L Marcucci

L Truskinovsky

Muscle contraction: A mechanical perspective

In this paper we present a purely mechanical analog of the conventional chemo-mechanical modeling of muscle contraction. We abandon the description of kinetics of the power stroke in terms of jump processes and instead resolve the continuous stochastic evolution on an appropriate energy landscape. In general physical terms, we replace hard spin chemical variables by soft spin variables representing mechanical snap-springs. This allows us to treat the case of small and even disappearing barriers and, more importantly, to incorporate the mechanical representation of the power stroke into the theory of Brownian ratchets. The model provides the simplest non-chemical description for the main stages of the biochemical Lymn-Taylor cycle and may be used as a basis for the artificial micro-mechanical reproduction of the muscle contraction mechanism.

Introduction

The sliding-filament hypothesis [1][2][3] assumes that during contraction actin filaments move past myosin filaments while actively interacting with them through myosin heads known also as cross-bridges. The interaction is powered by the hydrolysis of ATP and the motor part of the myosin head acts as an enzyme which, simultaneously, increases the hydrolysis reaction rate and converts the released chemical energy into useful work. The whole system remains in permanent disequilibrium because the chemical potentials of the reactant (ATP) and the products of the hydrolysis reaction (ADP and P) are kept out of balance by a steadily operating exterior mechanism (e.g. [START_REF] Alberts | Molecular Biology of the Cell[END_REF][START_REF] Howard | Mechanics of Motor Protein and the Cytoskeleton[END_REF][START_REF] Jülicher | Transport and Structure,e d i t e db yS[END_REF]). From an engineering point of view, it is of interest to understand how the energy input due to a non-equilibrium chemical reaction and a random thermal noise are incorporated in this ingenious design to produce a highly efficient force-producing machinery. A purely mechanical model of an attached cross-bridge has been recently introduced in [START_REF] Marcucci | [END_REF]. Here we develop this model further by including also the description of attachment-detachment process.

A simplified four-state biochemical scheme of the cross-bridge cycle, known as Lymn-Taylor model [3], contains the description of the most important chemical states involved in muscle contraction. These states, commonly known by their chemical compositions as M-ATP, A-M-ADP-P, AM-ADP and AM, can be associated with particular mechanical configurations of the globular rea Present address: Graduate School of Frontier Biosciences, Osaka University, 1-3 Yamadaoka, Suita, Osaka 565-0871 Japan; e-mail: lorenzo.marcucci@gmail.com gion. The Lymn-Taylor model assumes that when attached to actin, each myosin head uses stored potential energy to stretch (or move) the actin filament through a rotation of the lever arm representing the terminal part of the myosin head (power stroke). The underlying conformational change, advancing the system from the pre-powerstroke state to the post-power-stroke state, is assumed to be related to a chemical transition A-M-ADP-P → AM. To return in its original pre-power-stroke configuration the head needs to split an ATP molecule. In the process, it first detaches from the actin filament (chemical transition AM → M-ATP) and then uses the energy of hydrolysis to recharge (reprime) the power stroke mechanism and to bind a new active site on the actin filament (chemical transition M-ATP → A-M-ADP-P). Subsequently, the whole process repeats itself (e.g., [START_REF] Howard | Mechanics of Motor Protein and the Cytoskeleton[END_REF][START_REF] Hill | Free Energy Transduction and Biochemical Cycle Kinetics[END_REF][START_REF] Geeves | [END_REF][10]).

The chemo-mechanical approaches to muscle contraction were developed to complement purely biochemical schemes. They mostly follow the ideas first proposed in the classical papers [11,12] that can be seen as complementary since the first paper describes the attachmentdetachment process and the events related to the slow time scale (hundreds of milliseconds), while the second paper deals with the power stroke and the events related to the fast time scale (milliseconds). There have been a considerable number of attempts to extend (e.g. [START_REF] Hill | Free Energy Transduction and Biochemical Cycle Kinetics[END_REF][13][14][15][16]) and to unify these two types of models (e.g. [START_REF] Jülicher | Transport and Structure,e d i t e db yS[END_REF]15,[17][18][START_REF] Duke | Proc. Natl. Acad. Sci. U.S.A[END_REF][START_REF] Vilfan | [END_REF][21][22][23][24]). In the chemo-mechanical framework, muscle contraction is modeled as a set of reactions among discrete chemical states of a motor molecule. These states form a network describing, on one side, various chemical compositions of the catalytic domain and, on the other side, different mechanical configurations of the myosin head. A variety of crystallographically identified configurations (attached and detached [11], strongly and weakly attached [25], pre and post power stroke [12], associated with the first or second myosin head [26], etc.) are then identified with particular sites of the chemical network.

The mechanics enters the chemical models through phenomenological assumptions regarding the dependence of the chemical rate constants on a single continuous variable characterizing the state of the force-generating spring. It is implied that the discrete chemical states are replaced by one-parametric configurational spaces. The functional freedom remaining in these models after the local conditions of detailed balance are fulfilled originates from the lack of information about the energy barriers separating different states. In the chemo-mechanical models this freedom is used to fit the experimental curves and therefore the best of these models reproduce available data quite well. However, such approach gives only limited insight regarding the actual micro-mechanical functioning of the force-generating mechanism (e.g. [24] and references therein).

The development of the theory of Brownian motors which are fundamentally involved in muscle contraction, has followed a more mechanically explicit path (e.g. [27][28][29][30][START_REF] Peskin | Cell Mechanics and Cellular Engineering,e d i t e db yV[END_REF][START_REF] Prost | [END_REF][33][34]). In this class of models the motion of the myosin II is typically represented as a biased diffusion of a particle on a periodic asymmetric potential subjected to a colored noise. The white component of the noise reflects the presence of a heat reservoir while the correlated component mimics the non-equilibrium chemical environment. A direct application of these models to muscle can be found in the papers [21,35,[START_REF] Esaki | Proc. Jpn. Acad[END_REF] that are focussed on the attachment-detachment process at the expense of the analysis of the short time scales, requiring a detailed description of the power stroke. Such reduced models collapse the four-state Lymn-Taylor cycle onto a two-state cycle by absorbing the configurational changes associated with the transitions M-ATP → A-M-ADP-P and AM → M-ATP into more general transitions M-ATP → AM-ADP and AM-ADP → M-ATP. On the mechanical level the complexity of the structure of the myosin head is once again reduced to a single degree of freedom representing a stretch of a series elastic spring.

Despite the unquestionable heuristic value of such minimal models, the crystallographic reconstructions suggest that the myosin head is formed by at least two irreducible domains which act in synchrony to generate force. Thus, in the classical lever arm theory, the myosin is divided into a motor or catalytic domain (CD), which contains the actin binding site together with the ATP binding pocket. This domain is attached to the actin filament during the power stroke. The power stroke is generated by the combined action of the converter region, which ensures rotation (in the range of 70 degrees), and the long terminal part of the myosin head, which amplifies the rotation acting as a lever arm and causing the relative motion of the two filaments. Since the lever is formed of a heavy chain helix which carries two light chains, it is sometimes called light chain domain [START_REF] Howard | Mechanics of Motor Protein and the Cytoskeleton[END_REF][START_REF] Cooke | [END_REF]38]. Mechanically, the converter and the lever arm can be seen as one bi-stable molecular complex and in what follows we refer to it as the "snap-spring" (SS) [START_REF] Marcucci | [END_REF]. Notice that most of the converter related activity (associated with a millisecond time scales) is lost in the Brownian ratchet theories where the SS domain and the CD domain are usually combined into one collective variable which represents the motor location on the actin filament.

In this paper we argue, following [17,39], that in order to deal simultaneously with the fast time scale response, which implies an almost mechanical discharge of a power stroke mechanism [12], and the slow time scale isometric contractions, necessarily implying the attachmentdetachment cycle [40], the minimal theory must operate with at least two non-trivial mechanical degrees of freedom describing separately the state of the CD and the state of the SS. The CD related internal variable, responsible for the attachment-detachment process, should be exposed to a periodic asymmetric potential generated by the actin filament. On the contrary, the SS related variable, responsible for the power stroke, is expected to see a bi-stable energy landscape. In addition to these two degrees of freedom one also needs a trivial mechanical variable describing a series elastic spring. This spring creates micro-metastability (see [START_REF] Marcucci | [END_REF]) and allows one to bridge the cases of isotonic (soft device) and isometric (hard device) contractions.

Following these heuristic ideas, we develop in the present paper a minimal purely mechanical analog of the conventional chemo-mechanical approach. To achieve uniformity of the mathematical language we abandon the Huxley-Simmons-type description of kinetics of the power stroke in terms of a jump process and instead attempt to resolve the continuous stochastic evolution on an appropriate energy landscape. In general physical terms, we replace a conventional description of the active-force-generating system in terms of hard spin variables (indicating different mechanical states in the chemical approach) by a fully mechanical description in terms of soft spin variables imitating bi-stable snap-springs. This allows us to treat the cases of small and even disappearing barriers which are not amenable to classical Kramers theory [41] and, more importantly, to incorporate seamlessly the mechanical representation of the power stroke into the theory of Brownian ratchets (see other advances in the same direction in [START_REF] Jülicher | Transport and Structure,e d i t e db yS[END_REF][21][22][23]42]).

To represent all three main collective variables characterizing skeletal Myosin II -position of the motor domain, configuration of the lever domain and the stretch state of the series elastic element-we use three continuous coordinates and project a more general MD picture into the stochastic Langevin dynamics on a threedimensional phase space. For determinacy and mechanical transparency we utilize a particular representation of the Brownian ratchet proposed by Magnasco [29,[43][44][45]. Although this ratchet has low efficiency due to considerable diffusion which accompanies the drift, it has unambiguous mechanical meaning. We expect that other ratchet models could also be used as the building blocks of similar mechanical models (e.g. [33][34][35]).

The paper is organized as follows. In sect. 2 we introduce the system of stochastic ordinary differential equations describing the behavior of N cross-bridges posed in parallel. The parameter selection and scaling are discussed in sect. 3. In sect. 4 we simulate the behavior of an effective sarcomere and show that the model can successfully reproduce the observed mechanical response of the muscle fiber to sudden length and force increments and produces a realistic force-velocity curve. In sect. 5 we focus on the detailed description of the sequence of mechanical processes behind isotonic contractions. The final sect. 6 contains our conclusions.

The model

We view a half-sarcomere as a parallel connection of N cross-bridges. Each cross-bridge is represented as a chain containing a linear elastic spring, which generates the force, a bi-stable contractile element, responsible for the power stroke (SS), and a molecular motor representing the ATP regulated attachment-detachment process (CD) (see fig. 1). As in [35], each cross-bridge is attached to a common rigid backbone (thick filament), however, we substitute the rigid links adopted in [35] with the bi-stable snap-springs. The system is loaded either by a given force f representing a cargo or is constrained by the fixed displacement of the backbone.

