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Allometric scaling law in a simple oxygen exchanging network: possible implications on the biological allometric scaling laws
ARTICLE IN PRESS
Journal of Theoretical Biology 223 (2003) 249–257
Allometric scaling law in a simple oxygen exchanging network: possible implications on the biological allometric scaling laws Moise! s Santilla! n* ! Depto. de F!ısica, Esc. Sup. de F!ısica y Matematicas, Instituto Polit!ecnico Nacional, Edif. 9, U. P. Zacatenco 07738 M!exico D. F., Mexico Received 29 October 2002; received in revised form 3 February 2003; accepted 26 February 2003
Abstract A simple model of an oxygen exchanging network is presented and studied. This network’s task is to transfer a given oxygen rate from a source to an oxygen consuming system. It consists of a pipeline, that interconnects the oxygen consuming system and the reservoir and of a fluid, the active oxygen transporting element, moving through the pipeline. The network optimal design (total pipeline surface) and dynamics (volumetric flow of the oxygen transporting fluid), which minimize the energy rate expended in moving the fluid, are calculated in terms of the oxygen exchange rate, the pipeline length, and the pipeline cross-section. After the oxygen exchanging network is optimized, the energy converting system is shown to satisfy a 3=4-like allometric scaling law, based upon the assumption that its performance regime is scale invariant as well as on some feasible geometric scaling assumptions. Finally, the possible implications of this result on the allometric scaling properties observed elsewhere in living beings are discussed. r 2003 Elsevier Science Ltd. All rights reserved. Keywords: Allometry; Performance optimization; Biological network
1. Introduction
species, the following relation is observed:
Power laws are ubiquitous in nature. They appear in phenomena as diverse as the magnitude distribution of earthquakes, the fractal geometry of tree branches and roots, the fractal geometry of cardiovascular and respiratory networks, critical thermodynamic states, the self-similarity of biological and non-biological time series, etc. (Wiesenfeld, 2001; Changizi, 2001; Brown et al., 2000). There are some theoretical studies concerning the wide range of biological phenomena where power laws arise (West et al., 1997; Banavar et al, 1999; West et al., 1999; Torres, 2001; Brown et al., 2002). A problem in such direction which has been recently addressed to by several authors (West et al., 1997; Banavar et al., 1999; West et al., 1999, 2000; Kurz and Sandau, 1998; Brown et al, 1998; Nagy et al, 1999; Bishop, 1999; Dodds et al., 2001) is the, experimentally observed, allometric relation between metabolic rate B and mass M: According to this, when we compare the metabolic rate and the mass of individuals of different
BpM x ;
*Tel./fax: +52-57-29-60000. E-mail address: [email protected] (M. Santill!an).
ð1Þ
where the symbol p means proportional to. The value of x was first estimated to be close to 2=3: Later observations of deviations from x ¼ 2=3 eventually lead to its replacement by x ¼ 0:72 0:73; which was then substituted by the simpler approximation x ¼ 3=4 (Dodds et al., 2001, and references therein). However, according to Dodds et al., a statistic analysis of the experimental evidence (on birds and mammals) does not support in all cases the 3=4 B–M allometric law. Conversely, they suggest that for birds and small mammals ðMo10 kgÞ a 2=3 allometric relation is observed, while for larger mammals they could not reject the 3=4 relation. On the other hand, plenty of reports supporting the 3=4 allometric law in species ranging from arthropoda to fish, amphibians, and reptilians have been published over the years (West and Brown, 2000). Thus, although there is no general agreement concerning the generality of the 3=4 allometric relation, it can still be considered as a paradigm for many animal species.
0022-5193/03/$ - see front matter r 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0022-5193(03)00097-3
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The first attempt to explain the B–M allometric law was due to Rubner (1883). He asserted that the rate of heat exchanged with the environment should be proportional to the individuals surface; and since this surface is proportional to M 2=3 ; then BpM 2=3 (The heat exchange rate is assumed to be proportional to B). More recently, West et al. (1997, 2000), studied a fractal-like circulatory network and derived the 3=4 B–M power law by minimizing the network resistance to blood flow. The same allometric law was obtained by Banavar et al. (1999) by looking for the local connection pattern of the circulatory network that minimizes the total blood volume. Finally, West et al. (1999) recovered the same allometric relation by maximizing the scaling of the circulatory network area. The most recent (West et al., 1997, 2000; Banavar et al., 1999; West et al., 1999) of the previously mentioned approaches, for explaining the appearance of biological allometric scaling laws, focus on the study of distribution networks like the circulatory system of reptilians, birds, mammals, etc. All of them emphasize the fractal properties of the system and rely upon the optimization of adequate static properties (the network resistance to flow, the total blood volume, and the scaling of the network circulatory area), ignoring the dynamics of the transporting fluid. The performance of any dynamic system is determined by both its static and dynamic properties. In the case of biological distribution networks, properties like the power required to move the transporting fluid and the efficiency of transport and distribution of nutrients, oxygen, etc. depend on the network geometric characteristics, but also on the fluid velocity through the network pipes. This leads us to wonder whether the allometric scaling properties of biological systems depend only on the distribution network geometry, or they also depend on the fluid dynamics. In order to show that the fluid dynamics can indeed be important to explain the allometric properties of distribution networks we introduce a simple oxygen exchanging network and find that, by minimizing the power expended in moving the transporting fluid (which demands the optimization of both the system design and the transporting fluid volumetric flow) and using some feasible geometric scaling properties, it obeys an allometric scaling law similar to the B–M biological one. Finally, the feasibility of this network as a model of real biological distribution networks and the possible implications of our results on the understanding of the, experimentally observed, biological allometric relations are discussed.
