*Review* **Kramers' Theory and the Dependence of Enzyme Dynamics on Trehalose-Mediated Viscosity**

#### **José G. Sampedro 1,\*, Miguel A. Rivera-Moran <sup>1</sup> and Salvador Uribe-Carvajal <sup>2</sup>**


Received: 29 April 2020; Accepted: 29 May 2020; Published: 11 June 2020

**Abstract:** The disaccharide trehalose is accumulated in the cytoplasm of some organisms in response to harsh environmental conditions. Trehalose biosynthesis and accumulation are important for the survival of such organisms by protecting the structure and function of proteins and membranes. Trehalose affects the dynamics of proteins and water molecules in the bulk and the protein hydration shell. Enzyme catalysis and other processes dependent on protein dynamics are affected by the viscosity generated by trehalose, as described by the Kramers' theory of rate reactions. Enzyme/protein stabilization by trehalose against thermal inactivation/unfolding is also explained by the viscosity mediated hindering of the thermally generated structural dynamics, as described by Kramers' theory. The analysis of the relationship of viscosity–protein dynamics, and its effects on enzyme/protein function and other processes (thermal inactivation and unfolding/folding), is the focus of the present work regarding the disaccharide trehalose as the viscosity generating solute. Finally, trehalose is widely used (alone or in combination with other compounds) in the stabilization of enzymes in the laboratory and in biotechnological applications; hence, considering the effect of viscosity on catalysis and stability of enzymes may help to improve the results of trehalose in its diverse uses/applications.

**Keywords:** trehalose; viscosity; enzymes; protein dynamics; Kramers' theory; protein stabilization; enzyme inhibition

#### **1. Trehalose in Biology**

The disaccharide trehalose has been used as food or therapeutic agent by humans since ancient times [1]. Trehalose is naturally found in the sweet cocoon (the capsule named trehala-manna) that protects the larvae of weevils, insects in the genus *Larinus* [1–4]. Pure trehalose crystals were isolated first from the ergot of rye by Wiggers (1832), then the carbohydrate was identified as the main component of trehala-manna (from which it takes the name trehalose) by Berthelot (1858) and identified as a non-reducing disaccharide formed by two glucose units [5].

Trehalose (α-D-glucopyranosyl-1, 1-α-D-glucopyranoside) was found at high concentrations in diverse organisms [6] and under stress conditions [4], including yeast and fungi [7,8], plants [9], and others [6]. Trehalose, in addition to forming part of cellular structures [6], has been claimed to behave as a reserve carbohydrate as it is usually found in spores of yeast, fungi, and bacteria [4,6–8,10], and also as a biostructure stabilizer, namely of proteins and membranes during harsh environmental conditions [4], and even in the pathogenesis of virulent bacteria (like actinomycetes and mycobacteria) and fungi [11,12]. Currently, the metabolic pathways for trehalose biosynthesis and hydrolysis, and their regulation are well known [8–10,13,14], and in some organisms, trehalose metabolic pathways seem to have a major role in the regulation of other metabolic processes [7,8].

Among the diverse stress conditions where trehalose is synthesized and accumulated, dehydration (or the phenomenon of anhydrobiosis) has been that which has attracted the most interest (because of its potential industrial applications) [5,15–17], namely in food and pharmaceutics. Van Leeuwenhoek (1702) first described the phenomenon in bdelloid rotifers [18,19]. Since then, other organisms have been recognized to endure the absence of water, such as nematodes, brine shrimp cysts, yeast, and some plants [4,6,18].

Experiments have demonstrated the effectiveness of trehalose to preserve the structure of proteins and membranes in the dry state [16,20,21]. Nonetheless, the role of intrinsically disordered proteins (stress proteins) protecting organisms that synthesize little to no trehalose seems to be important [22,23]. In nature, the dehydration process in anhydrobiotic organisms seems to be slow, i.e., water is removed in minute amounts leading to cell desiccation [16]. At the same time, trehalose begins to be synthesized, reaching high concentrations in the cytoplasm, while a small amount is exported (trehalose is required at both surfaces of the plasma membrane) [24]. Although cell survival does correlate with the amount of trehalose present in the cell [16,19], other biochemical responses contributing to anhydrobiosis (or acting synergically with trehalose) seem to take place simultaneously to trehalose accumulation [16], e.g., the synthesis of heat shock proteins [25].