The elastic energy of the linear spring can be written as

E e (ǫ e )= 1 2 K(ǫ e -l) 2 , (1)
where ǫ e is the deformed length, K is the elastic modulus and l is the reference length characterizing the pre-strain.

The energy of the bi-stable lever arm mechanism (light chain domain or SS) is assumed to be piece-wise quadratic

E c (ǫ c )= ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ 1 2 k 1 (ǫ c ) 2 ,ǫ c ≥ b 1 , - 1 2 k 3 (ǫ c -b) 2 + c, b 2 ≤ ǫ c <b 1 , 1 2 k 2 (ǫ c -a) 2 + d, ǫ c <b 2 , (2)
where ǫ c is the elongation, k i , i =1 , 2 are the curvatures of the distinct energy wells representing pre-power-stroke and post-power-stroke configurations, respectively; a<0 is the characteristic size of the power stroke. The driving force d sets the bias towards the post-power-stroke state and is somewhat arbitrarily chosen to ensure that the two wells have the same energy in the isometric contraction. We model the catalytic domain (CD) as the Brownian ratchet of Magnasco type [29]. It can be viewed as a particle moving in an asymmetric periodic potential and subjected to a correlated noise. The periodic potential is assumed to be piece-wise linear in each period n

E m (ǫ m )= ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ Q λ 1 L (ǫ m -nL), 0 <ǫ m -nL < λ 1 L, Q λ 2 - Q λ 2 L (ǫ m -nL),λ 1 L<ǫ m -nL < L, (3)
where Q is the amplitude, L is the period, λ 1 -λ 2 is the measure of the asymmetry of the potential and λ 1 +λ 2 =1. The variable ǫ m indicates the location of a particle in the periodic energy landscape: by our convention it is attached if it is close to one of the minima and is detached if it is close to one of the maxima.

It is convenient to identify the coordinates of the nod points on fig. 1 as x, y and z, implying that ǫ m = x, ǫ c = y -x and ǫ e = z -y. Using these notations the system of N cross-bridges connected in parallel can be described by the set of Langevin equations

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩ η x ẋi = -E ′ m (x i )+E ′ c (y i -x i )+F (t+t i )+ √ η x k B θΓ (t), η y ẏi = -E ′ c (y i -x i ) -K(y i -z -l i )+ η y k B θΓ (t), η z ż = N i=1 K(y i -z -l i )+f + η z k B θΓ (t).
(4) Here the random term Γ (t), representing thermal fluctuations, has the standard properties

Γ (t 1 ) =0, Γ (t 1 )Γ (t 2 ) =2δ(t 1 -t 2 ),
where θ is the temperature and k B is the Boltzmann constant. All elements experience viscous resistance characterized by the corresponding drag coefficients η x , η y , η z . The correlated component of the noise F (t), imitating the activity of the ATP, is assumed to be periodic and piecewise constant

F (t)= +A, 0 <t-mT < T, -A, T < t -mT < 2T. ( 5
)
m =0 , 2, 4,..., where A is the amplitude and 2T is the period.

Since the ATP activity at different sites is expected to occur independently, we introduced temporal de-synchronization in (4) through the phase shift parameters t i . We assume t i as independent random variables uniformly distributed in the interval [0, 2T ]. Following [18] we also assume that the pre-strains l i are different for different i which introduces spatial de-synchronization. Again, we assume that l i are independent random variables that are now uniformly distributed in the intervals [iL -a/2,i L+ a/2]. Such temporal and spatial desynchronization eliminates spontaneous oscillations that were observed in muscles under some special conditions (e.g. [46]) and later explained in the collective Brownian ratchet framework by introducing chemo-mechanical feedback effects [47].

Normalization and parameter selection

To obtain quantitative predictions, we need first to nondimensionalize the system of equations ( 4). For normalization we choose the (power-stroke-related) length scale a and the (viscosity related) time scale τ = a 2 η y /(k B θ).

It is also natural to choose the energy scale k B θ and the force scale k B θ/a. The dimensionless system depends on several non-dimensional parameters which fully define its qualitative behavior.

Knowing that K ∼ 2pN/nm, a ∼ 8nm and k B θ ∼ 4pN/nm (e.g. [START_REF] Howard | Mechanics of Motor Protein and the Cytoskeleton[END_REF]), we put for the main measure of elasticity the value Ka 2 /k B θ = 32; the values of relative elasticities of the two conformations k 1 /K =5 ,k 2 /K = 2 then guarantee that we can reproduce the linear regimes in the T 1 and T 2 data. The last elastic parameter k 3 /K =2 .45, defining the energy barrier for the power stroke, was chosen to ensure the appropriate rate of fast force recovery relatively to the rate of slower movement of the CD controlled by parameter Q (see below). The assumptions b/a = -0.247 and c/a =1 .61 make the barrier between the two energy wells sufficiently localized to exclude its interference with the equilibrium behavior associated with the two energy minima. We have chosen the value η x /η y = 10 to make the process of detachment and reattachment of the myosin head sufficiently slower than the power stroke. The parameter η z /η y = 100 describes even slower movement of the backbone.

The period of the actin potential L was taken to be twice as large as the characteristic amplitude of the power stroke L/a = 2. It implies a much longer distance traversed by the CD in the attachment-detachment process comparing to the characteristic amplitude of the power stroke. This assumption is crucial for the possibility of a purely mechanical recharge of the power stroke mechanism and at this stage of the schematization, we did not attempt to link it with the actual geometry of the actin and myosin molecules.

The normalized period of the external force imitating the ATP hydrolysis T/τ =210 4 was chosen to ensure sufficient number of tilting oscillations in the ratchet between two attachment-detachment events. We fixed the asymmetry parameter at λ 1 =0 .3. To speed up numerical exper-iments we artificially increased the rate of the processes reducing the energy barrier and choosing Q/(k B θ) = 60. For the ATP-related force amplitude we assumed the value A =1 .4 Q/(1 -λ 1 )L between the minimal and the maximal values of the force generated by the asymmetric potential. Given the size of the energy barrier in the power stroke mechanism, the above "accelerating" assumptions do not affect the normalized curves T (δ)a n dV (t). We studied the system (4) numerically by using the simplest Euler discretization of the Langevin equations. We assumed that N = 2000 implying that experimental data are obtained for many sarcomeres working in unison. The dimensionless time scale of discretization 5 10 -5 was chosen to be below the smallest physical time scale in the model.

Length clamp and force clamp experiments

The mechanical behavior of the contracting muscle in the short time scale (milliseconds) revealing the quasimechanical nature of the power stroke mechanism was first studied in [12] and our model allows one to reproduce numerically these classical length clamp experiments. In the simulations we applied to the backbone z a sudden increment of length δ at time t. The increments varied in the interval [-1.4a, 0.1a] and the force generated by the system was evaluated from

T (t)= N i=1 K(y i (t) -z(t) -l i ). ( 6
)
The normalized response curves T (t) (see fig. 2), closely resemble the experimentally observed behavior (e.g., [12]). Thus, in the first few microseconds after the application of the displacement increment one observes an elastic response usually denoted by T 1 (δ), which is followed by a recovery of the initial value of the isometric force (T 0 ). Between T 1 (δ)a n dT 0 one can distinguish a plateau, known as T 2 (δ). The simulated functions T 1 (δ) and T 2 (δ), shown in fig. 3, quantitatively agree with experimental observations (experimental points from [48] normalized for a = 8 nm), including the prediction of a pronounced softening region around δ =0.

A particularly important prediction concerns the difference in slopes of the linear segments of the curves T 1 (δ) and T 2 (δ) at large values of the applied shortening. Such divergence of the two curves agrees well with recent precise measurements [48] and indicates the sensitivity of the amplitude of the power stroke to the value of the applied force. We stress that this effect, which is due to the different curvatures of the two wells in the power stroke potential E c , cannot be obtained in the hard spin model with only two chemical states.

In another typical experiment, illustrating the mechanical response of skeletal muscles, one records the velocity of contraction at different external loads (e.g. [40]). It is known that a myofibril contracting against a constant load lower than T 0 , assumes, after a transient related to the

Mechanical phenomena behind isotonic contractions

In this section we study the details of the mechanical functioning of the model at different time scales by looking more closely at the case of isotonic contraction against a fixed load.

To interpret correctly the behavior of the individual mechanical units we first fix the parameter z =0a n d write the total energy of an attached cross-bridge as a Fig. 4. Force-velocity curve generated by the model (triangles) compared to the experimental points (dots). The triangles are obtained from the trajectories z(t) corresponding to different values of f (see the insert). The velocity of shortening is computed as (z(t f ) -z(ti))/(t fti) and normalized to V0.T h e experimental points are obtained from [48].

function of two remaining mechanical variables y and x:

E t (x, y)=E m (x)+E c (y -x)+E e (-y). (7)
The corresponding energy landscape is shown in fig. 5.

Here the upper two local minima A and B indicate the prepower-stroke and the post-power-stroke configurations of an attached motor, respectively. The two lower local minima A ′ and B ′ correspond to the pre-power-stroke and the post-power-stroke configurations of a motor that has shifted to a new attached position on actin potential. We associate the detached state with an unstable position around the maximum separating the minima (A, B)a n d (A ′ ,B ′ ). One can see that the system cannot remain in a detached position without an appropriate stimulation.

Using fig. 5 we can visualize the combined working of the power stroke and the attachment-detachment mechanism. Suppose that the upper right minimum A corresponds to the initial pre-power-stroke configuration with the attached portion of the head fixed in the first preferred site on the actin filament. In this state the system can lower the energy by rotating the lever arm and reaching the post-power-stroke configuration B. This process does not require the energy supply from outside and can be viewed as a purely mechanical relaxation.

Suppose next that external stimulation, representing the ATP activity, can shift variable x towards smaller values (transition B → B ′ ). If the corresponding ratchet is sufficiently strong it can simultaneously advance the representative point on fig. 5 further into the pre-power-stroke position corresponding to the point A ′ . The new transition B → A ′ would then mean mechanical recharging of the power stroke mechanism in the course of transition to the next preferred actin site. If in the meanwhile the muscle shortens, making the power stroke again energetically preferable, the process can be potentially repeated again starting with a mechanical relaxation A ′ → B ′ .Infig.6we show that this scenario indeed takes place in our numerical simulations when we reproduce load clamp experiment at f =0.5 T 0 .

