# From Cycling Between Coupled Reactions to the Cross-Bridge Cycle: Mechanical Power Output as an Integral Part of Energy Metabolism

## Abstract

**:**

^{2+}]. Muscular fatigue is triggered when ATP consumption overcomes ATP delivery. As a result, the substrate of the cycle, [MgATP

^{2−}], is reduced. This leads to a switch off of cycling and ATP consumption, so that a recovery of [ATP] is possible. In this way a potentially harmful, persistent low energy state of the cell can be avoided.

## 1. Introduction

^{+}, and Mg

^{2+}, are of interest under these conditions. This is achieved by formulating, in particular, the ATP splitting reaction according to Alberty [20] as a function of both [H

^{+}] and [Mg

^{2+}].

## 2. Results and Discussion

#### 2.1. How Negative Conductances Are Generated

_{1}= L

_{c}((λ

_{1}+ 1)A

_{1}+ A

_{2}), and J

_{2}= L

_{c}(A

_{1}+ (λ

_{2}+ 1)A

_{2})

_{1}and J

_{2}designate fluxes through affinities A

_{1}and A

_{2}, respectively, and L

_{c}represents the coupling conductance. Under totally coupled conditions (λ

_{1}= λ

_{2}= 0) both fluxes are equal. The dissipation function, Ф, of a coupled process is composed of two parts, Ф

_{1}for the output, and Ф

_{2}for the input reaction, with:

_{1}+ Ф

_{2}

_{1}= J

_{1}A

_{1}, and Ф

_{2}= J

_{2}A

_{2}

_{1}= L

_{c}(A

_{1}+ A

_{2})A

_{1}, and Ф

_{2}= L

_{c}(A

_{1}+ A

_{2})A

_{2}(total coupling).

_{1}usually is negative, Ф

_{1}must also be negative. Expanding the right hand terms yields:

_{c}, which is associated with A

_{1}, while

_{2}. They relate to the usual different forms of energy being processed through the coupling reaction. Obviously, when A

_{1}is negative, L

_{c1}must also be negative to yield a negative Ф

_{1}.

_{1}, L

_{c1}has to be negative for a negative A

_{1}.

_{1}and A

_{2}are in series, hence, L

_{c}can be regarded as the equivalent conductance of both in series conductances L

_{c1}and L

_{c2}, yielding:

#### 2.2. Conductances in Cycles between Coupled Reactions

_{1}

^{I}, may be used by the second reaction as an input force A

_{1}

^{II}(A

_{2}

^{I}is the input affinity, A

_{1}

^{II}denotes the load affinity). In such a cycle between two coupled reactions, both forces must be equal but of opposite sign. The output power of the first reaction delivers the input power for the second reaction by flowing through A

_{1}

^{I}, and A

_{2}

^{II}= - A

_{1}

^{I}, and back to A

_{1}

^{I}(at zero power). At steady state, fluxes through A

_{1}

^{I}and A

_{1}

^{II}are equal, and hence, both dissipation functions of the cycle, Ф

_{1}

^{I}and Ф

_{2}

^{II}, must vanish. From Ф

_{1}

^{I}= - Ф

_{2}

^{II}, and L

_{c1}

^{I}(A

_{1}

^{I})

^{2}= -L

_{c2}

^{II}(A

_{2}

^{II})

^{2}, L

^{I}

_{c1}= - L

^{II}

_{c2}is obtained. That is, the partial conductances of two coupled reactions in series are opposite and equal, if under conditions of steady state cycling the magnitudes of the output force of reaction I and the input force of reaction II are equal.

_{1}

^{I}= -Ф

_{2}

^{II}can also be expressed in terms of the steady state flux of cycling and of resistances R

_{c1}

^{I}= 1/L

_{c1}

^{I}, and R

_{c2}

^{II}= 1/L

_{c2}

^{II}, yielding R

_{c1}

^{I}J

^{2}+ R

_{c2}

^{II}J

^{2}= 0. It follows that at steady state cycling between two coupled reactions, the sum of the resistances in that cycle must vanish, i.e., that the steady state flux through a cycle between two coupled reactions always occurs at zero overall resistance. Cycling is driven solely by A

_{2}

^{I}+ A

_{1}

^{II}(A

_{1}

^{II}negative). Because both reactions are coupled, the conductance (resistance) of the whole cycling process is brought about by both partial (in series) conductances associated with the input and load affinity, respectively.

_{red}and FAD

_{red}) reactions of the respiratory chain (J

_{NA}and J

_{FA}) equals the back flow of a given fraction (Q

_{H}) of protons through ATP synthase (J

_{SY}) and ATP/ADP exchange (J

_{AE}) plus H/Pi symport (J

_{Pi}). Both partial conductances,

_{R}= v

_{s+1}(total coupling; A

_{R}

^{o}= affinity of J

_{NA}plus J

_{FA}; v

_{R}= v

_{s+1}= 4protons/extent of reaction).

_{H}) flows back (driven by ) through several parallel conductances given by the proton leak flux, J

_{PL}, mitochondrial Na

^{+}/Ca

^{2+}exchange (with Na

^{+}/H

^{+}contracted to H

^{+}/Ca

^{2+}exchange), 2J

_{HCE}, and the malate-aspartate shuttle, J

_{MS}. The partial conductance of this residual proton efflux and the sum of conductances of back flowing fluxes, are also of opposite equality.

_{m}) production plus ATP transport (contracted to A

_{1}

^{I}), and of cytosolic ATP splitting (A

_{2}

^{II}) can be formulated. Opposite equality of partial conductances is also fulfilled for this cycle (the above results were obtained by using the simulation SIM

_{GlOx}from reference [1]).

_{Ld}= L

_{c}) is needed to achieve a maximal power output [1]. At total coupling, the output power is given by:

_{out}is found by differentiation with respect to the variable A

_{1}, while A

_{2}remains constant, and by setting the derivative equal to zero:

_{1max}into equation (2e) yields L

_{c1}= - L

_{c}and because - L

_{c1}= L

_{Ld}, it follows L

_{Ld}= L

_{c}.