Physically, under optimum growth conditions (nutrients, temperature, etc.) the cell cytoplasm is a crowded space containing a great diversity of molecules, hence inherently highly viscous [26–29]. Then, during dehydration, the accumulation of trehalose increases viscosity even further as water-loss proceeds [30–34]. Hence, viscosity increases up to finally reach the glassy state [34,35]. Here, although the interaction of trehalose with proteins has been subjected to some controversy [36], the effect of trehalose on the maintenance of the integrity of membranes (via the interaction with phospholipid headgroups) is relatively well understood [16,37]. Nonetheless, what is certain is that viscosity (the glassy state) generated by trehalose is high. Here proteins and other bio-structures are "arrested", which in turn preserves both the structure and function of proteins [34,38–40].

In microorganisms, trehalose is also synthesized in the stationary phase of growth. In yeast cultures when nutrients are exhausted, cells enter a quiescent state known as the stationary phase [41], where the metabolic rate is slowed and cell division stops (arrested cell cycle) [42,43]. Notably, during entry to the stationary phase, trehalose is accumulated at high concentrations (up to 20% in dry weight), as well as glycogen (up to 8% in dry weight) [42]. The cells at the stationary phase thus become denser, heavier, and cytoplasm viscosity increases [31,32,42]. In this regard, trehalose seems to have a role in the exit of yeast from the stationary phase besides [42]; however, no specific role of trehalose in cell metabolism under the quiescent state has been described. Instead, cells in the stationary phase display high thermotolerance, possibly due to trehalose accumulation [4,21].

#### **2. Trehalose, Water, and Proteins**

Trehalose/water interaction is the basis for protein stabilization by trehalose [36,44–46]. Water is the medium where most biochemical processes take place and the hydration water has a fundamental role in protein structure and function [47–49]. In this regard, water molecules seem to be more uniformly distributed on the surface of globular proteins when compared to those on intrinsically disordered proteins (IDPs) [48,50]; despite that, the arrangement of tetrahedral water molecules is more disordered on the surface of globular proteins than on IDPs [50]. The above is probably due to the highly irregular surface of globular proteins. Interestingly, the dynamics of hydration water on proteins are different from bulk water molecules [47,48,51–53]. In solution, trehalose molecules display a relatively rigid conformation and slow translational motion (self-diffusion) [46,54]. Besides, trehalose molecules interact through trehalose–trehalose hydrogen bonds [44,55,56]; while trehalose–water hydrogen bonds are weaker [56]. This seems to facilitate vitrification [46], i.e., by allowing water molecules to escape from the system. Additionally, trehalose appears to be physically (preferentially) excluded from protein surfaces [57,58], thus promoting preferential hydration [45,59–61]. This is while maintaining the disordered arrangement of hydration water molecules [50]. Nonetheless, the dynamics

of hydration water appears to be slowed down by trehalose and consequently, dynamics of the amino acid residues in proteins decrease [45,48,61–67], e.g., in C-phycocyanin (CP), trehalose slows down the dynamics of the hydration water by an order of magnitude when compared to the dynamics of bulk water [68]. As a result, protein processes where structural dynamics are a major component become affected by the presence of trehalose in the medium, namely folding, unfolding, thermal and cold inactivation, oligomerization, aggregation, catalysis, and even the equilibrium between catalytic states (Figure 1). The thermodynamic effect of trehalose would be to increase the energetic barrier between different structural states (Figure 1).

**Figure 1.** Energy landscape of diffusion-dependent protein processes. High medium viscosity (η) increases the free energy height (red line) of the interconversion between conformational states in proteins (blue line). (**a**) Structural conformational states in enzyme catalysis, (**b**) conformational equilibrium between the *T* and *R* states in a cooperative enzyme regarding the model of Monod, Wyman, and Changeux [69,70], (**c**) three-state protein unfolding (N ↔ I ↔ U); formation of the intermediary (I) may involve the dissociation of monomers from the oligomer, (**d**) two-state protein unfolding (N ↔ U).