From fig. 6 one can see that the CD of an individual head, x, goes through three different attachment positions (see fig. 6a). In fig. 6b we show the time history of the variable x -y characterizing the internal configuration of a given head (SS) during the cycle. The first vertical line shows the moment in which the power stroke A → B takes place. The second vertical line shows the motion of the attached portion from the active site i on the actin filament to the next site i ′ = i + 1 corresponding to the transition B → A ′ . This motion induces a change of state in the bi-stable element which brings the lever arm into the pre-power-stroke position. Due to the evolution of the variable z (isotonic contraction, see fig. 6d) the elastic element whose configuration can be read in fig. 6c, relaxes and the post-power-stroke minimum B ′ becomes preferable. The third vertical line shows the moment in which the new power stroke A ′ → B ′ takes place. Then the cross-bridge cycle starts again.

Observe that the position of the backbone can be considered constant during the recharging of the power stroke. In this situation, the key factor for the possibility of recharging (after the variable x has overcome the barrier the periodic potential) is that the total energy E t (x, y) has a minimum when SS is in the pre-power-stroke state. This is true providing (Q/(k B θ))/(λL) >d/awhich places an important constraint on the choice of parameters.

One can see that the model is able to qualitatively reproduce some of the elementary steps of the basic crossbridge cycle. Thus, the fast power stroke of the head stretches the elastic element which in turn pulls the myosin backbone and causes contraction. In the meantime a slower process allows the myosin head to come back to the pre-power-stroke state (recharge), leaving the tension in elastic element almost constant. Finally, due to the motion of the myosin backbone, the head becomes capable of producing another power stroke and the cycle repeats itself. We observe that the power stroke recharges at exactly the same times when the main force producing motor is detached. This suggests that the important role of the power stroke in this cycle is to smoothen the functioning of the force-generating mechanism.

A direct comparison of the simulated mechanical cycle with the conventional Lymn-Taylor cycle shows that while the two attached configurations are represented by the model rather faithfully, the detached configurations appear only as transients. In fact, one can see that our (slow) transition B → A ′ represents a combined description of the detachment, of the power stroke recharge and then of another attachment. In the biochemical cycle such a transition would be described by at least two distinct chemical states. To achieve such splitting in the framework of our model one would need to have a more adequate representation for the detached state. More specifically, an assumed simple passing through a single energy barrier may be in reality a complex series of events which include different stages along the way towards the final change in the attachment position and the recovery of the power stroke. This suggests the necessity of an additional fine structure of the areas near maxima of the actin potential.

The reasons for the observed discrepancies between mechanical and chemical cycles may also lay deeper. Recall, for instance, the implicit assumption in all chemomechanical models that the ATP energy can be decomposed into two parts. One part drives the attachmentdetachment process by sustaining the out-of-equilibrium transition rates (e.g. [START_REF] Prost | [END_REF]). The remaining part ensures the recovery of the power stroke and our model provides a simple mechanism of how it may be working: the power stroke is mechanically recharged in the course of the change of the attached position of the CD. It is clear, however, that the effect of the ATP hydrolysis may not be fully reducible to the periodic rocking of the actin potential. For instance, the ATP-induced mechanical signal may also affect the bi-stable potential and this will considerably change the structure of the effective cycle.

Concluding remarks

In this paper we argue that a mechanical approach to muscle contraction, initiated by the theory of Brownian ratchets, offers certain important advantages comparing to the purely biochemical approach. First, it allows one to deal in a consistent manner with distributed chemical states, each characterized by several continuous degrees of freedom. Second, it lets one insert the essential information about the energy barriers in an explicit and coherent way. Third, it offers the possibility to include into the study the states that are only minimally stable in the sense that they are surrounded by arbitrarily small barriers.

Despite its rather elaborate mechanical nature, our model remains mostly schematic and can be considered only as prototypical. A particularly serious simplification consists in the decoupling of the chemical and mechanical pictures in relation to the power stroke. In this way we neglect the fact that ADP has to be released during the discharge and that ATP has to be hydrolyzed during the recharge of the power stroke mechanism. Instead, the power stroke is recharged and discharged due to the mechanical stretch assisted by thermal fluctuations. Another simplification is that we associate the detachment with the state of the weakest interaction between the crossbridge and the actin filament without trying to distinguish the sub-states (M-ATP, A-M-ADP-P) and match the observed duty ratio [START_REF] Howard | Mechanics of Motor Protein and the Cytoskeleton[END_REF]. Finally, the non-equilibrium chemical reservoir enters the picture only through the correlated external noise imitating simultaneously both binding and hydrolysis.

The reported numerical experiments raise important questions regarding mechanical interdependency between different biochemical states involved in muscle contraction. Thus, in the framework of chemo-mechanical models the force generation is usually linked exclusively to the power stroke. The power stroke, indeed, plays a major role in the fast recovery of tension, however, at slower time scales the attachment-detachment mechanism also contributes to force generation by effectively recharging the power stroke mechanism. Since the ATP activity is required for such recharging, one can say that the hydrolysis of ATP plays an essential role in the working of the power stroke despite its apparently "mechanical nature". This idea is supported by the fact that even the fast tension recovery in rigor is much smaller than in tetanus [49].

Here it is appropriate to mention that already Eisenberg and Hill [39] associated the difference in the level of the minima in the energy of the bi-stable element with the free energy liberated by the ATP hydrolysis.

The recharging mechanism proposed in the present paper could be made compatible with other choices of Brownian ratchets provided that certain conditions on E t (x, y) detailed in the previous section are satisfied. For instance, rocking ratchet used in this paper could be replaced by the flashing ratchet (e.g. [21,35]) generating correct duty ratio and producing interesting cooperative effects [50]. Flashing ratchet models, however, introduce a jump switching process which is not easy to interpret in purely mechanical terms.

Due to the minimal nature of the present model many other important features of the phenomenon of muscle contraction, including the possibility of additional conformational states [12], the double-head structure of the myosin head [26], the elasticity of the filaments [51], the elastic interaction among many sarcomeres in series [START_REF] Vilfan | [END_REF], the contractional instabilities and waves [52] and the strong temperature dependence of the entropic elasticity of the elements [53], have been left outside the scope of the analysis. Despite these limitations our mechanical model offers a consistent description of the ATP-driven motor undergoing periodic attachment detachment which is concurrent with a mechanically driven configurational change. The use of the mechanical framework allows one to resolve the long standing controversy regarding the load dependence of the magnitude of the power stroke and provides the first non-chemical description for the main stages of the biochemical Lymn-Taylor cycle. Most importantly, the mechanical perspective opens the way to recreate the intricate machinery behind muscle contraction artificially and to potentially build larger devices exhibiting active elasticity.

Fig. 1 .

 1 Fig. 1. The mechanical representation of a cross-bridge as a series connection of a linear elastic element Ee, a bi-stable snap-spring E c and a Brownian ratchet Em.

  The energy barrier is characterized by the position b,t h e height c and the curvature k 3 of the maximum. The values of parameters b 1 and b 2 are chosen from the condition of continuity of the energy function.

Fig. 2 .

 2 Fig. 2. Force versus time response curve for δ = -1.1a.T h e value of T 2 is obtained by averaging in the dashed column. The insert shows the functions T (τ ) at different values of δ (see also fig. 3).

Fig. 3 .

 3 Fig. 3. Triangles: the values of T1 and T2 generated from the force trajectories shown in fig. 2. Dots: experimental data from [48] scaled with a =8nm.

Fig. 5 .

 5 Fig. 5. Contour plot of the effective energy Et(x, y; z0)a tz0 = 0. Inserts illustrate the states of various mechanical subunits.

Fig. 6 .

 6 Fig. 6. The numerical simulation of the time histories for different mechanical units in a load clamp simulation at zero external force: (a) the behavior of the CD; (b) the behavior of the SS; (c) the behavior of the elastic element; (d) the total displacement of the thick backbone.

The authors thank J.M. Allain, J.F. Joanny, F. Jülicher, V. Lombardi and G. Piazzesi for helpful discussions. The work was supported by the EU contract MRTN-CT-2004-505226.


Quentin Roy
email: quentin@quentinroy.fr

Sébastien Berlioux
email: sebastien.berlioux@etu.univ-nantes.fr

Géry Casiez
email: gery.casiez@univ-lille.fr

Daniel Vogel
email: dvogel@uwaterloo.ca

Typing Efficiency and Suggestion Accuracy Influence the Benefits and Adoption of Word Suggestions

The use of word suggestion in a simple se

INTRODUCTION

Text input is one of the most common tasks on desktops, laptops 1 , tablets, and phones. Physical and soft keyboards remain the main input modality for text input even as speech-to-text is an increasingly viable alternative [START_REF] Ruan | Comparing Speech and Keyboard Text Entry for Short Messages in Two Languages on Touchscreen Phones[END_REF]. However soft keyboards, as implemented on most modern phones and tablets, have much lower entry speeds [START_REF] Palin | How Do People Type on Mobile Devices? Observations from a Study with 37,000 Volunteers[END_REF]. As an attempt to improve this, intelligent text entry techniques are integrated into typing interfaces; the two most common are auto correction and word suggestions. This paper focuses on the latter.

A word suggestion interface has two common forms: an inline suggestion to complete a partially typed word or a prediction for the next word; or buttons to enable selection among multiple word suggestions (e.g. Fig. 1). Inline suggestions are more frequently used on desktops. Multiple suggestions presented in a "bar" of buttons just above the keyboard is common on mobile devices, but also exist on desktops, for example using the Touchbar on some Apple models. Regardless of interface, suggestions are typically updated after each keystroke.

Millions of users are daily exposed to word suggestions. Despite their ubiquity, our understanding of their use, the performance gain they offer on different devices, and how accurate they need to be useful, remains incomplete. Understanding how people use these suggestions, and how their use can be improved, has a tremendous impact not only for industry, but for the text entry research community. Previous work has shown that current implementations are infrequently used, with people picking one suggestion every 63 characters [START_REF] Buschek | ResearchIME: A Mobile Keyboard Application for Studying Free Typing Behaviour in the Wild[END_REF]. As Palin et al. noted, the efficacy of using word suggestions is unclear since it has a cognitive and perceptual load: users have to switch their attention from the keyboard and the text being typed to attend to the suggestions [START_REF] Palin | How Do People Type on Mobile Devices? Observations from a Study with 37,000 Volunteers[END_REF][START_REF] Quinn | A Cost-Benefit Study of Text Entry Suggestion Interaction[END_REF]. They further argue that the usefulness of word suggestions depends on many factors, including the efficiency of the text entry method, accuracy of word suggestions, and user experience. Previous work has mixed results regarding word suggestion benefits [START_REF] Anson | The Effects of Word Completion and Word Prediction on Typing Rates Using On-Screen Keyboards[END_REF][START_REF] Horstmann | Effect of a word prediction feature on user performance[END_REF], some even found them detrimental [START_REF] Palin | How Do People Type on Mobile Devices? Observations from a Study with 37,000 Volunteers[END_REF][START_REF] Quinn | A Cost-Benefit Study of Text Entry Suggestion Interaction[END_REF][START_REF] Trnka | User Interaction with Word Prediction: The Effects of Prediction Quality[END_REF]. However, the effect of suggestion accuracy on usage, performance, and satisfaction remains unclear, nor has the impact of text entry method efficiency on suggestion usage been formally investigated in a controlled experiment.

