0.2

0.3

0.4

Cost (J

mg

-1)

orga

nic

fract

ion

(wt%

)

2.5

5

010

Sensitivity (%)

0.25

25

16

0.1

0.5

1.0

1.5

Cost (J

mg

-1)

orga

nic

fract

ion

(wt%

)

2.5

5

0

10

Sensitivity (%)

3

1.70.25 0.5

ii

“CalcDraft” — 2015/5/25 — 9:48 — page 4 — #4 ii

ii

ii

We suppose that all of the CaCO3

within each calcifyingcompartment comes out of solution at once at the end point.In terms of the variables [4], the end-point is reached at adimensionless time,

t

0end

=x

2

2 (✏ � 1)[6]

where

x

2

⌘ [HCO�3

]sw

[DIC]sw

[7]

is the bicarbonate fraction of seawater. For ✏ < 1, the end-point time is negative, reflecting the fact that for low pump-ing rates, CO

2

di↵uses into the compartment and dissociatesfaster than ion pumps can neutralize the resulting protons.Accordingly, below a certain pumping capacity, calcificationby means of heightened ⌦ becomes impossible.

Given a time t

0end

we compute the cost of CaCO3

per unitmass,

C =12

⌘

µ

C

x

2

✏

✏ � 1, [8]

where µ

C

= 100g mol�1 is the molar mass of CaCO3

and⌘ ⇡ 30 kJ mol�1 (see above). An important consequence isthat the faster an organisms pumps (the higher is ✏), the

cheaper its skeleton becomes. Thus there is an ever-presentenergetic benefit in pumping rapidly. For ✏ ! 1 the costC ! 1/2 (⌘/µ

C

)x2

⇡ 0.13 J mg�1, a number consistent withobservations of larvae [76], but adults appear to expend moreenergy than this minimum [58], suggesting lower ✏. Mechanis-tically, the infinite pumping case is where one H+ is pumpedfor each HCO�

3

ion present in the original seawater.The above discussion accounts for the inorganic, CaCO

3

part of the skeleton, but the total cost of a skeleton includesthe metabolic investment associated with the organic portion[57, 47, 58, 49, 12]. The organic matrix makes up about 1-5wt% gastropod [57] and mollusc [49] skeletons, but its costis poorly constrained. An approximate experimental valuestands at ⌫ ⇠ 30 J mg�1 [57, 58], roughly 200 times the mini-mum inorganic cost C. The pure cost of amino acid synthesishas been inferred from other work to sit closer to ⇠ 3 J mg�1

[56, 59], however, the matrix itself is an organised structureof glycoproteins which likely costs much more than its con-stituent amino acids. Accordingly, here we display results foran organic cost of ⌫ ⇠ 30 J mg�1 but show in the Supplemen-tal information that our results remain qualitatively similarfor smaller organic costs.

The total cost E of a given mass of shell comprising an or-ganic matrix of mass fraction f

p

is

E(x2

, f

p

) =12

µ x

2

✓✏

✏ � 1

◆(1 � f

p

) + ⌫ f

p

. [9]

For ease of presenting, we now define a function that measuresthe sensitivity of whole shell cost to changes in seawater chem-istry. This function is expressed as the fractional increase inE for a given increase �x

2

of the parameter x

2

:

S ⌘ 1E

@E@x

2

�x

2

. [10]

It is important to note that the value of x

2

is a measure ofacidification state and increases under ocean acidification, i.e.the bicarbonate ion becomes more abundant at the expense ofcarbonate ions. At present x

2

⇡ 0.85 [17, 76], with the IPCCreport predicting an increase of �x

2

⇠ 0.1 within a century[18, 17](does IPCC5 HAVE a guesstimate?), approximatelyequivalent to a 10% rise.

Instead of presenting our results in terms of ✏, we favourthe fraction of carbon within the CaCO

3

that is metabolically-derived fM . Such a fraction has been inferred from numerousmeasurements of �

13C within a variety of organisms acrossmultiple ontogenetic stages [51, 2, 27, 76]. Within the frame-work of our model, fM may be computed from the quantityof light CO

2

that di↵uses into the calcifying space from sur-rounding cells in time t

0end

. Using the solutions to equations 3fM evolves according to,

f

M

(t0) =t

0

1 + t

0 . [11]

Substituting t = t

0end

yields the relationship,

fM|t0end=

x

2

2 (✏ � 1) + x

2

, [12]

relating the fraction of metabolic carbon in the final shell tothe pumping parameter ✏. Note that a lower �

13C signaturecorresponds to slower ion-pumping.