#### **3. Protein Dynamics and Catalysis**

Proteins, like most polymers, display inherent movements inside their structure [71,72]. Some movements participate in the function, e.g., catalysis [71,73], while others are part of the protein vibrational state specific to the medium temperature [74–76]. Indeed, the function of proteins is defined by the dynamics of the three-dimensional structure [77–81]. The inter-conversion between different structural states determines the function of the protein [82]. For example, the motions of the enzyme structure are a major component in catalysis [73,78,83–86]. There exists ample evidence on the role of enzyme structural dynamics in catalysis [73,87–89]; currently, experimental and computational techniques have demonstrated enzymes are highly dynamic entities [90]. Importantly, the rate of the structural dynamics in enzymes whether internal, specific (like loops and domain), as a structural network, or global (collective) has been determined as the limit for the turnover rate (*k*cat) [71,91]. In this regard, protein crystallography has provided the structural details of enzymes at the atomic level [74,92,93]. Unfortunately, such a structure is just a snapshot of the protein structure that yields limited information about function [77,94]. Nonetheless, efforts have resulted in the successful crystallization of enzymes in complex with ligands which demonstrates the existence of structural dynamics related to catalysis [74,95]. In some fortunate cases, the crystallographic structures of enzymes obtained under different conditions allow the depiction of the catalytic cycle [71,73,96–98]; however, it is worth noting that these states usually are a (small) fraction of the infinite conformations the protein structure may sample [75,82]. Experimental methods to study protein dynamics include NMR relaxation, hydrogen-deuterium exchange, fluorescence, UV-Vis spectroscopy, Raman spectroscopy, infrared spectroscopy, and molecular dynamics simulations (MDS) [73,77,99]; nonetheless, the timescale of measurement usually is variable, e.g., MDS ranges in the ns scale and less, while hydrogen-deuterium exchange ranges between ms and s scale [77]. Importantly, most biological processes (such as enzyme catalysis) lay on the μs to ms time scale [71,73,77]. Here protein dynamics have been evaluated by different methods, e.g., in rotary proteins like FoF1-ATPase [100,101]. MDS has emerged as a very helpful technique to study protein dynamics and function [72,88,102]. The position of any given atom in the polypeptide chain may be known with more precision at any given moment [72], although, a high-resolution three-dimensional (3D) structure is a requisite [103]. Still, when the 3D-structure is not available, modeling is an option [104]. However, the in vitro experimental validation of the in silico results is desirable [49,80,105].

In enzymes, it is known that under isobaric conditions the structural dynamics are modulated by variations in temperature (thermal energy) and solvent composition (cosolvents, ions, polymers, etc.) [38,49,76,106–108], i.e., in Michaelian enzymes, some conformational states may be favored as the kinetics of state interconversion are affected [77,109,110]. The diverse conformational states in a "resting" protein/enzyme (in the absence of ligands) in solution result from side-chain fluctuations, movements of loops and secondary structures, structural domains, and collective global arrangements [82]; thus, before catalysis, enzymes probe multiple conformations [77,109,111]. In this sense, even the whole catalytic cycle may occur in some enzymes in the absence of ligands [73,112]. Notably, in the kinetics of cooperative enzymes, the Monod, Wyman, and Changeux model is used on the assumption that enzymes fluctuate between two states with high ("relaxed", *R*) and low substrate affinity ("tight", *T*) [69,113], while binding of ligands, allosteric effectors, and even phosphorylation of some amino acid residues, changes the equilibrium towards the high- or low-affinity state, accordingly [114].