We investigate the relationship between device typing efficiency and accuracy of word suggestions during text entry in a 36 participant mixed-design experiment. Participants performed a text transcription task. To control for typing efficiency, they used three different devices, desktop, tablet, and phone. We controlled for accuracy between subjects by manipulating how frequently a beneficial suggestion is presented as the participant types.

Our results show suggestion usage does increase with higher accuracy, but the resulting text entry speed does not improve much, and only for the highest accuracy value on mobile devices. Satisfaction, however, was greatly influenced by accuracy. We also found that natural entry speed without suggestions was a better predictor of suggestion use than device: fast typists use fewer suggestions than slow typists, even though fast typists could theoretically save the same number of keystrokes to further increase their speed. These results demonstrate that even highly accurate word suggestions will not compensate for inefficient text entry methods, and are of little benefit for accomplished typists.

BACKGROUND AND RELATED WORK

Before describing our experiment, we summarize related work about word suggestions and a related topic of acceptable accuracy.

Word Suggestion Interfaces

Text prediction systems attempt to predict the next word, or the next few words that a user will type. They may be used for automatic error correction ("autocorrect") [START_REF] Banovic | The Limits of Expert Text Entry Speed on Mobile Keyboards with Autocorrect[END_REF][START_REF] Dunlop | Multidimensional Pareto Optimization of Touchscreen Keyboards for Speed, Familiarity and Improved Spell Checking[END_REF][START_REF] Garay | Text prediction systems: a survey[END_REF][START_REF] Weir | Uncertain Text Entry on Mobile Devices[END_REF], but our focus is on their application to word suggestions. Anson et al. differentiate between two types of suggestions: word completions, which are suggestions for a partially typed word, and word predictions, which are suggestions for the next word each time a typist completes a word [START_REF] Anson | The Effects of Word Completion and Word Prediction on Typing Rates Using On-Screen Keyboards[END_REF]. Most word suggestion systems today include both forms, but most early systems focused on word completion.

Word suggestions were originally designed for Augmentative and Alternative Communication systems (AAC) to help people with disabilities to communicate, and most of the early literature focuses on users with special needs [START_REF] Patrick | Generating Text from Compressed Input: An Intelligent Interface for People with Severe Motor Impairments[END_REF][START_REF] Garay | Text prediction systems: a survey[END_REF][START_REF] Trnka | User Interaction with Word Prediction: The Effects of Prediction Quality[END_REF]. However, word suggestions are now ubiquitous on phones [START_REF] Palin | How Do People Type on Mobile Devices? Observations from a Study with 37,000 Volunteers[END_REF][START_REF] Quinn | A Cost-Benefit Study of Text Entry Suggestion Interaction[END_REF], and increasingly implemented in applications for desktop computers too.

On desktop computers, suggestions are generally displayed inline with user's input, after the insertion point (like Google Mail), or underneath it (like Apple Pages). Fig. 1 shows examples of word suggestions from commercial systems. If several suggestions are proposed, they may also be shown in a contextual menu, which is common in code editors. Most frequently, they are grouped in a dedicated area, for example a bar displayed at the top of the keyboard of mobile devices, or on the touch bar of recent Apple laptops. Typically, three words are suggested, the most likely one in the middle. However, some systems show more; for example the Microsoft SwiftKey keyboard2 on Windows 10 can suggest more than ten words (see Fig. 1e).

One important aspect of the design of word suggestions is how frequently they should be updated. Quinn and Zhai investigated this, and found users prefer suggestions to be updated after each keystroke even though it slows them down [START_REF] Quinn | A Cost-Benefit Study of Text Entry Suggestion Interaction[END_REF]. We implemented this update strategy because it is how most commercial systems work today.

Another important aspect of their design is how many suggestions should be shown. Swiffin et al. investigated the effect of the number of words suggested [START_REF] Swiffin | Adaptive and predictive techniques in a communication prosthesis[END_REF]. They found keystroke saving started to plateau after 5 suggestions. Later, Venkatagiri found that while 15 word suggestions reduced the number of keystrokes on an Augmentative and Alternative Communication program compared to only 5 suggestions, it had no effect on entry speed [START_REF] Venkatagiri | Effect of window size on rate of communication in a lexical prediction AAC system[END_REF]. While it remains unclear how many word suggestions is optimal, most modern systems propose 3, with the notable exception of Windows 10's SwiftKey keyboard. In our study, we implemented a 3-word suggestion bar.

Word Suggestion Benefits

When correct, a word suggestion provides a shortcut to typing the whole word, saving keystrokes. However, word suggestions incur cognitive and perceptual loads that lower these benefits [START_REF] Horstmann | Effect of a word prediction feature on user performance[END_REF][START_REF] Koester | Modeling the speed of text entry with a word prediction interface[END_REF][START_REF] Palin | How Do People Type on Mobile Devices? Observations from a Study with 37,000 Volunteers[END_REF][START_REF] Quinn | A Cost-Benefit Study of Text Entry Suggestion Interaction[END_REF]. In fact, analyzing data from 37,370 participants, Palin et al. found using word suggestions resulted in lower text entry speed on average [START_REF] Palin | How Do People Type on Mobile Devices? Observations from a Study with 37,000 Volunteers[END_REF]. Similar results had been previously observed for people with physical impairments. For example, in a 1996 study, Koester and Levine found that even for mouth-stick typing, word suggestions actually decreased text entry speed for participants with spinal cord injuries [START_REF] Horstmann | Effect of a word prediction feature on user performance[END_REF]. The authors conclude that the cognitive cost of word suggestions overwhelmed their benefits. One must note that text prediction algorithms improved since the time of their study. A more recent study from Wobbrock and Myers found word suggestions created a significant improvement with their EdgeWrite trackball text entry technique for special need users [START_REF] Jacob | From Letters to Words: Efficient Stroke-based Word Completion for Trackball Text Entry[END_REF].

A different, but interesting effect that word suggestions do have is that they can change the way users write. For example, Arnold et al. found the use of word suggestions, or more generally text prediction, tends to encourage predictable writing [START_REF] Arnold | Predictive Text Encourages Predictable Writing[END_REF].

To summarize, previous work indicates that the use of word suggestions tends to vary across text entry systems, but the effect of device typing efficiency on suggestion use has not been formally investigated in a controlled experiment.

Acceptable Accuracy

A reasonable question to consider is, "how accurate must word suggestions be so they are perceived as acceptable and useful?" A perfectly accurate suggestion system would certainly be very acceptable. But this implies even the first word would be suggested before typing anything, and the second word immediately suggested after that, and so on. Obviously such a perfect system requiring no user input beyond selecting every first ranked suggestion is impossible. Consequently, users always face a certain level of inaccuracy, but how much is acceptable before suggestions are ignored completely and no typing efficiency gained? This question is complex, since factors other than accuracy can impact word suggestion use, for example, even a user's emotions [START_REF] Ghosh | Does Emotion Influence the Use of Auto-Suggest during Smartphone Typing?[END_REF].

Word suggestion systems rely on a dictionary of possible words, and in their simplest form, present words from this dictionary that are near the word prefix already entered. This basic approach can be improved further by weighting the likelihood of a word as a function of its frequency in the target language, and even further by estimating its likelihood as a function of the previous words using language modelling [START_REF] Jurafsky | Speech & language processing[END_REF][START_REF] Christopher D Manning | Foundations of statistical natural language processing[END_REF].

Bi et al. proposed a tool to assess the efficiency of word suggestions and correction algorithms by automatically replaying user input [START_REF] Bi | Octopus: Evaluating Touchscreen Keyboard Correction and Recognition Algorithms Via[END_REF]. However, while it is useful to evaluate accuracy, it cannot be used to investigate user behaviour when faced with a new typing system, or more or less accurate suggestions.

The effect of word suggestion accuracy has received little attention in the literature. Notably, Trnka et al. [START_REF] Trnka | User Interaction with Word Prediction: The Effects of Prediction Quality[END_REF] investigated the effect of two different word prediction algorithms, one "basic" that relies on frequency of word series, and one "advanced" that employs natural language modelling. They found the advanced algorithm significantly improved entry speed with a soft keyboard. In addition, participants saved 93.6% of the keystrokes that could be saved with the advanced algorithms, but only 78.2% with the other. Said another way, the potential of the advanced algorithm was more utilized than the simpler one. However, Trnka et al.'s experiment only included word predictions, not word completion. Also, they did not formally control accuracy, and did not take into account the effect of the text entry method.

Suggestion algorithms have improved substantially, and more recent works show that better algorithms lead to better performance, making them worth their cost for people with special needs [START_REF] Trnka | User Interaction with Word Prediction: The Effects of Prediction Quality[END_REF]. Still, these results do not appear to generalize for other populations, even if many users report they like word suggestions [START_REF] Quinn | A Cost-Benefit Study of Text Entry Suggestion Interaction[END_REF]. Banovic et al. investigated the effect of autocorrect accuracy for spelling mistakes on phones [START_REF] Banovic | The Limits of Expert Text Entry Speed on Mobile Keyboards with Autocorrect[END_REF]. They found higher accuracy enables users to make more typing mistakes, and increase typing speed. They conclude that improving autocorrect accuracy is worth pursuing. However, Banovic et al. did not investigate word suggestions, or the effect of typing efficiency. It remains unclear how accurate word suggestions need to be to become useful.

From a higher-level perspective, word suggestions are a classic form of automation. Parasuraman and Riley define automation as "the execution by a machine agent (usually a computer) of a function that was previously carried out by a human" [START_REF] Parasuraman | Humans and Automation: Use, Misuse, Disuse, Abuse[END_REF]. When automating a task, some amount of inaccuracy is practically unavoidable. Low accuracy unavoidably impacts user trust in the automation, and as a result, reduces how often users rely on the automation [START_REF] Lee | Trust in automation: Designing for appropriate reliance[END_REF][START_REF] Parasuraman | A model for types and levels of human interaction with automation[END_REF][START_REF] Yin | Understanding the Effect of Accuracy on Trust in Machine Learning Models[END_REF]. Kay et al. proposed a survey instrument to measure the acceptable accuracy of a classifier [START_REF] Kay | How Good is 85%?: A Survey Tool to Connect Classifier Evaluation to Acceptability of Accuracy[END_REF]. They notice that the accuracy of classifiers is perceived quite differently depending on the function of their application. For example, the perceptions of a house alarm texting the owner when a possible intrusion is detected is quite different than an alarm calling the police. However, they did not consider the impact of any manual interface control.