ResultsIn Figure 2, we present a contour plot denoting the theoreti-cal value of sensitivity S for a range of organic shell fractionsfp and metabolic carbon fractions fM . For the sake of def-initeness, we choose �x

2

/x

2

= 0.1 and x

2

= 0.85, roughlyconsistent with the forecasts, with the results qualitativelysimilar for slightly di↵erent numbers. High ✏ is equivalentto low f

M

(see equation 12; Figure ). The greatest sensitiv-ities occur at low organic fractions, when S ⇡ �x

2

/x

2

, in

f M

0.5

0Pumping parameter, ε 20 1

Observed For Juvenile Oysters

Observed For Larval Oysters

Theoretical (This study)

ε Max, i

x2=0.85x2=0.9x2=0.95

{Drop in maximum rate for a species

of given ε Max, i

10 20 εMax

Larvae pump ions at physiological capacity

ii

“Calc” — 2014/12/14 — 14:35 — page 4 — #4 ii

ii

ii

There exists a large number of uncertainties in this problem.Namely, the membrane permeability v, the vesicle volume andsurface area, together with the pumping rate. All of these con-stitute free parameters in the problem. We may remove suchparameters by scaling the variables appropriately. Specifically,the concentration of DIC in the seawater, [DIC]

sw

largely dic-tates the mass of calcium carbonate that results and the in-flux of [DIC] into the fluid (F ⌘ v A �[CO

2

]) sets a timescale,T over which all of the DIC is replaced by cell-derived DIC.Accordingly, we parameterize the pumping rate R

p

as somefraction, ✏ of F . In summary,

F = v A �[CO2

]

R

p

= ✏ F

T =V [DIC]

sw

F . [7]

Once the end point is reached, we assume that all of theDIC within the sequestered fluid comes out of solution at once.Thus, the time taken gives an energy required and the volumeof carbonate precipitated allows a determination of the costper mole of CaCO

3

. In terms of the variables 7, the end-pointis reached at a dimensionless time,

t

0end

=x

2

2 (✏ � 1)[8]

where x

2

= [HCO�3

]sw

/[DIC]sw

is the bicarbonate fraction ofseawater. Notice that for this time to be positive, ✏ > 1, mean-ing that below a certain pumping rate, the DIC concentrationgrows faster than the carbonate concentration.

Put the rest in the Methods section.

The Cost of Shell.Integrating all of the equations, we end up with an analyti-cal expression for the energy required per mole of inorganicCaCO

3

precipitated. Specifically, the cost is

C =12µx

2

✏

✏ � 1. [9]

This expression approaches the cost assumed elsewhere (e.g.[25]) for large ✏, with the cost being grater for smaller ✏. Ac-cordingly, the faster an organisms pumps, the cheaper calcium

carbonate becomes and so there is a naturally, evolutionarydrive to pump faster, assuming the goal of the organism is tominimise the cost of CaCO

3

. The cost C (⇠ 0.3 J mg�1) is verysimilar to that determined experimentally, both for adults[22]and larvae[25].

A shell is comprised of more than simply the inorganicCaCO

3

required in building it. A substantial fraction, of theorder of ⇠ 0.5 � 5 wt% is comprised of organic matrix com-prised of post-translationally-modified proteins which act asa template ([21, 4]). The total cost of the shell must theninclude the cost of this protein fraction. However, the costof protein, though poorly constrained, is about 50-100 timesthat of the inorganic fraction[22] at ⌫ ⇠ 30 J mg�1. Thus,when considering the total cost per given mass of shell, E withprotein fraction f

p

we obtain

E(x2

, f

p

) =12

µ x

2

✓✏

✏ � 1

◆(1 � f

p

) + ⌫ f

p

. [10]

Metabolic Carbon FractionFrom Figure ??, the only DIC entering the fluid is in theform of respired CO

2

derived from the cell cytoplasm. SuchDIC will consist of isotopically light carbon (CITE: coral pa-per). Owing to the di↵usive nature of CO