#### **4. Dependence of Enzyme Catalysis on Medium Viscosity (Kramers' Theory)**

The viscosity of the crowded cell cytoplasm is higher than pure water [26,27,115–117]. Viscosity varies by cell type, e.g., in *Saccharomyces cerevisiae* cytoplasm it has been calculated to be ~10 cP [115]. Certainly, the viscosity of the cytoplasm is considerably higher in organisms at the quiescent state and during dehydration, osmotic stress, and at heat-shock as they accumulate viscosity generating solutes such as trehalose, glycerol, and others [115]. In the physics of fluids, viscosity is defined as the friction between molecules or the resistance to flow (molecular diffusion) in a liquid (or gaseous) system [115]. In this environment, the three-dimensional (3D) structure of enzymes fluctuates between different conformations [71,118,119]. Protein structural fluctuations are coupled to the movement of water molecules at the hydration shell, and these in turn, to the dynamics of the molecules present in the bulk solution [49,120]. Recently, studies on the translational diffusion of hydration waters concluded that protein function may be modulated by sugar molecules [48,64]. Hence, the inhibition or hindering of protein motions, whether by a decrease in temperature or by changing the composition of the suspending medium, leads to decreased catalysis [85,121–124].

Specifically, trehalose promotes the hydration of the protein surface (preferential hydration) [60,61,125], but it also diminishes the dynamics of hydration water [61,64]. As a result, the motion of amino acid residues is slowed down, and the rate constant of enzyme structural fluctuations becomes dependent on the viscosity of the medium [47]. Certainly, the effect of medium viscosity on enzyme catalysis has already been considered [118], when the dependence of catalysis on viscosity was rationalized to the inhibition of enzyme fluctuations according to Kramers' theory [106,118,120,123,126–128], as described by Equation (1):

$$k = \frac{\mathbf{A}}{\eta} e^{-\Delta/k\_B T} \tag{1}$$

where Δ is the height of the potential barrier, *k* is the rate constant of catalysis (*k*cat), *A* is a function of structural parameters characterizing the potential energy profile, *k*<sup>B</sup> is the Boltzmann constant, and η is the viscosity of the medium. Equation (2) is an actualized version of Equation (1) when using the enzyme kinetic parameter *V*max (*V*max = *k*cat·[*E*t]).

$$V\_{\text{max}} = \frac{\mathbf{A}}{\eta} e^{-\Lambda Ul/RT} \tag{2}$$

where −Δ*U* is the height of the potential barrier, and *R* is the gas constant. Notably, Equation (2) has been useful to analyze the inhibition of enzyme catalysis by the presence of trehalose [31,32,129,130]; thus indicating that trehalose does inhibit the rate of catalysis by hindering the structural fluctuations of enzymes during the catalytic cycle [31,32,108,130]. *V*max becomes inversely proportional to the viscosity (*V*max α 1/η) of the medium, and a straight line is usually observed when plotting *V*max versus η−<sup>1</sup> [31,32,108,130]; i.e., as viscosity increases, enzyme activity decreases [32]. Importantly, the viscosity generated by trehalose in the bulk solution would affect mainly the largest structural motions involved in catalysis (Figure 1) [31,129,130]. For example, in the plasma membrane H+-ATPase from yeast, the motion of the N-domain (a globular structure of ~16.5 kDa) toward the P-domain is required to transfer the γ-phosphate of ATP to the Asp residue [131]. Notably, the activity of the H+-ATPase decreases linearly as the viscosity generated by trehalose increases (Figure 2a) [31]. This behavior is in agreement with Equation (2), as described by Kramers' theory [31]. Similarly, the H+-ATPase from plants is inhibited linearly by the viscosity generated by the presence of sucrose (the natural disaccharide in plants) [132]. Additionally, the kinetics of pyruvate reduction by lactate dehydrogenase (LDH) from rabbit muscle is dependent on medium viscosity, as LDH catalysis depends on the movement of a loop on the active site [130,133]. The LDH *V*max is dependent on medium viscosity as expected from Kramers' theory (Figure 2b) [130]. Notably, the viscosity generated by trehalose increases the activation energy (*E*a) in both enzymes, H+-ATPase, and LDH [31,130] (Figure 3), as suggested in Figure 1a. Therefore, enzyme inhibition by trehalose is expected when a relatively large structural motion is involved in the catalytic mechanism.