vessel’’ splits into various generations of branches. Examples include the exchanges of respiratory gases between gills or lungs and tissues, exchanges of nutrients between gusts and tissues, and exchanges of metabolic waste products between tissues and kidneys or other excretory organs (Jr. and Liem, 1994; Snordgrass, 1965). All of these systems involve flows of materials and their overall design is quite similar from a geometrical point of view. In the following paragraphs we introduce a simple parallel oxygen-exchanging network, which is intended to account for the basic properties of real biological distribution systems. It is a deliberately oversimplified model. Nevertheless, real biological distribution or collection networks often seem to have properties of both fractal and parallel networks. For example, the gills of fish and many other aquatic organisms and the book lungs of land crabs and spiders are fundamentally parallel structures that are supplied by blood vessels that show a few generations of branching (Jr. and Liem, 1994; Snordgrass, 1965). Moreover, many biological networks may resemble hybrids between the one modelled here and fractal networks like those of West et al. (1997, 2000). Consider the oxygen-exchanging system depicted in Fig. 1. It consists of an oxygen consuming system of mass M; an oxygen reservoir, a pipeline that crosses and interconnects them, and an oxygen transporting fluid that passes through the pipeline with a volumetric flow Q: The oxygen consuming system requires a continuous supply to maintain, in the steady state, a constant oxygen concentration mb : The oxygen reservoir has a constant concentration me : The pipeline is made up of two identical sections, one of them passing through the oxygen consuming system and the other passing through the oxygen reservoir. Every section consists of n identical parallel pipes of length l and radius r; which
µb
Q
µc
µh
µb < µe
Q
µe
2. A simple oxygen exchanging system Many biological processes rely upon exchanges carried out by distribution networks in which a ‘‘main
Fig. 1. Schematic representation exchanging system.
of
a
simple parallel oxygen
ARTICLE IN PRESS ! / Journal of Theoretical Biology 223 (2003) 249–257 M. Santillan
connect into a single pipe before entering and after leaving the oxygen consuming system and the reservoir. Both pipeline sections connect each other so that the fluid that gains oxygen, after passing through the reservoir, then passes through the oxygen consuming system, loosing it, and so forth. The oxygen exchange between the transporting fluid and both the oxygen consuming system and the source takes place through the parallel pipe walls. Let a be a constant, proportional to the pipe-wall oxygen permeability, such that the oxygen exchange rate ðdJÞ through the wall of a pipe differential element of surface dS is given by dJ ¼ aDm dS; where Dm is the oxygen concentration difference across the pipe wall. In this model, the number ðnÞ of oxygen-exchanging parallel pipes can change, but their length ðlÞ and their total cross-section ðc ¼ npr2 Þ are assumed to be functions of the system mass ðMÞ and thus, they are constant for a given system. In the steady state, this oxygen exchanging system transports oxygen at a constant rate J from the reservoir to the consuming system. This oxygen exchange rate is assumed to be a function of the system mass M too. So, it is also constant for a given system.