2

entry, the quan-tity of metabolic carbon in the fluid vs “heavy” carbon at time

t

0 = t

0end

will be larger for longer times. We see from equation 8that longer times translate to smaller values of ✏, or equiva-lently, weaker pumping. It is simple to quantify the fractionof metabolic carbon using the solutions to equations 3 - 5.Specifically,

f

M

=t

0

1 + t

0 [11]

where f

M

is the fraction of metabolic carbon in the fluid atdimensionless time t

0 = t/T . Substituting the expression fort

0end

yields a quantitative, measurable proxy for dimensionlesspumping rate, ✏, in the form of a metabolic carbon fraction.Such a metabolic carbon fraction manifests in real-world sam-ples as a well-defined carbon isotope signature ([18, 1, 10, 25]).Accordingly, the metabolic carbon fraction of the final, pre-cipitated CaCO

3

will be

fM(t0end

) =x

2

2 (✏ � 1) + x

2

. [12]

Sensitivity to Ocean ChemistryWith the above functional forms for shell costs in hand, wecan begin to evaluate what kinds of selective pressures may actupon calcifying organisms on both short (ecological) and long(evolutionary) timescales3. Specifically, on long timescales,it may be advantageous to minimise the cost of a shell (E)which from equation 10 may be achieved by reducing the pro-tein fraction, f

p

within the shell. However, throughout thecourse of evolution, a species must adapt to changing environ-mental conditions[5]. The environmental factor entering theequations here is the bicarbonate fraction of seawater, x

2

. Atpresent x

2

⇡ 0.8 [29, 25], but ocean acidification will causeit to increase, with the IPCC report predicting an increaseof �x

2

⇠ 0.15 within a century (CITE). Such a change willlead to change in the cost E with fractional change in cost be-

ing dependent upon f

p

. Specifically, we define a measure ofsusceptibility,

S ⌘ 1E

@E@x

2

, [13]

which quantifies the fractional change in cost for a givenchange to seawater chemistry. We plot the susceptibility infigure ?? for a range of protein fractions and metabolic car-bon fractions4

Organisms with little metabolic carbon, who are pumpingfast, are both minimising their inorganic cost (C) and theirsusceptibility (S). Therefore, both from an energy-saving andevolutionary stand-point, it pays to pump fast.5 In lightof this, we also display (Figure ??) the fractional increasein cost resulting from an increase of x

2

from the modernvalue to unity for various protein fractions in the infinite-pumping case (f

M

! 0). The maximum increase in cost is⇠ 15% (= 100 �x

2

/x

2

), with that number dropping rapidlywith larger protein fractions. GIVE TWO EXAMPLES.

Hence, whereas pumping fast is beneficial on two fronts,protein fraction acts as a balance between the costly and thesusceptible. Thus, we may imagine that times of turbulentenvironmental conditions favour the evolution of organisms

3The distinction between such timescales is breaking down with recent realisations that evolutionarychange may proceed at much more rapid rate than once thought - even approaching the timescalesof ecological change (CITE CITE CITE:)4A proxy for pumping rate, �. High � is equivalent to low fM as faster pumping shortens to timefor precipitation.5Provided minimisation of energy is paramount, which might not always be the case, as discussedbelow.

4 www.pnas.org/cgi/doi/10.1073/pnas.0709640104 Footline Author

Max

imum

Rat

e

High

Low

Fig. 3. Upper Panel: The metabolic carbon fraction in the final CaCO3

as afunction of pumping parameter ✏. Higher ✏ translates to shorter calcification times,allowing less time for light carbon to di↵use in from the cell. As a result, faster pump-ing leads to lower metabolic carbon fractions. Lower Panel: The influence of x

2

onthe calcification rate for a given value of ✏. The upper panel suggests that, owing tothe very low metabolic carbon values of larval shells, they pump at or near physiolog-ical capacity. The lower panel demonstrates the consequences of ocean chemistry onthe maximum calcification rate attainable using this physiological limit. Acidificationreduces the capacity, meaning that a given physiology is not able to reach as a high acalcification rate. Owing to kinetic constraints within the larval stage, such a reducedcalcification capacity may have a di↵erential a↵ect on mortality.