**Figure 2.** Dependence of enzyme catalysis (30 ◦C) on medium viscosity as described by Kramers' theory. (**a**) Viscosity dependence of *V*max of plasma membrane H+-ATPase from *Kluyveromyces lactis*. Adapted with permission from Sampedro et al. [31]. Copyright © 2020, American Society for Microbiology; (**b**) viscosity dependence of *V*max of lactate dehydrogenase (LDH) from rabbit muscle. Adapted with permission from Hernández-Meza and Sampedro [130]. Copyright © 2020, American Chemical Society.

**Figure 3.** Plot of the activation energy (*E*a) of enzyme catalysis versus trehalose concentration. (**a**) Plasma membrane H+-ATPase [31]. Reprinted with permission from Sampedro et al. [31]. Copyright © 2020, American Society for Microbiology; (**b**) LDH [130]. Reprinted with permission from Hernández-Meza and Sampedro [130]. Copyright © 2020, American Chemical Society. The activation energies were calculated as described by the Arrhenius equation (*kcat* = <sup>A</sup>·*e*−*Ea*/*R*·*T*) using the enzymes *<sup>V</sup>*max values (*V*max = *k*cat·[*E*t]).

Enzymes do display catalysis dependence on viscosity as described by Kramers' theory even when using viscogens others than trehalose; e.g., a decrease in the rate constants (*k*cat) is observed in the CO2 hydration and HCO3<sup>−</sup> dehydration by carbonic anhydrase (CA) [134], the ATPase hydrolysis in chloroplast coupling factor (CF1), myosin and meromyosin [135,136], the deacylation step of subtilisin BPN catalyzed ester hydrolysis [137], the thioesterase activity of fatty acid synthetase (FAS) [138], the carboxypeptidase-A (CPA)-catalyzed benzoylglycyl-L-phenyl lactate hydrolysis [139], the cysteine protease activity of human ribosomal protein S4 (RPS4) [140], the DNA polymerizing activity of polymerase β (POLB) [141], the sugar cleaving enzyme oligo-1,6-glucosidase 1 (MalL) [142], the indole-3-glycerol phosphate synthase (IGPS) from *Sulfolobus solfataricus* [105], and in the electron transfer reactions of sulfite oxidase (SO), respiratory and photosynthetic complexes [143–145]. In extreme situations of viscosity, i.e., enzymes embedded in a glass-state environment where viscosity is infinitely high, the absence of protein motions leads halts catalysis [120]. The modulation

of protein/enzyme dynamics by viscosity has been also recently demonstrated by molecular dynamics simulation (MDS) [146] using the small protein factor Xa [147,148], and ssIGPS [105].

Viscosity generating solutes also affects the enzyme Michaelis-Menten constant (*K*<sup>m</sup> = *<sup>k</sup>*−1+*k*<sup>2</sup> *<sup>k</sup>*<sup>1</sup> ) or the binding (*k*1) and dissociation (*k*−1) of the substrate from the binding site, when in the enzyme a structural diffusion is involved in either of these processes, e.g., glycerol increases the *K*<sup>m</sup> in phosphorylase B [149], sucrose increases the *K*m of polymerizing myosin [135], and trehalose decreases the *K*<sup>m</sup> of glucose oxidase (GOx) and glucoamylase from *Aspergillus niger* [150,151]. Therefore, the catalytic efficiency (*k*cat/*K*m) of enzymes varies as substrate affinity is decreased/increased by viscogens [149–151] by hampering structural motions in either ligand binding or dissociation. Interestingly, enzyme activation by trehalose has been reported in thermophilic enzymes like neutral glucoamylase from the mold *Thermomucor indicae-seudaticae* [152]. Similarly, in pyruvate kinase (PK) from *Geobacillus stearothermophilus* (a cooperative enzyme), trehalose increases the velocity of the enzyme reaction (Figure 4a) by increasing substrate affinity (*K*<sup>R</sup> and *K*T) in both "tight" and "relaxed" (T and R) states, and hindering the population of catalytically unproductive conformational states; nonetheless, trehalose does affect the equilibrium between the R and T states as described in Figure 1b, without changing the enzyme cooperativity (Hill number, *n*) (Figure 4b). This phenomena (enzyme activation) may be explained either by the effect of viscosity on the enzyme structural fluctuations (i.e., in the transition between R and T states) [70], by favoring the population of a given oligomeric state [153], by decreasing the protein volume [154] thus leading to the formation of the binding site [155], by preventing the sample of catalytically unproductive enzyme conformations [85], or by a combination of all of them. In this regard, the compressing effect of protein structure by viscosity generating solutes has already been observed with glycerol [155,156], sucrose [155,157], and trehalose [67,155], e.g., the volume of lysozyme decreases ~2% in the presence of trehalose [67].