Roy, et al. investigated the trade-off between machine automation and user manual control [START_REF] Roy | Automation Accuracy Is Good, but High Controllability May Be Better[END_REF]. They introduced the notion of controllability of an automated task, which they defined as "how much a user is 'in control' of the process", and to "what extent they can control the automation or alter its result". In other words, controllability is strongly related to how difficult it is to execute a task manually. They use a simple synthetic robot placement operation as the automated task, and manipulated the accuracy of the automation, and the amount of effort required to fix its inaccuracies using manual controls. Their participants demonstrated a strong tendency to rely on manual controls to fix the automation inaccuracies, rather than trying the automation again. We build on this previous work, but focus on a task where users can choose to use the automation or not. This is different than Roy et al.'s task where automation performs the task initially, and users can choose how to fix its inaccuracies. Our work also explore controllability in a more ecological automatable task.

OPERATIONALIZING THE ACCURACY OF WORD SUGGESTIONS

In this section, we create a user-centred definition of accuracy for a word suggestion interface and the method we use to "operationalize" word suggestion accuracy as an independent variable in a transcription task. We adopt the general strategy of Roy et al. who operationalized the accuracy of their synthetic automation system by controlling how often it produces a result that achieved the user's goal [START_REF] Roy | Automation Accuracy Is Good, but High Controllability May Be Better[END_REF]. For example, 50% accuracy means the system could complete the task for the user half of the time, or could complete half of the user's task. In the case of a text transcription task, the user's goal is to enter characters. So a word suggestion interface with 50% accuracy should be able to accurately suggest half of the characters a user needs to type.

Accuracy as Keystroke Saving

A common metric to evaluate word suggestion benefits is "keystroke saving" (KS) [START_REF] Swiffin | Adaptive and predictive techniques in a communication prosthesis[END_REF][START_REF] Trnka | Word prediction and communication rate in AAC[END_REF][START_REF] Trnka | User Interaction with Word Prediction: The Effects of Prediction Quality[END_REF]. This is the ratio of the number of keystrokes (excluding edits) that are avoided by using suggestions (𝑁 𝑠𝑘 ), to the number of characters in the phrase (|𝑃 |):

𝐾𝑆 = 𝑁 𝑠𝑘 |𝑃 | (1)
3.1.1 Defining word suggestion accuracy. In a transcription task, we know in advance exactly what will be typed, so we can operationalize the accuracy of word suggestions around this keystroke saving definition. We define accuracy 𝐴 𝑃 as the maximum keystroke savings offered by the suggestions for a phrase 𝑃 with |𝑃 | characters (including spaces):

𝐴 𝑃 = max 𝐾𝑆 = max 𝑁 𝑠𝑘 |𝑃 | (2)
Table 1: Example of distribution of keystrokes that can be saved for the phrase "the rationale behind the decision " with the corresponding computation of phrase accuracy and SD. max 𝑁 𝑠𝑘 is determined by simulating a user entering text without errors, and selecting correct suggestions as early as possible (i.e. as soon as they are available). A word suggestion system with a high accuracy will show the correct suggestion sooner (ideally after entering the first letter of a word), compared to a system with a low accuracy.

In a similar way, the accuracy of a specific suggestion for a word (𝐴 𝑤 ) can be defined as the ratio of the maximum number of keystrokes that can be saved, to the number of characters in that word. For convenience, we include a space entered at the end of the word to be a required character. The main reason is that we automatically insert a space after each selected suggestion, and this eases the comparison with words entirely typed using a keyboard (more details in the results section). Also, in contrast to previous work [START_REF] Trnka | User Interaction with Word Prediction: The Effects of Prediction Quality[END_REF], keystrokes used to accept suggestions are ignored (for example tapping on the suggestion bar on mobile devices). This is because we are interested in measuring the contribution of suggestions independently from the interface technique used to trigger them. Ignoring suggestion interface keystrokes makes our results easier to compare with other ways of triggering word suggestions, for example using a technique that may require several keystrokes, or the use of a mouse.

Controlling word suggestion accuracy.

To control the accuracy of word suggestions in an experiment, we have to decide how many keystrokes should be saved for each word a participant will have to type. These potentially saved keystrokes determine when the correct suggestion is first shown as a word is typed. As an example, if three keystrokes should be saved from the word "calm␣", the correct suggestion would need to appear once the user typed "ca", effectively saving keystrokes for "l", "m", and the following whitespace character. If four keystrokes should be saved for the word "yet␣", the correct suggestion should be shown as soon as the previous word is completed. Conversely, if no keystrokes can be saved, the correct suggestion should never be offered.

How many saved keystrokes should be offered for an entire phrase (max 𝑁 𝑠𝑘 ) depends on the expected accuracy for this phrase (Equation 2). Note that it is often not possible to reach the exact expected accuracy for a phrase. For example, if there is an odd number of characters to type, one cannot save exactly half of them.

Distributing the potentially saved keystrokes among the words of a phrase is not trivial either. Our first approach was to ensure all words benefit from approximately the same accuracy of word suggestions: for each word 𝑤, 𝐴 𝑤 ≈ 𝐴 𝑃 . Specifically, we made the standard deviation of the accuracy for words, SD(𝐴 𝑤 ), close to zero. However, unlike in the real world, the appearance of the correct suggestion was then highly predictable. Indeed, if for each word 𝐴 𝑤 = .75, the correct suggestion always appeared when approximately a quarter of the word is typed. Enforcing a higher standard deviation, SD(𝐴 𝑤 ), makes it uneven, increasing external validity. Fixing this standard deviation for each phrase, for example SD(𝐴 𝑤 ) = 0.2, increases internal validity. Table 1 shows an example of a distribution of potentially saved keystrokes.

To summarize, potentially saved keystrokes need to be distributed among the words of each phrase so that: 1. The accuracy of the phrase 𝐴 𝑃 is as close as possible to the expected accuracy; 2. The standard deviation of the accuracy for words (SD(𝐴 𝑤 )) is as close as possible to a chosen non-zero value, for example 0.2. A brute force tree traversal algorithm is sufficient to find the distributions optimizing these criteria.

Suggestion Ranking Evolution

When more than one suggestion is displayed, the position of the correct suggestion among all the suggestions also needs to be realistically operationalized. There are two ways to analyze this behaviour on commercial systems: video recording with image processing, which is tedious; or directly using a word suggestion API. We analyzed the behaviour of the Apple macOS Catalina word suggestions using the NSSpellChecker API [START_REF]NSSpellChecker: An interface to the Cocoa spell-checking service[END_REF].

Using the API, we wrote a small utility to log the ranking of suggested words after each keystroke when a simulated user types every phrase from Mackenzie and Soukoreff's set [START_REF] Scott | Phrase Sets for Evaluating Text Entry Techniques[END_REF] without errors. To avoid potential adaptation of the suggestion engine, a default blank macOS user account was used. Analysis of the logs show that a correct suggestion is often immediately ranked first the moment it appears, or it very quickly moves up in the ranking. Also, the length of the suggested word has an effect. the shorter the length, the fewer the keystrokes required before the suggestion reaches first rank.

To model the general behaviour of how suggestions transition through rankings, we compute a matrix of probabilities capturing the likelihood of a correct suggestion moving from one rank to another. Given the impact of word length on suggestion rank, we compute this matrix for each suggestion word length (see examples in Fig. 2). This created 13 matrices since all words in Mackenzie and Soukoreff's phrases are between 1 and 13 characters.

These transition matrices can be used to operationalize the evolution of a correct suggestion's ranking. For example, if a maximum of 4 keystrokes may be saved from the word "doctor␣", "doctor" will first be suggested after the three letters "doc" have been typed. According to the second transition matrix of Fig. 2, there is a 39% probability it is immediately ranked first, 35% second, and 26% third. If it is ranked third, and the user types "t", there is a 45% probability it moves second. Interestingly, at any point in time, there is virtually no chance the rank of the correct suggestion decreases if the user types a correct character.

The transition matrices computed from the logs, and source code for the logger, are available from an associated project page 3 .

EXPERIMENT

The goal of our experiment is to investigate the relationship between typing efficiency, and the accuracy of word suggestions. A transcription task with word suggestions grouped in a bar interface is used. Typing efficiency is controlled by varying the device used for the task: a desktop computer, a landscape-oriented tablet, or a phone used with one hand. Accuracy is the theoretical maximum number of saved keystrokes from suggestions using the method just described, with the placement and evolution of a correct suggestion in the interface based on transition matrices.

Our analysis focuses on how often suggestions are used as a function of the two main factors, and also their effect on task performance and user satisfaction. Inspired by Roy et al. 's results [START_REF] Roy | Automation Accuracy Is Good, but High Controllability May Be Better[END_REF], we made the following hypotheses: (H1) Word suggestion usage increases with suggestion accuracy; (H2) Word suggestion usage is greater on a phone than a tablet, and greater on a tablet than a desktop.

Due to emergency measures resulting from the 2020 COVID-19 pandemic, the experiment was facilitated remotely using a web application and live video-conferencing.

Participants

We analyze the data of 36 participants recruited using mailing lists and social networks (ages 18 to 53, average 29.8, sd = 8.3, 13 selfdeclared as female, and 23 as male). Participants were required to have access to a smartphone, a touchscreen tablet, and a laptop or desktop computer equipped with a physical keyboard using a QWERTY mapping. They received a $10 gift card for the one hour study.

Based on a pre-questionnaire, 31 (86.1%) participants reported spending more than 4 hours typing on a desktop computer in the past 7 days. Only 2 participants (5.6%) reported similar habits for tablets, and 12 (33.3%) for phones. Six (16.7%) participants reported spending more than 1 hour typing with one hand on a phone in the 3 https://ns.inria.fr/loki/WordSuggestions past 7 days. Regarding their estimated word suggestion usage, 7 (19.4%) participants reported using more than 30 word suggestions when typing on a desktop over the past 24 hours. On a tablet, no participants reported having used more than 30 word suggestions in the past 24 hours, 30 (83%) reported using less than 10. On a phone, 10 participants (27.8%) reported using more than 30 word suggestions during the past 24 hours.

At the start of the session, the participant transcribed 10 phrases without suggestions to measure their natural typing speed. On average, participants typed 76.5 words per minutes (wpm) on desktop (sd = 0.6), 35.0 wpm on tablet (sd = 1.2), and 29.9 wpm on phone (sd = 1.1).