4 www.pnas.org/cgi/doi/10.1073/pnas.0709640104 Footline Author

metabolic carbon fraction,0 0 0.25 0.5

metabolic carbon fraction,

ii

“CalcDraft” — 2015/5/25 — 9:48 — page 4 — #4 ii

ii

ii

We suppose that all of the CaCO3

within each calcifyingcompartment comes out of solution at once at the end point.In terms of the variables [4], the end-point is reached at adimensionless time,

t

0end

=x

2

2 (✏ � 1)[6]

where

x

2

⌘ [HCO�3

]sw

[DIC]sw

[7]

is the bicarbonate fraction of seawater. For ✏ < 1, the end-point time is negative, reflecting the fact that for low pump-ing rates, CO

2

di↵uses into the compartment and dissociatesfaster than ion pumps can neutralize the resulting protons.Accordingly, below a certain pumping capacity, calcificationby means of heightened ⌦ becomes impossible.

Given a time t

0end

we compute the cost of CaCO3

per unitmass,

C =12

⌘

µ

C

x

2

✏

✏ � 1, [8]

where µ

C

= 100g mol�1 is the molar mass of CaCO3

and⌘ ⇡ 30 kJ mol�1 (see above). An important consequence isthat the faster an organisms pumps (the higher is ✏), the

cheaper its skeleton becomes. Thus there is an ever-presentenergetic benefit in pumping rapidly. For ✏ ! 1 the costC ! 1/2 (⌘/µ

C

)x2

⇡ 0.13 J mg�1, a number consistent withobservations of larvae [76], but adults appear to expend moreenergy than this minimum [58], suggesting lower ✏. Mechanis-tically, the infinite pumping case is where one H+ is pumpedfor each HCO�

3

ion present in the original seawater.The above discussion accounts for the inorganic, CaCO

3

part of the skeleton, but the total cost of a skeleton includesthe metabolic investment associated with the organic portion[57, 47, 58, 49, 12]. The organic matrix makes up about 1-5wt% gastropod [57] and mollusc [49] skeletons, but its costis poorly constrained. An approximate experimental valuestands at ⌫ ⇠ 30 J mg�1 [57, 58], roughly 200 times the mini-mum inorganic cost C. The pure cost of amino acid synthesishas been inferred from other work to sit closer to ⇠ 3 J mg�1

[56, 59], however, the matrix itself is an organised structureof glycoproteins which likely costs much more than its con-stituent amino acids. Accordingly, here we display results foran organic cost of ⌫ ⇠ 30 J mg�1 but show in the Supplemen-tal information that our results remain qualitatively similarfor smaller organic costs.

The total cost E of a given mass of shell comprising an or-ganic matrix of mass fraction f

p

is

E(x2

, f

p

) =12

µ x

2

✓✏

✏ � 1

◆(1 � f

p

) + ⌫ f

p

. [9]

For ease of presenting, we now define a function that measuresthe sensitivity of whole shell cost to changes in seawater chem-istry. This function is expressed as the fractional increase inE for a given increase �x

2

of the parameter x

2

:

S ⌘ 1E

@E@x

2

�x

2

. [10]

It is important to note that the value of x

2

is a measure ofacidification state and increases under ocean acidification, i.e.the bicarbonate ion becomes more abundant at the expense ofcarbonate ions. At present x

2

⇡ 0.85 [17, 76], with the IPCCreport predicting an increase of �x

2

⇠ 0.1 within a century[18, 17](does IPCC5 HAVE a guesstimate?), approximatelyequivalent to a 10% rise.

Instead of presenting our results in terms of ✏, we favourthe fraction of carbon within the CaCO

3

that is metabolically-derived fM . Such a fraction has been inferred from numerousmeasurements of �

13C within a variety of organisms acrossmultiple ontogenetic stages [51, 2, 27, 76]. Within the frame-work of our model, fM may be computed from the quantityof light CO

2

that di↵uses into the calcifying space from sur-rounding cells in time t

0end

. Using the solutions to equations 3fM evolves according to,

f

M

(t0) =t

0

1 + t

0 . [11]

Substituting t = t

0end

yields the relationship,

fM|t0end=

x

2

2 (✏ � 1) + x

2

, [12]

relating the fraction of metabolic carbon in the final shell tothe pumping parameter ✏. Note that a lower �

13C signaturecorresponds to slower ion-pumping.