Kinetics of protein processes others than catalysis are also affected by viscosity [158] as described by Kramers' theory, some of these are the escape of O2 from respiratory proteins [159] and hemerythrin [160], the heme pocked relaxation in hemoglobin after CO photolysis [161], the dissociation of CO from horse ferrocytochrome *c* (FCC) [162], and the CO binding to horse myoglobin [163].

**Figure 4.** Activation of pyruvate kinase (PK) from *G. stearothermophilus* by trehalose [164]. (**a**) Cooperative kinetics of PK in the absence (-) and presence (•) of trehalose. Considering the model of Monod, Wyman, and Changeux [69] in enzyme kinetics analysis: *V*max = 112 and 118 μmoles PEP·(min·mg protein)−<sup>1</sup> in the absence and presence of trehalose, respectively. Affinity constants: *<sup>K</sup>*<sup>T</sup> <sup>=</sup> 328 and 11, and *<sup>K</sup>*<sup>R</sup> <sup>=</sup> <sup>40</sup> <sup>×</sup> <sup>10</sup>−<sup>4</sup> and 5 <sup>×</sup> <sup>10</sup>−<sup>4</sup> mM phosphoenolpyruvate (PEP) in the absence and presence of trehalose, respectively. The Hill number (*n*) was ≈1.5, both in the absence and presence of trehalose. The ratio [*T*0]/[*R*0] (*L*) increases ~13 times in the presence of trehalose. (**b**) Hill plot of PK cooperative kinetics of data in a (symbols as in a). The slope (*n*) value in both straight lines is ≈1.5. Adapted with permission from Rivera-Moran [164]. Copyright © 2020, M.A. Rivera Moran.

#### **5. The Application of Kramers' Theory on Heat-Mediated Enzyme Inactivation, Protein Folding, and Unfolding**

Heat-shock is another stress condition where trehalose may accumulate in the cytoplasm of some organisms [4,6,8,14,24,129,165], e.g., in yeast, trehalose accumulation (~0.5 M) has been related to the ability of the cells to survive at relatively high temperatures [21,165]. The number of conformational states a protein may sample is related to the surrounding temperature [75]. When raising the temperature, the structural fluctuations increase critically up to unfold/denature the three-dimensional (3D) structure of the protein as described in Figure 1c,d; in the denatured state, the number of possible conformations becomes immense [23,166,167]. The first event to occur in enzymes when undergoing unfolding is the loss of catalytic activity [168–171]. It has been proposed that the high viscosity generated by trehalose stabilizes proteins by inducing preferential hydration of the protein surface [60,61], preserving the hydration shell and decreasing its dynamics [172], increasing surface tension on the trehalose-water phase [59], and hampering protein motions that lead to unfolding [61,129,173]. Hence, thermal stabilization mediated by trehalose has been observed in a diversity of enzymes, such as bovine intestine alkaline phosphatase (BIALP) [174], human brain-type creatine kinase (hBBCK) [175], ornithine carbamoyltransferase (OCTase) [176], polyphenol oxidase (PPO) from yacon roots (*Smallanthus sonchifolius*) [177], rabbit muscle phosphofructokinase (PFK) [153], citrate synthase (CS) [25], cutinase from *Fusarium solani pisi* [178], larval *Manduca sexta* fat body glycogen phosphorylase B (GPb) [179], *Renilla* luciferase (Rluc) from *Renilla reniformis* [180], yeast cytosolic pyrophosphatase (PPi) [181], and others. In this regard, the MDS of thermal inactivation and unfolding of Rluc in the presence of trehalose showed that Rluc is stabilized through the inhibition of the structural fluctuations generated by high temperature [180], while in thermal inactivation of lysozyme and glucoamylase (GA) from *A. niger* and α-amylase (AA) from *Bacillus sp*, it was found that trehalose prevents the loss of protein hydrophobic interactions and helical structure, thus avoiding protein aggregation [150,182,183].