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the pipeline is too high, the oxygen exchange rate results to be proportional to the pipeline oxygen permeability and is independent of the fluid velocity. In other words, in the regime of very high velocities, the oxygen exchange rate depends only on the pipeline wall properties. On the other hand, given that tanhðxÞC1 when xb1; the oxygen exchange rate is proportional to b when sbb: This means that when the fluid velocity is too small, the oxygen exchange rate is proportional to the transporting fluid volumetric flow. That is, in the regime of very low fluid velocities, the oxygen exchange rate is determined by the flow properties. After solving for s in Eq. (5) we get 1 bþo : ð6Þ s ¼ b ln 2 bo
The dynamics of the system sketched in Fig. 1 are studied with detail in Appendix A. In there, the steadystate oxygen exchange rate between the oxygen consuming system and the reservoir is shown to be s J ¼ ðme mb Þb tanh ; ð2Þ b
This last equation permits us to plot the o level curves in the s vs. b space. Such level curves are shown in Fig. 2 for different values of o: Each one of these curves determines an infinite set of ðb; sÞ pairs that lead to the same oxygen exchange rate. Since we are assuming that the oxygen exchange rate ðJ ¼ oðme mb ÞÞ is a function of the system of mass M; it attains a constant value for a given oxygen consuming system. This constant value determines on it own a level curve in the s vs. b space, like those of Fig. 2. Every point in such curve represents one of the infinite number of possible combinations of s (total surface of the oxygen exchanging system) and b (oxygen transporting fluid volumetric flow) to exchange the oxygen rate determined by o: A natural choice would be to pick up, of all those ðb; sÞ pairs, the one that requires the minimum energy expenditure to move the fluid through the pipeline. The resistance to laminar flow of the present work’s oxygen-exchanging network (Fig. 1) is analysed in Appendix B and is given by
where s and b are respectively defined as
R¼
3. Optimization of the oxygen exchanging system
s ¼ pr ln a
ð3Þ
ð7Þ
where n is the fluid viscosity,
and b ¼ pr2 nv:
4p2 n 2 S ; c3 l
ð4Þ
From their definition, s is nothing else but one half of the pipeline surface times a; while b is the pipeline volumetric flow, Q: For the sake of decreasing the number of variables we have to dealt with, let us introduce a new variable o ¼ J=ðTb Te Þ; which is proportional to the oxygen exchange rate. From this definition and Eq. (2) it follows that s o ¼ b tanh : ð5Þ b Since tanhðxÞCx when x51; we have that oCs when s5b: This means that when the fluid velocity through
c ¼ npr2
ð8Þ
is the total cross-section of the n parallel pipes passing through the oxygen consuming system and the reservoir, and S ¼ n2prl
ð9Þ
is their total surface. The resistance of the network interconnecting pipes is not accounted for in Eq. (7). The reason for this omission is that our main goal is to find the optimal number and geometry of the oxygen-exchanging parallel pipes that minimize the power expended in moving the fluid. Thus, the inclusion of the interconnecting-pipe resistance would not alter the final results
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252 5
ω = 0.5 ω = 1.0 ω = 1.5 ω = 2.0
4
Direction of increasing ω
σ
3
2
1
0
0
1
2
3
4
5
β Fig. 2. Plots of the o level curves in the s vs. b space for different values of o (look at the main text for the definition of variables o; s; and b). These plots asymptotically approach to the b ¼ o and s ¼ o lines. The direction of increasing o is shown close to the level-curve bending points.
5
lines of constant P line of constant ω
4
σ
3
Direction of increasing P 2
σ* 1
0
0
1
β*
2
β
3
4
5
Fig. 3. In this graphic the P level curves are plotted with solid lines in the s vs. b space for different values of P: The o ¼ 1 level curve is also plotted (dashed line). See the main text for the definition of variables P; s; b; and o: The direction of increasing P is shown close to the P level-curve symmetry axis. The point ðbn ; sn Þ where the o level curve is tangentially touched by one of the P level curves is also indicated.
appreciably since it shall only play the role of an additive constant. The power P needed to move a flow o through a pipeline with resistance R is
By solving for s in Eq. (11) we obtain sffiffiffiffi P1 : s¼ wb
P ¼ Ro2 :
With the aid of Eq. (13) we can plot the level curves for P in the s vs. b space. These plots (which are hyperbolas whose symmetry axis is the s ¼ b line) are shown in Fig. 3 for different values of P; along with the o ¼ 1 level curve. As seen in there, every one of the P level curves either crosses the o level curve at two different points, is tangent to it, or does not touch it at all. The P level curves that do not touch the level curve for o correspond to values of power ðPÞ expended in moving the oxygen transporting fluid with which it is impossible to transfer the oxygen rate determined by o:
ð10Þ
Then, from Eqs. (7) and (10), and the definition of s (Eqs. (3)), it follows that P ¼ ws2 b2
ð11Þ
with w defined as w¼
16p2 n 1 : a2 lc3
ð12Þ
ð13Þ
ARTICLE IN PRESS ! / Journal of Theoretical Biology 223 (2003) 249–257 M. Santillan
The P value of the P-level curve that is tangent to the o level curve, is the minimum power we can expend in moving the oxygen transporting fluid in order to transfer such oxygen rate. From Eq. (11), this minimum power expenditure is given by Pmin ¼ ws * 2 b * 2 ;
ð14Þ
where bn and sn are the coordinates of the tangent point of the P and o level curves. For larger power expenditures there exist two possible combinations of s and b values (those corresponding to the points where the P and o level curves cross each other) to transfer the demanded oxygen rate. One of them corresponds to a high b and a low s values, while the other corresponds to a low b and a high s values. The optimal bn and sn values are calculated in terms of o and w in Appendix C rendering: bn C1:256o
ð15Þ
and sn C1:367o:
ð16Þ
Notice that neither bn nor sn depend on w; they only depend on o: This, in turn, means that the optimal bn and sn are independent of the network geometric characteristics ðl; cÞ; as well as of its material properties ða; nÞ; see Eq. (12). From the results of the present section we can conclude that, when we have an oxygen consuming system of mass M; which maintains a constant oxygen concentration mb by means of a continuous oxygen supply at rate J ¼ oðme mb Þ from a reservoir me ; and the oxygen is exchanged by a network as depicted in Fig. 1, we have to optimize both the pipeline surface (variable s) and the fluid velocity (variable b) in order to minimize the power expended in moving the fluid through the pipeline. The optimal sn and bn values are given by Eqs. (15) and (16). The minimum power expenditure is on its own given by Eq. (14). Recall that through the minimization process, the pipeline length ðlÞ and total cross-section ðcÞ; as well as the oxygen exchange rate ðJÞ; were kept constant, since they are assumed to be functions of the system of mass ðMÞ: The precise form of these functions will be discussed in the following section.