_{1}

^{I}with associated L

^{I}

_{c1}, whereas the affinity of the reverse reaction in myofibrils corresponds to A

_{1}

^{II}(with L

^{II}

_{c2}). To ensure diffusional flow of PCr and Cr between both locations, an additional driving force (corresponding to U

_{e}; see (A4)) with associated conductance must be present. Under such conditions partial conductances do not match. Only when the additional conductance corresponding to the diffusional process (L

_{e}) is added to L

^{I}

_{c1}does this sum become opposite and equal to L

^{II}

_{c2}, as is shown in (A4). L

_{e}depends greatly on structural features. So, to achieve a high diffusional conductance, diffusional paths must be as short as possible, which in turn requires a high grade of structural organization [26,27,28].

^{2+}ATPases (SERCA) and Ca

^{2+}release channels of the sarcoplasmatic reticulum (SR) breaks off during activation of contraction. There is an enormous Ca

^{2+}efflux through release channels; meanwhile the pumping rate of SERCAs may be low. Under these conditions, respective conductances may greatly differ; however, when a new steady state cycling is reached, the partial conductance of SERCA must be opposite and equal to the conductance of the Ca

^{2+}release channels. The opposite has to be expected, when release channels close again, and the Ca

^{2+}pumping rate exceeds the release rate.

#### 2.3. From Chemical Potentials to Mechanical Force Generation

_{r}= K'

_{B}× K'

_{T}× K'

_{R}(k'

_{B}, k'

_{T}, and k'

_{R}are equilibrium constants of the binding, transition, and release reaction, respectively, whereas k'

_{r}denotes that of the non-catalysed reaction).

^{2-}in the diffusional space of myofibrils:

^{2-}bound to myosin (the bold point denotes binding to myosin). Two negative charges develop on the dissociated actin, which are neutralised by potassium ions, K

^{+}, stemming from free MgATP

^{2−}, which is now bound to myosin heads. On the dissociated myosin heads, it neutralises both emerging positive charges. This first actomyosin dissociation and binding of MgATP

^{2−}to myosin is followed by ATP splitting on the myosin heads. This transition reaction is described by

_{2}as conformational energy.

^{-}and H

_{2}PO

_{4}

^{-}from cross-bridges and then by releasing the stored conformational energy. During this reaction the cross-bridge tilts back by 60° towards the sarcomere centre, whereby free energy is transferred to the actin filament as mechanical energy.

_{1}to R

_{4}) two fluxes can be obtained, which are responsible on the one hand for the production of dissociated and energised myosin heads (J

_{En}), and on the other hand for the formation of cross-bridges and subsequent mechanical force generation by stroking (J

_{Str}). At steady state, a certain fraction of myosin heads of a half-sarcomere exists in a dissociated and energised state , while the residual fraction interacts as cross-bridges with actin. The resulting fluxes are given by:

_{En}= L

_{En}(A

_{En}

^{Ld}+ A

_{En}

^{P}), with

_{Str}= L

_{Str}(A

_{Str}

^{Ld}+ A

_{Str}

^{P}), with

^{ref}) see (A5); complete conductances (L

_{En}and L

_{Str}, respectively) are given in (A14) and (A15).)

_{Str}

^{Ld}+ A

_{ATP}). Here A

_{En}

^{Ld}(stored as conformational energy) denotes the affinity coupled to binding of MgATP

^{2−}to myosin heads (A

_{En}

^{P}), and A

_{Str}

^{Ld}the affinity which is coupled to the power stroke potential (A

_{Str}

^{P}). A

_{Str}

^{Ld}represents the mechanical work per mole of cross-bridges which has to be overcome during stroking. The quantity J

_{Str}× A

_{Str}

^{Ld}is directly related to mechanical power output P

_{Str}= F

_{Ld}× v(F

_{Ld}= load force in Newton (N) of all stroking cross-bridges of a given cross sectional area; v = velocity of shortening in m/s related to a given fiber length), which as such is conveyed to the surroundings.

_{ATP}is used at two mechanistically and temporally separated steps. They are given on the one hand by binding of MgATP

^{2−}and on the other hand by the release of MgADP

^{−}and H

_{2}PO

_{4}

^{−}. Here, most of the free energy of ATP splitting is associated with A

_{En}

^{P}, which by the coupling process on myosin heads is transformed first into A

_{En}

^{Ld}, and then is delivered as A

_{Str}

^{P}to the power stroke after cross-bridge formation. Therefore, the stroke potential in mechanical units (cross-bridge force × stroke length × N

_{A}; N

_{A}= Avogadro’s number) must be equal to A

_{Str}

^{P}(see below).

^{2−}binding can proceed only if actomyosin dissociates, whereas release of products becomes possible only when at the same time cross-bridge formation occurs. Moreover, the conformational change in the myosin head forces it into a new position, which favours an interaction with actin at a new actin binding site displaced a certain distance towards the Z-disc. During stroking, binding of a new MgATP

^{2−}molecule and detaching of cross-bridges may preferentially occur at the end of the power stroke, when cross-bridges form an angle of about 60° with the actin filament (see below for uncoupling by stroke shortening).

_{Ld}which has to be overcome during shortening.

_{Str}, the flux given in mM/s has to be converted into velocity with units of m/s. This is achieved by calculating the stroke frequency for a given concentration of stroking cross-bridges ([CB] = [CB]

_{tot}– [MH

_{En}], in mM) and by multiplying with the stroke length l

_{Str}(in m) and the number of in series half-sarcomeres N

_{hs}. The result is:

_{Str}

^{Ld}at constant A

_{Str}

^{P}. It represents a straight line (Figure 1A). Introducing a Michaelis-Menten like inhibition factor associated with L

_{Str}yields the desired hyperbolic dependency:

_{Ld}), affinities and Km

_{Ld}(both in J/mol) have to be converted into units of force. This is achieved by dividing by l

_{ Str}and by multiplying by the molar number of cross-bridges.