ResultsIn Figure 2, we present a contour plot denoting the theoreti-cal value of sensitivity S for a range of organic shell fractionsfp and metabolic carbon fractions fM . For the sake of def-initeness, we choose �x

2

/x

2

= 0.1 and x

2

= 0.85, roughlyconsistent with the forecasts, with the results qualitativelysimilar for slightly di↵erent numbers. High ✏ is equivalentto low f

M

(see equation 12; Figure ). The greatest sensitiv-ities occur at low organic fractions, when S ⇡ �x

2

/x

2

, in

f M

0.5

0Pumping parameter, ε 20 1

Observed For Juvenile Oysters

Observed For Larval Oysters

Theoretical (This study)

ε Max, i

x2=0.85x2=0.9x2=0.95

{Drop in maximum rate for a species

of given ε Max, i

10 20 εMax

Larvae pump ions at physiological capacity

ii

“Calc” — 2014/12/14 — 14:35 — page 4 — #4 ii

ii

ii

There exists a large number of uncertainties in this problem.Namely, the membrane permeability v, the vesicle volume andsurface area, together with the pumping rate. All of these con-stitute free parameters in the problem. We may remove suchparameters by scaling the variables appropriately. Specifically,the concentration of DIC in the seawater, [DIC]

sw

largely dic-tates the mass of calcium carbonate that results and the in-flux of [DIC] into the fluid (F ⌘ v A �[CO

2

]) sets a timescale,T over which all of the DIC is replaced by cell-derived DIC.Accordingly, we parameterize the pumping rate R

p

as somefraction, ✏ of F . In summary,

F = v A �[CO2

]

R

p

= ✏ F

T =V [DIC]

sw

F . [7]

Once the end point is reached, we assume that all of theDIC within the sequestered fluid comes out of solution at once.Thus, the time taken gives an energy required and the volumeof carbonate precipitated allows a determination of the costper mole of CaCO

3

. In terms of the variables 7, the end-pointis reached at a dimensionless time,

t

0end

=x

2

2 (✏ � 1)[8]

where x

2

= [HCO�3

]sw

/[DIC]sw

is the bicarbonate fraction ofseawater. Notice that for this time to be positive, ✏ > 1, mean-ing that below a certain pumping rate, the DIC concentrationgrows faster than the carbonate concentration.

Put the rest in the Methods section.

The Cost of Shell.Integrating all of the equations, we end up with an analyti-cal expression for the energy required per mole of inorganicCaCO

3

precipitated. Specifically, the cost is

C =12µx

2

✏

✏ � 1. [9]

This expression approaches the cost assumed elsewhere (e.g.[25]) for large ✏, with the cost being grater for smaller ✏. Ac-cordingly, the faster an organisms pumps, the cheaper calcium

carbonate becomes and so there is a naturally, evolutionarydrive to pump faster, assuming the goal of the organism is tominimise the cost of CaCO

3

. The cost C (⇠ 0.3 J mg�1) is verysimilar to that determined experimentally, both for adults[22]and larvae[25].

A shell is comprised of more than simply the inorganicCaCO

3

required in building it. A substantial fraction, of theorder of ⇠ 0.5 � 5 wt% is comprised of organic matrix com-prised of post-translationally-modified proteins which act asa template ([21, 4]). The total cost of the shell must theninclude the cost of this protein fraction. However, the costof protein, though poorly constrained, is about 50-100 timesthat of the inorganic fraction[22] at ⌫ ⇠ 30 J mg�1. Thus,when considering the total cost per given mass of shell, E withprotein fraction f

p

we obtain

E(x2

, f

p

) =12

µ x

2

✓✏

✏ � 1

◆(1 � f

p

) + ⌫ f

p

. [10]

Metabolic Carbon FractionFrom Figure ??, the only DIC entering the fluid is in theform of respired CO

2

derived from the cell cytoplasm. SuchDIC will consist of isotopically light carbon (CITE: coral pa-per). Owing to the di↵usive nature of CO