Apparatus

Participants used their own devices. The average screen diagonal of their desktop was 424 mm (16.7 inches, sd = 6 mm), 255 mm for their tablet (10.0 inches, sd = 2 mm), and 142 mm for their phone (5.6 inches, sd = 1 mm). We were not able to gather statistics about processing power because of browser security measures.

The experiment software was developed as three parts. A client web application ran on all devices. This dynamic page guided the participant through the protocol steps and presented the transcription and word suggestion interfaces to complete the measured tasks. Importantly, the tablet and phone version of the client provided an embedded keyboard that had to be used. This gave us needed control over the word suggestions and the bar interface, and controlled the type of keyboard that was used. An administration server implemented in JavaScript delivered static assets to the client, and enabled the experimenter to monitor and remotely control the client to facilitate the experiment and troubleshoot any issues. A suggestion server implemented in Go computed the word suggestions for the client. All source code is available from the project page 3 .

The embedded client keyboard was designed after iOS and iPa-dOS's keyboards (see Fig. 3). On the phone, the Shift technique [START_REF] Vogel | Shift: A Technique for Operating Pen-Based Interfaces Using Touch[END_REF] was implemented to reduce the effect of occlusion. This shows a preview of the key being pressed on top of the finger, and is consistent with most implementations of phone keyboards.

Task

Each trial was a transcription task using phrases from Mackenzie and Soukoreff's 500-phrases set [START_REF] Scott | Phrase Sets for Evaluating Text Entry Techniques[END_REF], which was designed to be representative of the English language. Participants were instructed to type as fast and precisely as possible. To insert or modify an existing character, participants had to delete all characters between the insertion point and that character using the backspace key (i.e. they could not use cursor keys or position the cursor by pointing). To complete a trial, participants had to transcribe exactly the stimulus phrase, which required all errors have to be fixed. While previous study about word suggestions fully prevented errors [START_REF] Quinn | A Cost-Benefit Study of Text Entry Suggestion Interaction[END_REF], recent work showed a user's aversion for errors has a significant effect on their typing behaviour [START_REF] Banovic | Quantifying Aversion to Costly Typing Errors in Expert Mobile Text Entry[END_REF][START_REF] Banovic | The Limits of Expert Text Entry Speed on Mobile Keyboards with Autocorrect[END_REF]. Consequently, we did not prevent errors in our experiment. Any characters could be entered, but the phrase set only uses alphabetical characters, mostly lowercase (99.8%).

Because a whitespace character is appended after each accepted suggestion, we added a terminal whitespace to key in at the end of every phrase. This ensures that the suggestion behaviour is equally beneficial for each word of the phrase, including the last one.

The same basic user interface was used for all devices (Fig. 3). For tablet and phone, the suggestions bar appeared just above the keyboard, like commercial interfaces. For desktop, the bar was positioned at the very bottom of the display, near the physical keyboard to approximate the position of the Apple Touchbar when used to display word suggestions. For all devices, the phrase stimulus and text entry field was positioned near the word suggestion bar. Selecting a suggestion on phone or tablet used direct touch. On the desktop, to avoid using the mouse or cursor keys, and to best approximate the position and action of direct input on a Touchbar, keys 1, 2 and 3 were used to select suggestions. Even if inline suggestions are more common on desktop, we used a suggestion bar on all three devices to avoid confounding the experiment.

When typing on the tablet, participants were told to lay it on a table in landscape orientation. The intention was to enable the most efficient 10-finger touch typing style for the tablet condition. When typing on the phone, participants were told to use only their dominant hand, both to hold the phone, and to type using a thumb.

In addition, they held the phone in the air without supporting their hand. These constraints were designed to reduce typing efficiency with phone, in an attempt to explore a larger segment of the typing efficiency scale. This condition is consistent with real-world habits of 12.7% of phone users according to a 2018 study [START_REF] Buschek | ResearchIME: A Mobile Keyboard Application for Studying Free Typing Behaviour in the Wild[END_REF], and 36% according to a 2012 study [START_REF] Gold | Postures, typing strategies, and gender differences in mobile device usage: An observational study[END_REF].

The accompanying video figure demonstrates the task on all three devices as well as a task tutorial each participant completed.

Suggestions

After each keystroke, the current word prefix is sent to a server that evaluates matching English words. We use Google Android Jelly Bean's dictionary [START_REF] Bean | List of English words with their adjusted frequency of appearance[END_REF], and replicate its scoring algorithm [START_REF] Bean | Word prediction scoring algorithm[END_REF]. The word with the best score is suggested, with the two following exceptions: 1. the target word 𝑤 is always suggested as soon as possible, in accordance to 𝐴 𝑤 , 2. any other suggestions that may save keystrokes are prevented. As an example to illustrate the second point, assume the next word the participant has to type is "select" and that this word has been assigned two potential saved key strokes (𝐴 𝑤 = 2/6). At first, suggesting "selected␣" is prevented as it could save 6 -3 = 3 keystrokes in total if accepted by the participant: 6 saved keystrokes because "select" would not need to be typed anymore, -3 for "e", "d" and the automatically inserted whitespace that will need to be deleted. However, once the participant has typed "sel", "selected␣" would not save keystrokes anymore as 3 -3 = 0. Therefore it would not be prevented if it is ranked high on the suggestion scoring algorithm.

Our software enforces a minimum 150 ms delay before updating suggestions after each keystroke. In addition, trials for which this delay exceeded 300 ms due to network latency were removed from the analysis. A 150 ms to 300 ms delay before suggestions is lower than the 350 ms to 850 ms delays we measured with popular commercial systems 4 , so our study generalizes to more responsive suggestion systems in the future.

We operationalized the accuracy of these suggestions as discussed in section 3. However, the ranking evolution slightly deviated from what we modelled due to an implementation inconsistency. This primarily affected the first ranking when the correct word moved from not being suggested to being suggested and the ranking of less frequent words with more than 8 characters. Since our transition modelling captures only one example of many word suggestion implementations, such a small deviation would not undermine the ecological validity. In addition, we found suggestions were as likely to be used regardless of their ranking (see Section 5.1.1). An explanation of the deviation and the exact transition matrices used in the experiment are available on the project's webpage 5 .

We targeted a standard deviation of the accuracy for words SD(𝐴 𝑤 ) = 0.2 as it is the highest standard deviation value that can be reached for 𝐴 𝑃 = 0.1 or 0.9, according to the König-Huygens theorem. This is consistent with the Apple macOS NSSpellChecker API for which we measured SD(𝐴 𝑤 ) = 1.79 on average with 1 suggestion, and SD(𝐴 𝑤 ) = 1.76 with 3 suggestions. We allowed a margin of error of 0.025 for the accuracy of the phrase (𝐴 𝑃 ), and of 0.1 for SD(𝐴 𝑤 ). Only 6 out of the 1,500 phrase × accuracy combinations did not allow these conditions to be met and were removed.

Procedure

The experiment was divided into two parts. During the first part, participants measured the size of their three displays, performed a short 10-trial natural typing speed test without word suggestions, completed a demographic questionnaire, then went through a short interactive tutorial demonstrating the task.

The second part of the experiment was divided into three sections, one for each device condition. Each section had 3 practice trials, 20 trials recorded for analysis, a subjective questionnaire (explain in results below), and a NASA-TLX questionnaire [START_REF] Hart | Development of NASA-TLX (Task Load Index); Results of Empirical and Theoretical Research[END_REF].

A dedicated interface was used by the experimenter to control each participant's progress, and remotely launch the different parts that composed the experiment.

Design

We used a mixed design with two independent variables: • device { desktop, tablet, phone } (within-subject),

• accuracy { 0.1, 0.5, 0.9 } (between-subject). 4 These estimates are by manually inspecting video screen recordings from iOS, iPadOS and macOS. On macOS, we used the accessibility keyboard to display a copy of the touch bar on the main screen, allowing it to be recorded. On iOS and iPadOS, we used the native screen recording feature. Our analysis revealed that 200 ms generally elapse from the moment a character is added to user's input, to the start of the animation updating suggestions. On iOS and iPadOS, the update animation lasted from 150 ms to 650 ms in our recordings; during most of this time, suggestions are not readable. On macOS, there is no animation. However, the suggestions are not updated after each keystroke but only when the user pauses for approximately 350 ms. 5 https://ns.inria.fr/loki/WordSuggestions device was within-subject to reduce inter-participant variability. Its order was counter-balanced using a balanced Latin square. accuracy was between-subject to avoid strong carryover order effects if administered within-subject: a participant first exposed to a low accuracy condition is less likely to use a high accuracy condition after, with the inverse if exposed to a high accuracy first. accuracy levels were chosen to explore the full accuracy spectrum. The potential keystroke saving of state-of-the-art word suggestion algorithms is less than 46% [START_REF] Fowler | Effects of Language Modeling and Its Personalization on Touchscreen Typing Performance[END_REF]. We measured 55% (sd = 14%) potential keystroke saving with Apple's NSSpellChecker API with one suggestion on Mackenzie and Soukoreff's phrase set [START_REF] Scott | Phrase Sets for Evaluating Text Entry Techniques[END_REF], 65% (sd = 12%) with two suggestions, and 69% (sd = 11%) with three suggestions.

Participants were assigned to accuracy levels to minimize between group differences of mean natural entry speeds. The greatest pairwise difference between accuracy groups was 1.7 wpm for phone, 2.0 for tablet, and 0.5 for desktop. We did not specifically balance age or gender. In practice, they were relatively well distributed among accuracy values. The largest average age difference was 4.4 years between accuracy 0.5 and accuracy 0.1. There were only 2 fewer females and 2 more males for accuracy 0.1 than across the two other conditions.

Phrases to copy were randomly sampled without replacement from the Mackenzie and Soukoreff phrase set [START_REF] Scott | Phrase Sets for Evaluating Text Entry Techniques[END_REF]. This was specifically designed to contain letter frequencies matching the English language. Our random sampling has similar letter frequencies to the entire phrase set, and there is little difference between conditions: 𝑓 ('e') = 0.13, 𝑓 ('t') = 0.09, 𝑓 ('o') = 0.08, and so on. This is also consistent with letter frequencies in the English language [START_REF] Robert | Frequency and versatility of letters in the English language[END_REF].

In summary: we recorded 20 phrase × 3 device × 3 accuracy × 12 participants per accuracy condition = 2,160 trials.

RESULTS

Before analysis, 11 trials out of 2,160 were removed because suggestions took more than 300 ms to be updated from the server. Four additional trials were removed because the webpage lost focus. All other trials were kept, including trials during which participants mistyped (errors had to be fixed). We provide the data and R-markdown analysis notebooks on the project's webpage 5 .