**Figure 1.**Flux as a function of load potential at 10.8 µM [Ca

^{2+}].

**A:**(grey dots) according to equation 9b; (light grey dots) according to equation 9c or 9d; (red line) according to equation 11a; (green line) according to equation 11b.

**B:**(light grey dots) according to equation 9c or 9d; (red squares) results from simulation SIM

_{GLYgen}.

_{Str}

^{Ld}being negative, F

_{Ld}must also be ≤0. Expressing shortening velocity as a function of a positive variable yields with F

_{Ld}= - F

_{Ld}

^{+}

**Figure 2.**Shortening velocity as a function of load force at two different Ca

^{2+}concentrations

**A:**[Ca

^{2+}] = 1.08 µM; (light grey dots) according to equation 10b; (red line) equation 10b plus uncoupling; (red circles) results from SIM

_{GLYgen}versus load force; (blue squares) results from SIM

_{GLYgen}versus load force as sensed by cross-bridges;

**B:**as in

**A**, but at [Ca

^{2+}] = 0.36µM.

_{0}denotes the maximal force obtained under isometric conditions, whereas F

_{P}in the latter equation is obtained from the input affinity (A

_{Str}

^{P}) of J

_{Str}by converting it into units of force (see below for a derivation of F

_{p}≡ F

_{0}).

_{max}[12]. From NET it is known that such a maximum is produced by uncoupling. To create such a maximal efficiency, uncoupling terms have to be incorporated into J

_{Str}. Variable, load dependent λ values (λ(A

_{Str}

^{Ld}) instead of constant λ's) are defined, to preserve the hyperbolic nature of the function. In this way, uncoupling becomes operative only when A

_{Str}

^{Ld}exceeds a certain value (Figure 1A). Both flux equations are given by:

_{Str}

^{Ld}, e.g.,

^{4 }J/mol, deviations from the hyperbolic (coupled) curve begin to arise. From the plots it can be seen that uncoupling leads to a shift of the intersection with the abscissa to less negative values of A

_{Str}

^{Ld}, whereas - (J

_{Str}

^{P}) is still maintained, even at A

_{Str}

^{Ld}= - (A

_{Str}

^{P}), where the coupled flux must be zero and only uncoupled fluxes are possible.

_{Str}

^{Ld}= - (A

_{Str}

^{P}), coupled reactions with associated actin filament movement come to a halt, because the driving force has vanished. As already mentioned above, now only uncoupled fluxes can occur. Such a situation may also be realised with isometric contraction, which is known to be associated with ATP splitting and heat production, but without power output. That is, a mechanism has to be found which explains the identity of the isometric force F

_{0}with F

_{P}, which was merely formally derived from the input affinity A

_{Str}

^{P}by a conversion factor. This is achieved by defining the uncoupling mechanism by a shortening of the stroke length l

_{Str}of the power stroke. Total uncoupling is reached when l

_{Str}= 0. This may be realised under isometric conditions. Free energy corresponding to A

_{Str}

^{P}≈ A

_{ATP}is delivered to actin filaments as mechanical work, i. e., F

_{0}× l

_{Str}× N

_{A}= A

_{Str}

^{P}. Shortening may be brought about through splitting of actomyosin bonds before the whole stroke length is transferred to an actin filament. When A

_{Str}

^{Ld}= - A

_{Str}

^{P}, as is realised under isometric conditions, actomyosin splitting already occurs at zero stroke length, so that no energy can be delivered to the actin filaments. Only force development by cross-bridges during the time interval between bond formation and bond splitting is possible under these conditions. This may be achieved by the torque every myosin head exerts on an actin filament after bond formation and release of H

_{2}PO

_{4}

^{−}and MgADP

^{−}. The associated force then acts on these filaments, but without being able to bring about filament movement, since this is hindered by the equal and opposite load force. Therefore, if the force remains constant over the whole stroke length, then F

_{P}is equal to the isometric force F

_{0}.

_{Str}

^{P}becomes dissipated as heat. Moreover, dissipative stroking under these conditions may occur in the presence of bound MgATP

^{2−}. The following derivation shows how stroke shortening may be involved with uncoupling.

^{P}

_{StrL}can be replaced by L

_{Str}, because this latter conductance may depend mainly on the formation mechanism of the actomyosin bond. The stroke reaction associated with conformational changes of the myosin head is assumed to proceed at a high conductance, since the energising reaction (J

_{En}), which is coupled to the same conformational change in the reverse direction, also proceeds at a very high conductance. So an increase by stroke shortening of a high conductance (stroking) in series with a low conductance (bond formation) may be negligible, so that Ф

^{P}

_{StrL}can be expressed as:

_{Str}(ΔA

_{StrL}

^{P})

^{2}is associated with output reactions, yielding:

_{Str}

^{Ld}= - A

_{Str}

^{P}, both λ values are equal. Moreover, if λ

_{Str}

^{Ld}= λ

_{Str}

^{P}= 1, (Δl/l

_{Str})

^{2}is also equal to 1.0, which means that now isometric conditions do exist.

_{Str}

^{Ld}as if there were a leak flux through A

_{Str}

^{Ld}. On the other hand, J

_{Str}

^{P}increases (equation (13g)) as if there were an additional leak flux through A

_{Str}

^{P}. The above derivations demonstrate that stroke shortening obviously leads to the same effects as uncoupling by leak fluxes. It seems justified, therefore, to also describe uncoupling by stroke shortening by lambda values, as was done previously mainly in the context of oxidative phosphorylation.

