles displaying a limited activity in terms of plastic rearrangements, there is a more remarkable variation of their local structure upon the change of amplitude of the deformation.

From Fig. 6 b) we see that a large fraction of 5% most immobile particles are characterized by n tet = 20, which is the number of tetrahedra that one would expect in an icosahedral cluster. We confirm this idea by inspecting Fig. 8 c) and e), where we notice that the 5% most mobile and 5% least mobile particles on average show significant difference regarding to the icosahedral ordering.

C. Glassy structure and the shear band

As discussed in the introduction, above yielding, the dynamics become heterogeneous and localized in space and, as a consequence, shear band formation is expected. There are numerous hints that the properties of the structure inside the shear band is different from what is observed outside. It has been observed, for a similar model to the one discussed here, that the density inside the shear band is lower 44 and that it may present a certain degree of hyperuniformity 71 . The question we want to answer next is whether its structure is different and if S 2 and n tet capture the differences inside and outside the shear band.

To detect the shear band, we bin the system along the axis perpendicular to the plane of the shear band. In this case this corresponds to the z-axis. Subsequently, we compute the mean square displacement (MSD) of the particles between the n th and (n + 1) th cycle as a function of z 44 . The position of the shear band can be clearly identified as the region where the particles have a larger MSD. The result, for the case of T = 0.7, is presented in Fig. 9 a). Along the same axis, we measure S 2 and n tet and show the results in Fig. 9 c) respectively. Both these structural features indicate strong structural differences between the portions of the system inside and outside of the shear band. On average, outside the shear band we observe higher structural order.

In Fig. 9 d), we show the distribution of the icosahedral clusters along the direction perpendicular to the shear band. We find a clear localization, where 30% of the particles outside the shear band are involved in an icosahedral cluster, whereas inside the shear band this fraction is much lower. This observation suggests that the lower density that has been observed inside the shear band 44 is related to a pronounced structural signature. In particular, the systems prefers to localize a great number of icosahedral structures, that are know to be more energetically favourable, outside the more mobile region. This localization is not found below yielding, where the icosahedral clusters are homogeneously distributed inside the simulation box.

IV. CONCLUSIONS

Predicting plastic re-arrangements based on local structures information is a challenging domain of research (see Ref. 30 and reference therein) that has attracted a lot of interest in recent years [72][73][74][75] . In this paper, we have studied the local structural rearrangements of particles undergoing cyclic shear deformation in a model glass former focusing on local tetrahedrality n tet that we introduced previously 19 and the well known two-body excess entropy S 2 . High values of n tet and low values of S 2 indicate a higher degree of local order. In particular, we consider two temperatures, above and below the onset temperatures, for which different levels of initial average order is observed. The inherent structure configurations were sheared following a cyclic athermal quasistatic deformation protocol that revealed the typical phenomenology of the weakly (high temperature) and deeply (low temperature) annealed glass 64 . We then investigated the links between local structure and local rearrangements at different strain amplitudes γ max below and above yielding. Below yielding, the system anneals, resulting in an increase of local structural order until the system reaches a periodic state and no longer diffuses. In contrast, above yielding both the low-temperature and high-temperature glasses become diffusive and exhibit shear banding. Moreover, above yielding, we found that both systems present on average the same values of S 2 and n tet irrespective of its preparation history. This is compatible with the phenomenology observed before in several different systems 41,43,64,65,[76][77][78] . During the deformation cycle in steady states below yielding, particles rearrange reversibly, meaning that at the end of the cycle they come back to their original positions. In contrast, above yielding the rearrangements are irreversible and particle, inside shear band, display diffusive motion. By investigating the local structure around the 5% most immobile and most mobile particles during a given shear cycle, we found that the immobile and mobile particles have on average different local order at the beginning of the cycle. In particular, particles with lower n tet and higher S 2 are more likely to rearrange, an observation which holds for all investigated γ max and temperatures. Moreover, the two temperatures explored here correspond to glass that would have a brittle (low temperature) and ductile (high temperature) mechanical response 64,78 corresponding to different levels of annealing. Interestingly, for the low temperature, the average local structures do not display significant evolution below yielding. However, if we consider only the immobile particles, we observe a clear evolution, for example, in the number of tetrahedra. This indicates that the main effect of annealing below yielding is the emergence of a population of particles with high local order, which are highly stable against further rearrangement. In line with these results, we closely investigated the structure inside and outside shear band. We showed that both of our local structure descriptors are spatially correlated to such shear band. In particular, outside it, the local structure is more ordered presenting higher values of n tet and lower of S 2 . Interestingly, we found that outside the shear band more than 30% of the particles are involved in icosahedral clusters whereas inside it this percentage is essentially negligible. In other words, the population of highly ordered particles that emerges upon annealing persists even beyond yielding, but remains localized outside the shear band, where they may have a stabilizing influence on their surroundings. Our results confirm that, as for the case of continuous shearing 30 , certain aspects of the response to cyclic shear deformation can be correlated to structural features.