Objective Measurements

In the analysis to follow, ANOVA was used. For each measure, trials were aggregated by participant and factors being analyzed. Because all measures exhibited non-normality of the residuals, and non-homogeneity of their variances, we applied an Aligned Rank Transform beforehand [START_REF] Wobbrock | The Aligned Rank Transform for Nonparametric Factorial Analyses Using Only Anova Procedures[END_REF]. Tukey HSD post hoc tests were used for pairwise comparisons of main effects. Interaction Contrasts were used for cross-factor comparisons in case of interaction.

5.1.1 Suggestion Usage. Suggestion Usage is the mean number of suggestions used during trials. As illustrated in Fig. 4, suggestions were used slightly more on phones (1.9 suggestions per trials) than tablets (1.5), and hardly used at all on desktop computers even when accuracy was high (0.3). Suggestion Usage increases from less than 0.5 suggestions per trials at 0.1 accuracy to more than 3.3 at 0.9 accuracy for both tablet and phone. It does not exceed 0.9 suggestions per trials on desktop, even at 0.9 accuracy. These observations are supported by statistical test. There is a significant main effect of device (𝐹 2,66 = 80.5, 𝑝 < .0001), with significant differences between every pairs, all 𝑝 < .0001 except [tablet, phone] (𝑝 < .05). There is also an effect of accuracy (𝐹 2,33 = 32.8, 𝑝 < .0001), with significant differences between all pairs, 𝑝 < .05 for accuracy [0.1, 0.5], 𝑝 < .001 for accuracy [0.5, 0.9], and 𝑝 < .0001 for accuracy [0.1, 0.9]. Finally, we found a device × accuracy interaction (𝐹 4,66 = 24.4, 𝑝 < .0001). The difference between desktop and tablet or desktop and phone is larger for accuracy 0.5 than accuracy 0.1 (𝑝 < .05), for accuracy 0.9 than accuracy 0.5 (𝑝 < .0001), and for accuracy 0.9 than accuracy 0.1 (𝑝 < .0001). No difference between tablet and phone were found regardless of accuracy.

We found no effect of a correct suggestion's first ranking on Suggestion Usage (𝑝 = 0.59). The correct suggestion was not more likely to be used regardless whether it was first ranked first, second, or third in the bar interface.

Keystroke

Saving. Each used suggestion does not contribute equally to the task as they may have been used to save a different amount of keystrokes. Keystroke Saving is the ratio of number of saved keystrokes using suggestions excluding editing (𝑁 𝑠𝑘 ), by the number of characters to type (|𝑃 |) [START_REF] Trnka | Word prediction and communication rate in AAC[END_REF]:

𝐾𝑆 = 𝑁 𝑠𝑘 |𝑃 | (3)
Note that the keystroke saving is at most equal to the accuracy of word suggestions for the phrase: max 𝐾𝑆 = 𝐴 𝑃 . Compared to Suggestions Usage, Keystroke Saving is a more accurate measure of the added value of word suggestions. As illustrated in Fig. 4, Keystroke Saving followed a pattern similar to Suggestions Usage, but tighter, blurring the differences between phone (26%) and tablet (22%). Only 5% of the keystrokes were saved on desktop. Keystroke Saving raises from 2% at 0.1 accuracy to an average 44% at 0.9. However, it barely reaches 15% on desktop for 0.9 accuracy, while it exceeds 55% for tablet, and 62% for phone.

These observations are supported by a significant main effect of device (𝐹 2,66 = 66.6, 𝑝 < .0001), with significant differences for [desktop, tablet], and [desktop, phone] (𝑝 < .0001). There is an effect of accuracy too (𝐹 2,33 = 49.8, 𝑝 < .0001), with significant differences between every pairs, 𝑝 < .0001 for accuracy [0.1, 0.9] and accuracy [0.5, 0.9], and 𝑝 < .01 for accuracy [0.1, 0.5]. Finally, there is a device × accuracy interaction (𝐹 4,66 = 28.9, 𝑝 < .0001).

The difference between desktop and phone is larger for accuracy 0.5 than 0.1 (𝑝 < .05), for accuracy 0.9 than accuracy 0.5 (𝑝 < .0001), and for accuracy 0.9 than 0.1 (𝑝 < .0001). The difference between desktop and tablet is larger for accuracy 0.9 than for accuracy 0.5 (𝑝 < .0001), and for accuracy 0.9 than for accuracy 0.1 (𝑝 < .0001). No difference between tablet and phone were found regardless of accuracy.

Entry Speed.

We measured Entry Speed in words-per-minute (wpm), where "word" means 5 characters [START_REF] Sabbir | Analysis of Text Entry Performance Petrics[END_REF][START_REF] Yamada | A historical study of typewriters and typing methods, from the position of planning Japanese parallels[END_REF], calculated as follows:

𝑆 = |𝑃 | -1 𝑇 × 60 5 (4)
|𝑃 | is the number of characters in the phrase to transcribe, and 𝑇 the interval of time in seconds from the moment the first input character is entered to the moment the transcribed text matches the target phrase. One character is subtracted from |𝑃 | in the calculation of Entry Speed because 𝑇 is only measured from the first input character.

As illustrated in Fig. 4, participants were more than twice as fast with their desktop (68.2 wpm) than with their tablet (32.6 wpm) or phone (30.8 wpm). Higher accuracy did slightly increase entry speed, in particular for the phone and tablet, but its effect was small, even at its highest. Entry Speed increased from (38.2 wpm) at 0.1 accuracy to (50.4 wpm) at 0.9.

Statistical support is from a significant effect of device on Entry Speed (𝐹 2,66 = 118.0, 𝑝 < .0001). Pairwise comparisons detected significant differences for [desktop, tablet], and [desktop, phone] (𝑝 < .0001). There is also an effect of accuracy (𝐹 2,33 = 8.6, 𝑝 < .0001), with a single pairwise difference for accuracy [0.1, 0.9] (𝑝 < .001).

Keystroke

Saving and Natural Entry Speed. We explored the correlation between Natural Entry Speed (without suggestions) measured at the beginning of the experiment, and Keystroke Saving.

Results are shown at the top of Fig. 5. Keystroke Saving decreases with Natural Entry Speed, regardless of device: Kendall's taus are 𝑟 𝜏 = -.33, 𝑝 < .01 for accuracy 0.1, 𝑟 𝜏 = -.62, 𝑝 < .0001 for accuracy 0.5, and 𝑟 𝜏 = -.56, 𝑝 < .0001 for accuracy 0.9. 5.1.5 Entry Speed Improvement. Finally, we investigated the effect of word suggestions on entry speed compared to natural entry speed. Entry Speed Improvement is the difference in Entry Speed with suggestions, as measured during the experiment; and Natural Entry Speed, as measured at the beginning of the experiment. Results are shown at the bottom of Fig. 5. Most of the time, word suggestions negatively impacted Entry Speed. They were only beneficial for slow typists and when very accurate. Except for accuracy 0.5, the faster the typist while typing without suggestions, the more their entry speed will be reduced with word suggestions: Kendall's taus are 𝑟 𝜏 = -.36, 𝑝 < .01 for accuracy 0.1, and 𝑟 𝜏 = -.50, 𝑝 < .0001 for accuracy 0.9.

Subjective Questionnaire

Table 2 provides the questionnaire's assertions with associated dependent variables. Answers were collected on a seven-point Likert scale, "Strongly Agree" to "Strongly Disagree". In the analysis below, "≪" indicates significant difference with 𝑝 < .05 or lower. The results of our questionnaire scale are shown in Fig. 6.

We also applied an Aligned Rank Transform on our subjective measurements, which are all ordinal, in order to investigate interactions. Tukey HSD post hoc tests were used for pairwise comparisons of main effects. Interaction Contrasts were used for cross-factor comparisons in case of interaction. The results of ANOVA significance tests are provided in Table 3.

Perceived Accuracy.

Our results demonstrate that the operationalization of accuracy matches the perception of users. We found significant Perceived Accuracy differences for the following accuracy values: 0.1 ≪ 0.5 ≪ 0.9.

Perceived Keyboard

Efficiency. This partially supports our operationalization of typing efficiency. Participants perceive a desktop computer as more usable for a typing task than both phone and tablet. Surprisingly, there is little difference between one-handed typing on a phone and typing with two hands on a tablet. We found { tablet, phone } ≪ desktop.

Satisfaction.

Our results show Satisfaction increases with accuracy, and participants were more satisfied on desktop than tablet or phone. We found accuracy { 0.1, 0.5 } ≪ 0.9 and { tablet, phone } ≪ desktop.

Suggestion Disruptivity.

Participants found accurate suggestions less disruptive than inaccurate suggestions: accuracy { 0.1, 0.5 } ≪ 0.9.

NASA-TLX

Few significant results were detected from our NASA-TLX data (see Fig. 6). The results of ANOVA tests after Aligned Rank Transform are provided in Table 3. Frustration Level and Physical Demand were

DISCUSSION

Our results demonstrate how reliance on word suggestions not only depends on their accuracy, but also on typing efficiency. We have shown that the use of word suggestions, and their contribution to the completion of a typing task, increases with suggestion accuracy but decreases relative to how fast it is to type on a device. This confirms H1 (word suggestion usage increases with their accuracy) and H2 (word suggestion usage is greater on the phone than tablet, and greater on tablet than phone). However, we expected participants to be able to type much faster with two hands on a tablet than with one hand on a phone, so we also expected suggestion usage on a phone to be well below the tablet. We did measure a significant difference, but we were surprised how small it is. Likely because of this, we found no significant differences in terms of keystroke saving. We believe this is explained because natural entry speed without suggestions was unexpectedly similar in the two conditions: 30 wpm for phone and 35 wpm for tablet. As shown in Fig. 5, we observed a strong correlation between natural entry speed and keystroke saving. The slower typing speed with two hands on tablet compared to one hand on phone is most likely due to experience and training: our demographic data revealed participants were much more used to typing on a phone than a tablet. This is also supported by how participants perceived phone and tablet typing efficiency similarly. An important take away from H2 is that acceptable accuracy results may not translate from one typing system to another. For example, while many studies on word suggestions were performed with Augmentative and Alternative Communication systems (AAC) [START_REF] Patrick | Generating Text from Compressed Input: An Intelligent Interface for People with Severe Motor Impairments[END_REF][START_REF] Garay | Text prediction systems: a survey[END_REF][START_REF] Koester | Modeling the speed of text entry with a word prediction interface[END_REF][START_REF] Trnka | User Interaction with Word Prediction: The Effects of Prediction Quality[END_REF], their results are not applicable to phones or desktops. Greater generalizability may only be achieved by considering typing efficiency.