_{Str}

^{Ld}= - A

_{Str}

^{P}; (Δl/l

_{Str})

^{2}= 1.0), q

_{Str}is given by:

^{4}< A

_{Str}

^{Ld}< −A

_{Str}

^{P}, λ

_{Str}

^{P}(A

_{Str}

^{Ld}) will be smaller than 1.0 but not zero, so that the process in this region would proceed at a higher degree of coupling (e. g. at A

_{Str}

^{Ld}= −4.5 10

^{4}, q

_{Str}= 0.833). At values of A

_{Str}

^{Ld}> ≈ -3 × 10

^{4}the process is totally coupled ((Δl/l

_{Str})

^{2}= 0), that is, cross-bridges work at full stroke length. Only this part of the performance curve (Figure 1 and Figure 2) is hyperbolic and fulfils Hill’s formalism. Between the intersection (A

_{Str}

^{Ld}= −4.756×10

^{4}) and A

_{Str}

^{Ld}= - A

_{Str}

^{P}, J

_{Str}

^{Ld}formally could be negative, which would mean that actin filaments were moving in the direction of stretching. This is, however, impossible, because actomyosin bonds would have to be broken by a load force, which is smaller than F

_{0}. Therefore, in this region of loads, J

_{Str}

^{Ld}cannot be negative; it must remain zero.

#### 2.4. Power Output and Efficiency

^{2+}]s (1.08 and 0.34 µM, respectively) are shown. Respective curves have similar shapes; however, F

_{0}and v

_{max}, and therefore power output values, are markedly increased at high [Ca

^{2+}].

^{2+}]s are nearly identical (Figure 3D). In panel B, efficiency of a totally coupled cross-bridge cycle is shown. Under these conditions the curve has no maximum.

_{En}, A

_{En}

^{Ld}, and A

_{En}

^{P}, as well as from L

_{Str}, A

_{Str}

^{Ld}, and A

_{Str}

^{P}. All results derived in the above sections could be verified by the simulation (SIM

_{GLYgen}). So,

_{En1}= -L

_{Str2}is fulfilled, and therefore, cross-bridge cycling at zero resistance.

^{2+}]

_{.}They are very similar; their maximum lies at about 0.18 v

_{max}. Because the appearance of the maximum is caused by uncoupling, the coordinates of η

_{max}are highly dependent on uncoupling parameters.

**Figure 3.**Power output and efficiency at two different Ca

^{2+}concentrations. (

**A**

**)**and

**(**

**C**

**)**[Ca

^{2+}] = 1.06 µM;

**(B)**[Ca

^{2+}] = 0.36 µM;

**C:**under totally coupled conditions; (

**D**

**)**(red squares) efficiency at 1.06 µM [Ca

^{2+}]

_{, }(blue circles) efficiency at 0.36 µM [Ca

^{2+}]. All plots are results from SIM

_{GLYge}.

#### 2.5. Calcium Ions and Force Development

_{Str}

^{Ld}at a given [Ca

^{2+}]. On the one hand, the driving force is changed by the load potential (see Figure 1, linear dependence), and on the other hand the conductance L

_{Str}depends on A

_{Str}

^{Ld}through the hyperbolic inhibition factor. At a given [Ca

^{2+}]

_{, }both effects are responsible for the characteristic appearance of the performance curve under coupled conditions.

^{2+}] with A

_{En}

^{P}as well as with L

_{Str}is necessary. In the latter case, [Ca

^{2+}] can activate J

_{Str}through a sigmoid activation factor (A15). This takes into account the fact that Ca

^{2+}binding to troponin C removes the inhibition of cross-bridge cycling, so that binding of myosin heads to actin binding sites becomes possible [32,33]. On the other hand, [Ca

^{2+}] is known to strongly activate force development. Here it is assumed that this may be caused by an increase in cross-bridge concentration [CB]. By introducing a [Ca

^{2+}] dependent K

_{B}

^{ref}(see (A14)), a sigmoid variation in both [CB] and force F by [Ca

^{2+}] can be obtained (Figure 4.).

**Figure 4.**Developed force and cross-bridge concentration [CB] and their dependence on [Ca

^{2+}]. (red squares) force; (blue circles) [CB]. Notice that at the given dimensioning of the right ordinate a matching of results is obtained.

^{2+}] activated J

_{Str}, and in addition by [Ca

^{2+}] inhibited J

_{En}(see (A14)). The inhibition of J

_{En}by [Ca

^{2+}] is brought about by a decrease of A

_{En}

^{P}with increasing [Ca

^{2+}]. This is possible because this reaction proceeds at a very high conductance and therefore, is close to equilibrium. So already a small variation of the driving force can produce a large change in the reaction velocity. In this way, a sensible, [Ca

^{2+}] dependent adjustment of [CB] and force can be achieved. An elevation of [Ca

^{2+}] thus increases both shortening velocity as well as force development.

_{En}] + [CB]) of a half-sarcomere amounts to 656 µM (see Methods). At a saturating [Ca

^{2+}] of 1.08 µM, fluxes J

_{En}and J

_{Str}are so adjusted as to yield a concentration of [CB] = 0.25 ([MH

_{En}] + [CB]), i.e., at this [Ca

^{2+}], 25% of myosin heads form cross-bridges and thus are involved with cycling and force generation. At [Ca

^{2+}] = 0.36 µM, only about 3% of cross-bridges are engaged, and at 0.09µM [Ca

^{2+}], [CB] is further markedly reduced, which means that now near resting conditions are reached.

^{2+}], four different groups may alternately be involved with contraction. The cycling frequency of an individual cross-bridge would then be much lower than the frequency of ATP splitting, which might be advantageous, especially at high velocities. Furthermore, an alternating involvement of groups may be absolutely necessary for a smooth shortening. How this might be accomplished is so far not known. An involvement of special filaments of the sarcomere cytoskeleton [34,35,36], which may be responsible for a subtle sensing of load forces and an undisturbed takeover of a given load by a new fraction of cross-bridges during synchronous stroking, seems indispensable.