2

entry, the quan-tity of metabolic carbon in the fluid vs “heavy” carbon at time

t

0 = t

0end

will be larger for longer times. We see from equation 8that longer times translate to smaller values of ✏, or equiva-lently, weaker pumping. It is simple to quantify the fractionof metabolic carbon using the solutions to equations 3 - 5.Specifically,

f

M

=t

0

1 + t

0 [11]

where f

M

is the fraction of metabolic carbon in the fluid atdimensionless time t

0 = t/T . Substituting the expression fort

0end

yields a quantitative, measurable proxy for dimensionlesspumping rate, ✏, in the form of a metabolic carbon fraction.Such a metabolic carbon fraction manifests in real-world sam-ples as a well-defined carbon isotope signature ([18, 1, 10, 25]).Accordingly, the metabolic carbon fraction of the final, pre-cipitated CaCO

3

will be

fM(t0end

) =x

2

2 (✏ � 1) + x

2

. [12]

Sensitivity to Ocean ChemistryWith the above functional forms for shell costs in hand, wecan begin to evaluate what kinds of selective pressures may actupon calcifying organisms on both short (ecological) and long(evolutionary) timescales3. Specifically, on long timescales,it may be advantageous to minimise the cost of a shell (E)which from equation 10 may be achieved by reducing the pro-tein fraction, f

p

within the shell. However, throughout thecourse of evolution, a species must adapt to changing environ-mental conditions[5]. The environmental factor entering theequations here is the bicarbonate fraction of seawater, x

2

. Atpresent x

2

⇡ 0.8 [29, 25], but ocean acidification will causeit to increase, with the IPCC report predicting an increaseof �x

2

⇠ 0.15 within a century (CITE). Such a change willlead to change in the cost E with fractional change in cost be-

ing dependent upon f

p

. Specifically, we define a measure ofsusceptibility,

S ⌘ 1E

@E@x

2

, [13]

which quantifies the fractional change in cost for a givenchange to seawater chemistry. We plot the susceptibility infigure ?? for a range of protein fractions and metabolic car-bon fractions4

Organisms with little metabolic carbon, who are pumpingfast, are both minimising their inorganic cost (C) and theirsusceptibility (S). Therefore, both from an energy-saving andevolutionary stand-point, it pays to pump fast.5 In lightof this, we also display (Figure ??) the fractional increasein cost resulting from an increase of x

2

from the modernvalue to unity for various protein fractions in the infinite-pumping case (f

M

! 0). The maximum increase in cost is⇠ 15% (= 100 �x

2

/x

2

), with that number dropping rapidlywith larger protein fractions. GIVE TWO EXAMPLES.

Hence, whereas pumping fast is beneficial on two fronts,protein fraction acts as a balance between the costly and thesusceptible. Thus, we may imagine that times of turbulentenvironmental conditions favour the evolution of organisms

3The distinction between such timescales is breaking down with recent realisations that evolutionarychange may proceed at much more rapid rate than once thought - even approaching the timescalesof ecological change (CITE CITE CITE:)4A proxy for pumping rate, �. High � is equivalent to low fM as faster pumping shortens to timefor precipitation.5Provided minimisation of energy is paramount, which might not always be the case, as discussedbelow.

4 www.pnas.org/cgi/doi/10.1073/pnas.0709640104 Footline Author

Max

imum

Rat

e

High

Low

Fig. 3. Upper Panel: The metabolic carbon fraction in the final CaCO3

as afunction of pumping parameter ✏. Higher ✏ translates to shorter calcification times,allowing less time for light carbon to di↵use in from the cell. As a result, faster pump-ing leads to lower metabolic carbon fractions. Lower Panel: The influence of x

2

onthe calcification rate for a given value of ✏. The upper panel suggests that, owing tothe very low metabolic carbon values of larval shells, they pump at or near physiolog-ical capacity. The lower panel demonstrates the consequences of ocean chemistry onthe maximum calcification rate attainable using this physiological limit. Acidificationreduces the capacity, meaning that a given physiology is not able to reach as a high acalcification rate. Owing to kinetic constraints within the larval stage, such a reducedcalcification capacity may have a di↵erential a↵ect on mortality.

4 www.pnas.org/cgi/doi/10.1073/pnas.0709640104 Footline Author

28organic cost, ν = 30 J mg-1

organic cost, ν = 3 J mg-1