Our results also show that the effect of accuracy on the contribution of word suggestions to the completion of a phrase is non-linear: the lower the accuracy is, the less the potential of suggestions is put into use (Fig. 4). In the highest accuracy condition of our experiment (accuracy 0.9) suggestions were used at 50% of their full potential across devices, up to 70% on the phone. However, for levels accuracy 0.1 and 0.5, only 15% of the keystrokes that could potentially be saved with suggestions were actually saved. And this was a very low 5% on desktop.

This hints that most users get a sense of how much suggestions can help them. They tend to disregard them if accuracy is low or if they can type quickly, but they use them more if accuracy is higher or if they type slowly. In a more systematic way, this confirms previous work. Buschek et al. observed that suggestion use was highly individual [START_REF] Buschek | ResearchIME: A Mobile Keyboard Application for Studying Free Typing Behaviour in the Wild[END_REF]. Later, Palin et al. note that slower typists use more suggestions [START_REF] Palin | How Do People Type on Mobile Devices? Observations from a Study with 37,000 Volunteers[END_REF].

Beyond usage, even if keystroke saving steadily increases with word suggestion accuracy, the effect is smaller than that observed by Banovic et al. with automatic error corrections [START_REF] Banovic | The Limits of Expert Text Entry Speed on Mobile Keyboards with Autocorrect[END_REF]. In fact, our results indicate that word suggestions are almost always detrimental to entry speed except for a slow typist when accuracy is very high (see Fig. 5). It is worth noting that the potential keystroke saving of state-of-the-art word suggestion algorithms from the literature is less than 46% [START_REF] Fowler | Effects of Language Modeling and Its Personalization on Touchscreen Typing Performance[END_REF]. However, we measured 55% potential keystroke saving with Apple's NSSpellChecker API when one suggestion was provided for phrases in the Mackenzie and Soukoreff set, and 69% with three suggestions. However, subjective ratings expose a slightly different pattern: satisfaction increases with accuracy, even in the lower accuracy range where it does not improve performance. This trend confirms previous observations from Quinn et al., who noticed that users value assistance, even when it impedes their performance [START_REF] Quinn | When Bad Feels Good: Assistance Failures and Interface Preferences[END_REF]. A higher-quality dictionary will increase accuracy, for example one that becomes tailored to phrases commonly used by the typists. Our results suggest that this may increase user satisfaction, but it may not create a real benefit in text entry performance.

From a higher level perspective, as discussed in section 2.3, automating, or partially automating a user's task is often done with the intention to improve usability. Word suggestion interfaces are an instance of such automation. The success of an automation depends on its purpose, but also its accuracy. The question is, how accurate does it need to be? Roy et al. showed that both automation accuracy and the "controllability" of a manual interface impact how users choose to fix automation inaccuracies [START_REF] Roy | Automation Accuracy Is Good, but High Controllability May Be Better[END_REF]. Our results can be integrated in this framework, and tend to confirm their results: the more usable a manual interface is, the less an automation system will be used even if it is highly accurate. In our case, the manual interface is the keyboard, and the automation is the word suggestions. We suspect this pattern likely generalizes to other common interfaces and tasks beyond word suggestions and text entry.

Overall, our results further show that dedicating resources to improving usability is always a winning bet for user performance. While improving accuracy will result in higher satisfaction, it may not ultimately increase performance, and more critically, it is unlikely to compensate for poor usability like poor typing interfaces or devices. In the context of word suggestions, academic and industry resources may be best allocated for optimizing input efficiencyfor example with the help of faster text entry techniques such as ShapeWriter gesture typing [START_REF] Per | SHARK2: a large vocabulary shorthand writing system for pen-based computers[END_REF]-rather than improving word suggestions. This has important implications for the design of related text entry techniques used on millions of devices.

LIMITATIONS

While the devices we used have strong ecological validity, it does not provide a formal control for levels of typing efficiency. Considering natural entry speed helps, but we could not control it either. A more controlled and synthetic experiment that tightly controls typing efficiency would further contribute to our understanding of the impact of typing efficiency on suggestion usage and entry speed. Some participants reported being less efficient than usual with the keyboard integrated in our mobile application because it differs from the one they are used to use with their device. While this does not put into question our pattern of results, and we did measure natural entry speed using our keyboard, it indicates the entry speed measured in our experiment under-represents the population of trained typists on phone and tablet.

Operationalizing accuracy may sometimes feel surprising to participants. In the 0.9 condition, some noted the word suggestions were "unnaturally accurate". On the contrary, in the 0.1 condition, some indicated they felt like the system was avoiding the correct word on purpose (it was). A few participants also noted the system would occasionally not suggest common words, like "the"; and surprisingly, it would correctly suggest much less common words, like "racketball". This reveals an unavoidable bias in our participants regarding their prior experience and expectations of word suggestion systems.

Finally, this experiment investigated a suggestion bar, but word suggestions may be presented in different ways. For example, on a desktop, suggestions are often inline with the document input, after or under the insertion point (see Fig. 1). It remains unclear if the use of inline suggestions follows the same pattern as our results. In addition, suggesting more than three words increases the chance of suggesting the right one, and as a result increases accuracy. However, a previous study showed it also increases cognitive load, quickly offsetting the benefits [START_REF] Swiffin | Adaptive and predictive techniques in a communication prosthesis[END_REF][START_REF] Venkatagiri | Effect of window size on rate of communication in a lexical prediction AAC system[END_REF]. We investigated a three-word suggestion bar in our experiment, though in practice, commercial systems often show only two suggestions. The first position on the bar is often used to validate the word prefix as it currently is, preventing automatic correction that would otherwise trigger if the user pressed space (see Fig. 1a and1b). It remains unclear how varying the number of suggestions would impact the trend we observed.

CONCLUSION

On desktop computers, we found word suggestions never improve entry speed, even when extremely accurate. They do on phones and tablets, but typically only when accuracy is very high. However, accuracy improves user satisfaction. These results have direct implications for the design of typing systems, and could justify prioritizing decisions and resources for industry and research.

An important take away from our work is that a useful accuracy for word suggestions strongly depends on the device used to type. This is implies that results for the acceptable accuracy of word suggestions does not generalize from one typing system to another, unless typing efficiency is considered.

Our results also open directions for future work. For example, the effect of standard deviation of the accuracy for words remains unclear. Smaller values may help users better estimate when an appropriate suggestion will become available. Likewise, some modern suggestion algorithms propose not only a single word, but an entire sentence [START_REF] Arnold | On Suggesting Phrases vs. Predicting Words for Mobile Text Composition[END_REF]. This is powerful but difficult to do accurately. The effect of this strategy, and its accuracy on suggestion use, have yet to be formally investigated.

  the accuracies for words in the phrase: SD(𝐴 𝑤 ) = SD ( 3 /4, 6 /10, 2 /7, 3 /4, 3 /9) = 0.22 Accuracy of the phrase: 𝐴 𝑃 = 17/34 = 0.5

Figure 2 :

 2 Figure 2: Examples of suggestion ranking transition matrices modelling the behaviour of macOS's suggestion engine. These matrices were calculated for each possible word length.

Figure 3

 3 Figure 3: a ○ User interface for the experiment, b ○ laptop condition, c ○ tablet condition, and d ○ phone condition. Phrases to copy appear in the dark grey area in the middle. Participant input appears in the blue box underneath it.

Figure 4 :

 4 Figure 4: Use and contribution of the suggestions, and entry speed. Effect sizes show .95 confidence interval. The keystroke saving ratio is the keystroke saving by the accuracy of the phrase.

Figure 5 :

 5 Figure5: Keystroke Saving (top) and Entry Speed Improvement (bottom) as a function of Natural Entry Speed. Participant S25, the fastest typist, lost 79 wpm with 10% accurate word suggestions. Unlike other fast typists, they saved 4% of their keystrokes, i.e. 36% of the keystrokes that could be saved, even as accuracy was very low.

Table 2 :

 2 Subjective Questionnaire

	Dependent Variable Assertion
	Perceived Accuracy The word suggestions are accurate
	Perceived Keyboard	The use of the keyboard is efficient in this
	Efficiency	task
		The controls (keyboard and word sugges-
	Satisfaction	tions) are satisfactory for the completion of
		the task
	Suggestion Disruptivity The word suggestions are distracting

Table 3 :

 3 Statistical significance of the questionnaire and NASA-TLX of Experiment 3

								device		accuracy	device × accuracy
								𝐹 2,99	𝑝		𝐹 2,99	𝑝	𝐹 4,99	𝑝
			Questionnaire
			Perceived Accuracy		0.32	.73	74.36 <.0001	2.09	.09
			Perceived Usability		22.09 <.0001		0.16	.85	0.23	.92
			Satisfaction			5.38 <.01		7.61 <.001	1.79	.14
			Suggestions' Disruptivity		2.66	.08		8.73 <.001	1.25	.29
			NASA-TLX
			Effort				3.56 <.05*		0.40	.67	0.47	.76
			Frustration Level			8.01 <.001		0.95	.39	0.75	.56
			Overall Performance		1.88	.16		3.05	.05	3.49 <.05
			Mental Demand			2.80	.07		1.56	.22	0.31	.87
			Physical Demand		8.64 <.001		3.04	.05	1.26	.29
			Temporal Demand		2.16	.12		2.76	.07	0.55	.70
							*No significant pairwise differences
		Perceived Accuracy	Perceived Keyboard Efficiency		Satisfaction		Suggestions' Disruptivity
	strongly agree
																	Device
																	desktop
																	tablet
																	phone
	strongly disagree
		0.1	0.5	0.9		0.1	0.5	0.9	0.1		0.5	0.9	0.1	0.5	0.9
								Accuracy
		Effort		Frustration Level	Overall Performance	Mental Demand	Physical Demand	Temporal Demand
	high
	low
	0.1	0.5	0.9	0.1	0.5	0.9	0.1	0.5	0.9	0.1	0.5	0.9	0.1	0.5	0.9	0.1	0.5	0.9
									Accuracy
		Figure 6: Subjective questionnaire (top) and NASA-TLX (bottom) of Experiment 1
	both lower with desktop than with phone and tablet: desktop ≪
	{ tablet, phone } for both measures.

In the rest of this paper, we will often use the term "desktop" to refer both desktop or laptop computers.

https://swiftkey.com

ACKNOWLEDGEMENTS

This work was made possible by NSERC Discovery Grant 2018-05187, the Canada Foundation for Innovation Infrastructure Fund 33151 "Facility for Fully Interactive Physio-digital Spaces, " Ontario Early Researcher Award ER16-12-184, and LAI Réapp.