^{2}= Pa, Pa = Pasqual) obtained from SIM

_{GLYgen}(A16) in the present study are comparable to experimental values. For instance, a value of 372 kPa (from F

_{0}= 7.756 × 10

^{−4}N, [Ca

^{2+}] = 1.06 µM, 37°C) found here, seems to be in reasonable agreement with about 320 kPa resulting from measurements with a fast-twitch mouse fiber at 25°C [37]. The extrapolated value of maximal shortening velocity, v

_{max}

^{HS}= 1.95 µm × HS

^{−1}× s

^{−1}([Ca

^{2+}]= 1.06 µM, A

_{L}= 0 J/mol, HS = half-sarcomere) compares to 1.6 µm × HS

^{−1}× s

^{−1}of frog fibers at 0°C [12]. A value of η

_{max}of about 50% at about 0.18 v

_{max}([Ca

^{2+}] = 1.06µM) results from adjustment. It compares to the experimental values of 35–45 % for the same value of v for frog muscles at 0 °C [12].

#### 2.6. [H^{+}], [Mg^{2+}], and Fatigue

^{2−}. By using a reference constant and binding polynomials, an [H

^{+}] and [Mg

^{2+}] dependent K'

_{ATP}of this reaction can be formulated (see A6 and A7). In simulations of fatigue, in addition to [H

^{+}], [Mg

^{2+}] has also been included as a variable, especially because this ion may interfere with ATP species and so may influence J

_{En}through a change in [MgATP

^{2−}], which in turn would alter [CB].

^{+}] in the sarcosol are brought about mainly by two different mechanisms, which are both related to metabolic activity. One source of protons is manifest when metabolism is switched from rest to high power output. Fluxes in ATP consumption and production, J

_{ATP}

^{Con}and J

_{ATP}

^{Pro}, respectively, must then both increase to the same extent to reach a new steady state. During the adjustment, a phase of disturbed steady state occurs, during which both fluxes do not match. When power output increases, J

_{ATP}

^{Con}always leads J

_{ATP}

^{Pro}, i.e., there is an uncompensated ATP splitting until a new steady state is reached, at which point ATP production again equals ATP consumption.

^{+}] and [Mg

^{2+}] (see (A6) and (A7)) for derivation of [H

^{+}] changes and pH buffering). In addition, the CK and adenylate kinase (AK) reactions are involved, because these equilibria are also changed under these conditions and, as with ATP splitting, H

^{+}and Mg

^{2+}binding species are involved. Buffering of both ion concentrations is brought about mainly by sites intrinsic to the sarcosol. For Mg

^{2+}binding sites, an additional release of Mg

^{2+}by interfering [H

^{+}] has to be expected.

^{+}] changes. Interestingly, [H

^{+}] production by ATP splitting is practically compensated by [H

^{+}] consumption by the CK reaction. The contribution by the AK reaction is negligible. A similar behavior is found for Mg

^{2+}(Figure 5B). A concentration increase in this ion is mainly brought about by acidification.

**Figure 5.**Time courses of [H

^{+}] and [Mg

^{2+}] during extreme power output.

**A:**[H

^{+}] fluxes; (brown) mainly LDH reaction and lactate transport; (red) ATP splitting; (blue) J

_{AK}; (yellow) J

_{CK}; (black line) resultant [H

^{+}] flux;

**B:**[Mg

^{2+}] fluxes of the same reactions.

^{+}accumulate in the glycocalyx (the outer aspect of the sarcolemma), the concentrations of these compounds also increase drastically in the sarcosol. This seems to be the main mechanism of sarcosolic acidification.

^{+}, and lactate accumulate during conditions of fatigue in a similar way as can be observed during ischemia or hypoxia, which are known to be the result of impaired ATP production, it seems justified to suggest that the preconditioning for fatigue may also be initiated by a deterioration of the energy metabolism of the muscle fibers. Whenever ATP delivery does not match ATP consumption, such a situation may arise.

_{GLYgen}

_{,}see (A16)), which is related to the energy metabolism of fast muscle fibers. At 1.08 µM [Ca

^{2+}] and a load of –1.5 × 10

^{4}J (constant glycogen content and glucose concentration [Glu] = 4.0 mM), efficiency of glycogenolytic ATP production is η

_{GLYgen}= 0.722, that of glycolytic ATP η

_{GLY}= 0.525. The higher efficiency is mainly caused by the stoichiometric coefficients of coupled ATP production of 3.0 and 2.0 for the glycogenolytic and glycolytic pathways, respectively.

^{2+}] and a load potential of −1.5 × 10

^{4}J/mol, [ADP] = 113, [Pi] = 8.32 × 10

^{3}, phosphocreatine concentration [PCr] = 9.7 × 10

^{3}, lactate concentration [Lac] = 3.0 × 10

^{3}, [Mg

^{2+}] = 832, and pH = 7.09).

_{e}and [H

^{+}]

_{e}, the flux through this pathway may become reduced. In addition, efficiency has been reduced by switching from glycogenolysis to glycolysis. The power output of ATP production is markedly reduced by these combined effects. As a result, the power of ATP production begins to fall, so that ATP consumption may overcome ATP production. Steady state cycling through ATP consuming and producing pathways can now no longer be maintained.

^{2−}] is followed by a phase of continuously enhanced reduction of this ATP species to low values ([MgATP

^{2−}] = 230.0 µM; [PCr] = 1.6 µM). Immediately after reaching a minimum, a rapid recovery of [ATP] (up to starting values) begins. [Mg

^{2+}] shows a corresponding behavior. During the first phase it increases because of acidification, and then a sharp peak is produced by the onset of an extreme uncompensated ATP splitting (Figure 6). An increased [Mg

^{2+}] may counteract the switch off of cross-bridge cycling and may aid recovery by increasing [MgATP

^{2−}].

**Figure 6.**Time courses of [MgATP

^{2−}] and [Mg

^{2+}] during development of fatigue. (red line) [MgATP

^{2−}]; (green points) [Mg

^{2+}].

^{2+}].