The kinetics of enzyme thermal inactivation in the presence of trehalose has been determined experimentally for some enzymes where usually first-order kinetics is observed [178,184,185]. Kinetic analysis showed that trehalose decreases the inactivation rate constant (*k*i) of xanthine oxidase (XO) from *Arthrobacter* M3 [184], PK from rabbit muscle [186], GOx from *A. niger* [151], and mushroom tyrosinase (MT) from *Agaricus bisporus* [185]. Therefore, trehalose increases the thermal stability of the enzymes by decreasing the inactivation rate constant (*k*i) [129]. Interestingly, thermal inactivation of the plasma membrane H+-ATPase from the yeast *Kluyveromyces lactis* is irreversible, showing biphasic kinetics with the presence of an active intermediary [129,173], i.e., a three-state mechanism (N → I → U) (Figure 1c) with respective inactivation rate constants for each phase, *k*i1, and *k*i2. Notably, physiological concentrations of trehalose stabilize the H+-ATPase by decreasing *k*i1 and *k*i2 [173,187]. The high viscosity generated by trehalose has been proposed as the mechanism of thermal stabilization of H+-ATPase [129,173]. Detailed analysis of the H+-ATPase thermal inactivation kinetics shows that the inactivation rate constants (*k*i1 and *k*i2) are inversely proportional to medium viscosity (*k*<sup>i</sup> α 1/η) as described by Kramers' theory and Equation (3) [129]:

$$k\_i = \frac{\mathbf{A}}{\eta} e^{-\Delta l I/RT} \tag{3}$$

where *k*<sup>i</sup> is the first-order inactivation rate constant, η the viscosity of the medium, *A* has the same meaning as in Equation (1) but regarding the enzyme inactivation process, Δ*U* is the potential energy barrier for inactivation, *R*is the gas constant, and *T* is the absolute temperature [129]. The plot of *k*<sup>i</sup> versus η−<sup>1</sup> usually shows a straight line when a diffusive component on the thermal inactivation mechanism of the enzyme exists [129,173,188], e.g., the dissociation of monomers from a large oligomer [188]. Analysis of the decrease of the inactivation rate constant by trehalose shows that trehalose increases the activation energy (*E*a) for thermal inactivation (Figure 1) of AA [183], H+-ATPase [173], and GOx [151].

Particularly, in GOx where catalysis is dependent on the presence of the coenzyme FAD in the active site [151], the thermal inactivation occurs first through the formation of a molten globule and after that FAD is released (Figure 5a) [189], resulting in the irreversible loss of GOx activity [151,189]. Interestingly, trehalose decreases the rate constant of the thermally generated large structural changes hence preventing the formation of the molten globule [151]. Due to the hampering of the FAD release, the inactivation rate constant (*k*i) of GOx decreases [151] (Figure 5b), the *T*1/<sup>2</sup> for FAD release is increased to higher temperatures (Figure 5b). In this unfolding mechanism, trehalose seems to act by compacting the protein structure of GOx, thus preventing the release of FAD, and as a consequence, GOx retains the catalytic activity [151]. A similar stabilizing pattern by trehalose was observed in OCTase against thermal inactivation [176].

The dependence of protein unfolding on the viscosity (Figure 1d) generated by trehalose presence has been observed in cytochrome *c* (Cytc) [140], phosphoglycerate kinase (PGK), β-lactoglobulin (βLG) [190], cutinase, and the small protein barstar (Bs) [169]. Like in thermal inactivation, trehalose decreases the rate constant of thermal unfolding (*k*U) by promoting the compactness of the unfolded state [191] and inhibiting large structural fluctuations [180]. Regarding protein folding, the high viscosity generated by trehalose affects the folding rate by inhibiting the structural dynamics as described by Kramers' theory [192–197].