FIG. 2 FIG. 3 .

 23 FIG. 2. a) Two-body entropy S 2 as a function of temperature T for liquid and inherent structure. b) Tetrahedrality n tet as a function of T for liquid and inherent structure.

FIG. 4 .

 4 FIG. 4. Behavior of a) S 2 and b) n tet in the steady state as a function of γ max , for cyclically sheared inherent states of equilibrium configurations obtained at T = 0.7 (blue) and T = 1.5 (red). The vertical dashed line shows the yielding amplitude γ y .

FIG. 5 .

 5 FIG. 5. Distribution of max(D 2min ) in the steady state at shear amplitude a) below yielding at γ max = 0.06 < γ y and b) above yielding at γ max = 0.08 > γ y . Above yielding we have longer tail. In both cases, the temperature of the initial configuration was T = 0.7.

FIG. 6 .

 6 FIG. 6. Distributions of a) S 2 and b) n tet for unsheared configurations in the steady state at temperature T = 0.7 and shear amplitude γ max = 0.06. The black line corresponding to the overall distribution. Distributions are shown for immobile (blue) and mobile (red) particles, corresponding to the lowest and highest 5% of displacements max(D 2 min ) during a shear cycle, respectively.

FIG. 7 .

 7 FIG. 7. Behavior of a) S 2 b) n tet as a function of γ max in the steady state, for the 5% most mobile (red circle) and 5% most immobile particles (blue squares). Filled symbols correspond to T = 1.5 and open symbols to T = 0.7.

FIG. 8 .

 8 FIG.8. TCC analysis for mobile (red) and immobile (blue) particles. In a) for T = 0.7 and γ max = 0.06 we report the fraction of these mobile and immobile particles in a particular type of cluster, where 13A is the icosahedral cluster. Next we plot these fractions against γ max for two particular classes b) 10B (defective icosahedral cluster) and c) 13A (icosahedral cluster) at a temperature of T = 0.7 and the corresponding d) 10B and e) 13A at T = 1.5.

FIG. 9 .

 9 FIG. 9. Shear band captured by the different properties a) meansquared displacement, b) S 2 , c) n tet and d) fraction of particles involved in icosahedral clusters N Ico /N as a function of the perpendicular axis to the shear band. All corresponding to sheared system at a γ max = 0.08 at a temperature T = 0.7.

ACKNOWLEDGMENTS

The authors would like to thank Davide Fiocco for the original production of the potential energy cartoons in Fig. 1. S.M., S.S and G.F. gratefully acknowledge IFCPAR/CEFIPRA for support through project no. 5704-1. S.M.-A. acknowledges support from the Consejo Nacional de Ciencia y Tecnología (CONACyT scholarship No. 340015/471710). SS acknowledges support through the JC Bose Fellowship (JBR/2020/000015) SERB, DST (India).