_{ATP}

^{Con}and J

_{ATP}

^{Pro}, become different now (J

_{ATP}

^{con}> J

_{ATP}

^{pro}; Figure 7A), both fluxes of the cross-bridge cycle, J

_{En}and J

_{Str}, have also changed. These fluxes determine concentrations of [MH

_{En}] and [CB], respectively. An increase in J

_{En}and a decrease in J

_{Str}would lower [MH

_{En}] (whereby [CB] would be increased). Both concentrations always change reciprocally (Figure 7B). A

_{Str}

^{P}and A

_{Str}

^{Ld}are also affected. A

_{Str}

^{P}in particular is rapidly reduced until it is equal to −A

_{Str}

^{Ld}. Now all fluxes of the cycle must vanish, because the driving force of J

_{Str}has become zero. As a result, ATP consumption by cross-bridge cycling is switched off.

**Figure 7.**Time courses of J

_{ATP}

^{Con}and J

_{ATP}

^{Pro}, of [CB] and [MH

_{En}], and potentials of the cross-bridge cycle during fatigue development. (

**A**

**)**(red) J

_{ATP}

^{Con}; (blue) J

_{ATP}

^{Pro}; (

**B**

**)**(black) [CB]; (blue) [MH

_{En}]; (

**C**

**)**(red points) A

_{ATP}; (brown line) A

_{Str}

^{P}; (blue line) A

_{Str}

^{Ld}; notice that after 7.5 s A

_{Str}

^{P}and A

_{Str}

^{Ld}become opposite and equal.

_{2}PO

_{4}

^{−}and MgADP

^{−}similar to an isometric contraction, but in contrast to those latter conditions, equilibrium of forces is now brought about at a much lower load force (A

_{Str}

^{P}=

^{−}A

_{Str}

^{Ld}= 0.375 × 10

^{4}J/mol at 1.08 µM [Ca

^{2+}]). A load-dependent actomyosin splitting by MgATP

^{2−}at the beginning of the stroke, that is uncoupling, is impossible under these conditions. So cross-bridge cycling with concomitant ATP consumption may be completely prevented. [ATP], therefore, can recover rapidly, even if the conditions leading to fatigue first remain unchanged.

^{+}] in the sarcosol is lowered by the Na/K pump to values below 10.0 mM, the reaction rate of this transport process is increasingly deactivated by the decreasing [Na

^{+}], so that ATP consumption also is reduced.

_{0}= b/v

_{max}(θ between 0.2 and 0.3). This latter equality is also fulfilled by the present model (

**θ**= 0.309). However, in contrast to the approach of the above authors, the hyperbolic form of Hill’s equation is produced here by introducing a Michaelis-Menten-like inhibition factor into the respective conductance, as already mentioned above. Moreover, uncoupling in the present model is produced through a load-dependent stroke shortening, which generates the maximum obtained with power plots.

## 3. Methods

_{red}] has to be reoxidised by the lactate dehydrogenase (LDH) reaction, and the lactate plus proton formed thereby is released to the extracellular space via Lac/H symport. Electrophysiological reactions at the cell membrane (sarcolemma) are omitted. Also, most reactions of the sarcoplasmatic reticulum (SR) are not addressed. Only Ca

^{2+}pumping by the sarco/endoplasmatic reticulum Ca

^{2+}ATPase (SERCA) as an ATP consuming reaction is included in simulations besides several other reactions of ADP production (see SIM

_{GLYgen}(A16)) taken over from reference [1]. Therefore, [Ca

^{2+}] is treated as an adjustable constant.

_{Cell}, a cylindrical geometry of the muscle cell is assumed. With radius R

_{Cell}= 25.76 µm, and a length L = 10

^{3}µm (fraction of whole fiber length), V

_{Cell}= 2.0847 × 10

^{6}µm

^{3}or 2.0847 nL, and A

_{Cell}= 2.0847 × 10

^{3}µm

^{2}. From data of Aliev et al. [39] for the heart, the volume of the sarcosol, V

_{c}, can be determined by adding the mitochondrial to the fibrillar volume, yielding V

_{c}/V

_{Cell}= (321 + 195 + 55)/758.5 = 0.7528 or V

_{c}≈1.57 nL. Then α

_{c}can be obtained using α

_{c}= 10

^{−12}/(F×V

_{c}) = 6.6024×10

^{−9}µM/C (F = Faraday’s constant, C = Coulomb). That is, to yield the corresponding flux in µM/ms from an electric current entering the sarcosol, this current in fA (= pS×mV = 10

^{−18}C/ms; pS = pico Siemens) has to be multiplied by α

_{c}.

_{Fibr}can be obtained. One hexagon is composed of six equilateral triangles of side length l

_{Tri}= 41.0 nm [12] and equal angles of 60°. The area of a hexagonal fibril (or HS) of radius R

_{Fibr}= 18.0 × 41.0 = 738 nm is given by:

_{Cell}(see reference [39]). The number N

_{Fibr}is then given by:

_{Tri}= 41.0 nm. A large triangle of the hexagonal HS with n-fold side length contains:

_{Tri}from 41.0 nm to 738 nm (n = 18), the resulting HS hexagon contains, with MF

_{Tri}= 190, MF

_{Hex}= 1027 myosin filaments per fibril.

_{Hex}, and in the whole fiber:

_{A}(= 6.022142 × 10

^{23}particles per mol), and by the volume of the water diffusible space in the filament lattice containing myosin heads of all HSs of the cross-sectional area of a fiber, V

_{Lat}. This volume is given by:

_{HS}= HS length at rest = 1.1 µm, f

_{MH}= length fraction of myosin filament containing myosin heads in terms of HS ([12]) = 0.62364, f

_{WDS}= volume fraction of water-diffusible space in the filament lattice volume = 0.7852).