**Figure 5.** Trehalose prevents the release of FAD during thermal inactivation/unfolding of glucose oxidase (GOx) from *Aspergillus niger* [198]. (**a**) Thermal inactivation mechanism of GOx [151,189]; (**b**) Effect of trehalose on the release of FAD from GOx during thermal inactivation; Trehalose (-) 0.0, and (•) 0.6 M. The data were fitted to the Pace equation [199] by non-linear regression; *T*1/<sup>2</sup> = 61.3 and 64.3 ◦C in the absence and presence of 0.6 M trehalose, respectively. Adapted with permission from Paz-Alfaro [198]. Copyright © 2020, K.J. Paz-Alfaro.

Finally, the effect of the high viscosity generated by trehalose on macromolecular folding is being currently extended to other important biopolymers such as DNA [200,201] and RNA, e.g., in human telomere sequence [202] and RNA tertiary motif of the GAAA tetraloop receptor (TLR) [203], human telomerase hairpin (hTR HP) and H-type pseudoknot from the beet western yellow virus (BWYV) [204].

In cells, trehalose synthesis and accumulation helps to cope with harsh stress conditions. Trehalose affects enzyme activity (catalysis) and stability by hindering protein mobility through the high viscosity generated. Viscosity-mediated effects of trehalose (and other solutes) on enzymes are explained by Kramers' theory, i.e., the rate constant (*k*) of a given diffusive process in the enzyme structure is inversely proportional to the media viscosity. Currently, trehalose is widely used in enzyme applications (in the laboratory and biotechnology), whether alone or in combination with other compounds; therefore, it would be safe to consider the effect of viscosity on enzyme structural dynamics to improve the results of the use of trehalose.

**Author Contributions:** Conceptualization, J.G.S.; Methodology, J.G.S.; Validation, J.G.S. and M.A.R.-M.; Formal Analysis, J.G.S., M.A.R.-M., and S.U.-C.; Investigation, J.G.S. and M.A.R.-M.; Resources, J.G.S. and S.U.-C.; Data Curation, J.G.S.; Writing—Original Draft Preparation, J.G.S.; Writing—Review & Editing, J.G.S., M.A.R.-M., and S.U.-C.; Visualization, J.G.S.; Supervision, J.G.S.; Project administration, J.G.S.; Funding Acquisition, J.G.S. and S.U.-C. All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was partially funded by FAI-UASLP México grant number C19-FAI-05-89.89 to J.G.S. and by a grant to S.U.-C.: DGAPA/PAPIIT Project IN203018.

**Acknowledgments:** M.A.R.-M. is a recipient of a Ph.D. fellowship of CONACyT México.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Abbreviations**

Molecular dynamics simulation: MDS; intrinsically disordered proteins: IDPs; C-phycocyanin: CP; lactate dehydrogenase: LDH; carbonic anhydrase: CA; chloroplast coupling factor CF1; fatty acid synthetase: FAS; carboxypeptidase-A: CPA; ribosomal protein S4: RPS4; polymerase β, POLB; oligo-1,6-glucosidase 1: MalL; indole-3-glycerol phosphate synthase: IGPS; sulfite oxidase: SO; glucose oxidase: GOx; pyruvate kinase: PK; ferrocytochrome *c*: FCC; bovine intestine alkaline phosphatase: BIALP; human brain-type creatine kinase: hBBCK; ornithine carbamoyltransferase: OCTase; polyphenol oxidase: PPO; phosphofructokinase: PFK; citrate synthase: CS; glycogen phosphorylase B: GPb; *Renilla* luciferase: Rluc; pyrophosphatase: PPi; glucoamylase: GA; and α-amylase: AA; xanthine oxidase: XO; mushroom tyrosinase: MT; cytochrome *c*: Cytc; phosphoglycerate kinase: PGK; β-lactoglobulin: βLG; barstar: Bs; tetraloop receptor: TLR; human telomerase hairpin: hTR HP; beet western yellow virus: BWYV.

#### **References**


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