_{tot}– [MH] ([MH] = the remaining myosin head concentration given by the simulation) is determined by two fluxes (see Results). It is adjusted to about 25% of [MH]

_{tot}at [Ca

^{2+}] = 1.08 µM and a load potential A

_{L}= 3.0×10

^{4}J/mol. The force generated by [CB] is given by:

_{CB}is the force of one single cross-bridge. It is obtained here from the stroke input potential A

_{P}(see Results) under these conditions, by assuming that the stroke length l

_{Str}= 12.0 nm. Then F

_{CB}is given by:

^{5}Pa (related to A

_{Cell}, Pa = Pasqual). The force is generated at 1.08 µM [Ca

^{2+}] by about 25% of myosin heads ([CB] = 164 µM). All force-generating cross-bridges are contained in the parallel HSs of the cross-sectional area of 1027 fibrils of one muscle fiber.

^{®}14.0 or 15.0 M011 solver AdamsBDF. The programs were run under Microsoft

^{®}Windows 7 and XP Professional.

## 4. Conclusions

_{r}S

_{i}) is always positive. When a reaction is forced against spontaneity, Δ

_{r}S

_{i}must become negative. All reaction parameters associated with entropy changes, like affinities and conductances, must inevitably follow a sign change of Δ

_{r}S

_{i}whenever such a change occurs. This is not a contradiction to Ohm’s law, but a consequence of the phenomenological definition of a conductance through ±L = J/±A. It can be concluded that the occurrence of negative conductances is realised not only with biochemical reactions in living cells, but represents a fundamental concept of coupled processes.

^{2+}], it seems highly unlikely, however, that this may be sufficient to allow normal locomotion of a subject. Only the control by the nervous system can bring about coordinated actions of several muscle fibers, groups of fibers, or even several different muscles. In this way, accelerated and decelerated motion becomes possible. To achieve this, the number of force generating cross-bridges is varied by a change in cross-sectional area, that is, by altering the number of fibers recruited for contraction. Thereby the locomotion at high efficiency or maximal power output can be controlled by will. Also, isometric contractions are indispensible for coordinated actions. They are produced by reducing the cross-sectional area to such an extent that a load dependent uncoupling is initiated to stop fiber shortening.

## Appendix

#### Negative Resistances in Simple Electric Circuits

_{R}of the redox reaction to the formation of an electrical potential difference Δϕ at the electrodes. Under open circuit conditions the reaction proceeds rapidly to equilibrium, at which A

_{R}+ Δϕ = 0. Taking A

_{R}as the positive input force, then Δϕ must be negative. A

_{R}can be expressed in electrical units using E = A

_{R}/zF (E = electromotive force in Volt V, z = charge number, F = Faraday constant in Coulombs/Volt, Δϕ in V). The coupled flow of charges (electrical current in ampere A) is then given by:

_{2}and output force A

_{1}, respectively. L

_{c}is the coupling conductance, and R

_{i}= 1/L

_{c}represents the inner resistance of the battery (R’s are given in Ω, L’s in 1/Ω). The partial conductances L

_{c1}and L

_{c2}(see equations 2e and 2f) are given by:

_{i}(0.2 Ω) and an outer resistance R

_{e}(0.4 Ω), the current I is given by E - I(R

_{i}+ R

_{e}) = 0, yielding:

_{e}= −Δϕ = +8 V.

_{c1}and L

_{c2}are given by –5/2 and 5/3 Ω

^{−1}, respectively, or R

_{i1}= −0.4, and R

_{i2}= 0.6 Ω, which fulfils R

_{i1}+ R

_{i2}= R

_{i}= 0.2 Ω. Setting R

_{e}= 4.0 Ω yields R

_{i1}= −4.0, R

_{i2}= 4.2 Ω, and again R

_{i}= 0.2 Ω. Moreover, in the electric circuit the overall resistance R

_{i1}+ R

_{e}vanishes. It should be noticed that partial resistances are not constant, although R

_{i}is a constant. They depend both on R

_{e}and L

_{e}, respectively.

_{circ}= Ф

_{1}+ Ф

_{2}+ Ф

_{e}= I(Δϕ + E + U), Ф

_{circ}= I × E(240 J/s or 0.24 kW), or

_{circ}= L

_{c2}× E

^{2}(1/0.6×144 = 0.24 kW)

_{e}is given by:

_{e}= R

_{i}= 0.2 Ω, p

_{out}

^{max}= 0.18 kW. Because P

_{out}can also be expressed as P

_{out}= -L

_{c1}× Δϕ

^{2}, this yields with L

_{e}= 1/0.2 1/Ω, p

_{out}

^{max}= −(−5.0)( −6)

^{2}= 0.18 kW.

^{II}is directed against E

^{I}(E

^{II}= −10.9 V, R

_{i}

^{II}= 0.5 Ω), Δϕ

^{II}is now positive and E

^{II}negative. The outer resistance R

_{e}stands for the resistance of the wires connecting both batteries. In this constellation, −Δϕ

^{I}is no longer equal to U

_{e}. Now − (Δϕ

^{I}+ U

_{e}) = −Δϕ

^{Ie}= Δϕ

^{II}is valid. Consequently, R

_{e}also has to be added to R

_{i}

^{I}yielding R

_{i}

^{Ie}= R

_{i}

^{I}+ R

_{e}.

^{Ie}= −11.4 V, and Δϕ

^{II}= 11.4 V is obtained.

_{i}

^{I}+ R

_{e}+ R

_{i}

^{II}= 1.1 Ω, therefore, is also given by R

_{i2}

^{I}+ R

_{i1}

^{II}= 12 + (–10.9) = 1.1 Ω.

#### ATP, ADP, and P_{i} Species as Functions of [H^{+}] and [Mg^{2+}]

^{2−}, are calculated according to the methods of Alberty [20]. When respective constants are known, which are dependent on temperature and ionic strength, so-called polynomials can be formulated, from which several parameters like species concentration, K'(biochemical equilibrium constant), or [H

^{+}] and [Mg

^{2+}] binding can be taken.

^{2−}, MgADP

^{−}, and H

_{2}PO

_{4}

^{−}, to make the reactions of the cross-bridge cycle directly dependent on these compounds. The reaction in chemical notation form is given by:

^{ref1}(= 6.267 × 10

^{5}), K3at, K3ad, P

_{ATP4−}, and P

_{ADP3−}were taken from [1]. At given [H

^{+}] and [Mg

^{2+}] values, the corresponding K'(= 4.9687 × 10

^{5}, pH = 7.1, [Mg

^{2+}] = 800 µM) is identical to formulations with other reference constants.

#### [H^{+}] and [Mg^{2+}] Buffering

^{+}] buffering of SMFs is treated here analogously to VMs (see [1]). It is given by:

^{+}] and [Mg

^{2+}], respectively, are given by:

_{H}

^{BU}if binding sites contain only a single site with only one proton dissociation constant.

^{2+}] buffering, it is suggested that during short time intervals Mg

^{2+}transport reactions across membranes can be neglected. Only intrinsic binding sites including [ATP] are present and, as with [H

^{+}] changes, [Mg

^{2+}] changes induced by ATP splitting, the CK reaction, and the AK reaction have been addressed. [Mg

^{2+}] buffering can be expressed as:

^{2+}binding depends on [H

^{+}]. A decrease of pH can liberate magnesium ions from intrinsic binding sites and from the predominant ATP species MgATP

^{2−}. The H

^{+}and Mg

^{2+}dissociation constants of both binding sites are set to the values of a simplified P

_{ATP4−}. The total concentration of Mg

^{2+}binding sites, , is adjusted to 9.0 mM plus a variable [ATP]. The change of [Mg

^{2+}] is given then by:

^{+}]/dt, only those fluxes producing or consuming protons are considered, because changes of [H

^{+}] depend mainly on these fluxes (see Figure 5A).

^{2+}] is introduced as a variable only in those simulations that deal with muscular fatigue. Because changes of [Mg

^{2+}] depend mainly on acidification, and pH does not change markedly even under conditions of high power output, this variable is set constant to 800 µM for all other simulations.

#### Simulation of Glycogenolysis and Glycolysis

_{Phosph}

^{max}= 4×10

^{−3}(µM/ms)×(mol/J), K

_{M}

^{Phosph}= 2.0 µM, K'

_{Phosph}= 0.286;

_{GPI}

^{max}= 2×10

^{−2}(µM/ms)×(mol/J K

_{M}

^{GPI}), = 300 µM, K'

_{GPI}= 0.276;

_{LDH}

^{max}= 2.4×10

^{−2}(µM/ms)×(mol/J), K

_{M}

^{ldh}= 50 µM, K'

_{LDH}= 2.497×10

^{4};

_{Lac}

^{max}= 2.866× 10

^{8}pS (pico Siemens = 10

^{−8 }Ω

^{−1}), K

_{M}

^{Lac}= 17 mM;

^{+}/H

^{+}exchange,

_{NaH}

^{max}= 10

^{5}pS, H

_{05}= 0.1 µM, S

_{[H}

^{+}

_{]}= 0.004 µM;

_{AnEx}

^{max}= 10

^{4}pS, H

_{05}= 0.05 µM, S

_{05}= 0.008 µM, K

_{M}

^{anex}= 13.0 mM.

_{En}

^{max}= 6.138×10

^{−2}(µM/ms)×(mol/J), f

_{corr}= ([CB

_{t}]−[CB

_{0}])/([CB

_{t}]−[CB]), [CB

_{t}] = 656 µM, [CB

_{0}] = 492 µM, ε = 24.0, A

_{L05}= 3.0×10

^{4}(J/mol), S

_{L}= 2.0×10

^{4}(J/mol), , A

_{En}= −5.866729×10

^{4}, .

_{0}] corrects L

_{En}

^{max}for changes of [CB]; the second factor is introduced to damp changes of [CB]. K

_{B}

^{ref}is not constant, but depends on [Ca

^{2+}].

_{Str}

^{max}= 4.6 × 10

^{−4}(µM/ms)×(mol/J) for changes of [CB

_{t}] – [CB]. The second factor introduces [Ca

^{2+}] dependence of L

_{Str}. The third factor is responsible for the hyperbolic character of the flux equation at constant [Ca

^{2+}] with K

_{I}

^{CB}= −1.8 × 10

^{4}J/mol, which represents the inhibition constant, K

_{R}

^{ref}= 1.310889 × 10

^{−4}. λ values are not independent; this interdependency is given in Results. Uncoupling is formulated to occur in two steps, expressed by λ

_{Str1}

^{P}= 0.15 and λ

_{Str2}

^{P}= 0.85.

^{2+}] ([Mg

^{2+}] = 0.8 mM = const.) is used to calculate the various points of figures (Figure 1B, Figure 2, Figure 3, and Figure 4) for a given [Ca

^{2+}] and various loads. As already mentioned, [Mg

^{2+}] is introduced as a variable only for conditions of very high power output leading to fatigue. From the output of the simulation many more variables, as shown here, can be obtained as functions of time, which may often be helpful in understanding underlying mechanisms.

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**MDPI and ACS Style**

Diederichs, F.
From Cycling Between Coupled Reactions to the Cross-Bridge Cycle: Mechanical Power Output as an Integral Part of Energy Metabolism. *Metabolites* **2012**, *2*, 667-700.
https://doi.org/10.3390/metabo2040667

**AMA Style**

Diederichs F.
From Cycling Between Coupled Reactions to the Cross-Bridge Cycle: Mechanical Power Output as an Integral Part of Energy Metabolism. *Metabolites*. 2012; 2(4):667-700.
https://doi.org/10.3390/metabo2040667

**Chicago/Turabian Style**

Diederichs, Frank.
2012. "From Cycling Between Coupled Reactions to the Cross-Bridge Cycle: Mechanical Power Output as an Integral Part of Energy Metabolism" *Metabolites* 2, no. 4: 667-700.
https://doi.org/10.3390/metabo2040667