253
the power expended in moving the fluid is minimized, given the rate of oxygen exchange ðJðMÞ ¼ oðMÞðme mb ÞÞ; the pipeline total cross-section ðcðMÞÞ; and the pipeline length lðMÞ; then from Eqs. (12), (14), (15), and (16), the power expenditure is related to o; l; and c by o4 P p 3: ð17Þ lc The proportionality constant implicit in Eq. (17) involves only fluid and pipeline wall properties: The fluid viscosity ðnÞ and pipeline wall oxygen permeability ðaÞ: The performance regime of biological and nonbiological energy converting systems is determined by the ratio of quantities such as the energy input rate, the power output, the rate of heat exchanged with the environment, etc. (Angulo-Brown et al., 1995; Santilla! n et al, 1997; Angulo-Brown et al., 2002; Hoffmann et al., 1997). Some authors suggest that there might be some optimal operation regimes which are attained by biological systems (muscle contraction, ATP synthesis by anaerobic and aerobic glycolysis, etc.) of living organisms of different sizes (Angulo-Brown et al., 1995; Santilla! n et al., 1997; Santilla! n and Angulo-Brown, 1997; Santilla! n, 1999). They also propose an explanation for this fact from evolutive considerations. Based on the above considerations, we assume a scale-invariant performance regime for our system, which in terms of variables P and o implies that, as the system mass changes, ð18Þ
PpopB;
with B the oxygen consuming system ‘‘metabolic’’ rate and M the system mass. Then, from Eqs. (17) and (18) Bpðlc3 Þ1=3 :
ð19Þ
If we assume that the system grows up proportionally in all directions and that the ratio of the system to the pipeline cross-sections is scale invariant, as well as the system mass density, that is, lpM 1=3 ;
cpM 2=3 ;
ð20Þ
then, the following B–M allometric relation is obtained from Eqs. (19) and (20): B pM 7=9 pM 0:78 :
4. Scaling properties derived from a scale-invariant performance regime If we have an oxygen system optimized in the way described in Section 3, i.e.: The pipeline surface (variable s) and the volumetric flow (variable o) are set up so that
ð21Þ
Finally, from Eqs. (3), (4), (8), (15), (16), (18), (20), and (21), it is possible to derive, after a little algebra, the following scaling relations for variables r; n; and v: rpM 2=9 ;
npM 2=9 ;
vpM 1=9 :
ð22Þ
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5. Concluding remarks We have studied a simple oxygen exchanging network, whose task is to transfer oxygen from a source (with a constant concentration me ) to an oxygen consuming system (with steady-state concentration mb ). Under the supposition that the oxygen exchange rate J ¼ oðme mb Þ; the cross-section available for the pipeline passing through the system c; and the pipeline length l are functions of the system mass (and thus fixed for a given system), it was shown that both the pipeline total surface and the transporting fluid volumetric flow have to be optimized, according to Eqs. (15) and (16), in order to minimize the power expended in moving the fluid through the pipeline. Then, we found that, if the oxygen consuming system performance regime is scale invariant then, the ‘‘metabolic’’ rate of the optimized system has to scale up according to the allometric relation given by Eq. (21), which is quite similar to the experimentally observed biological 3=4 allometric scaling law. The model here presented is deliberately oversimplified in order to be able to optimize its performance regime analytically. Nevertheless, despite its simplicity it is quite general in the sense that the results we obtain are independent of the nature of the transporting fluid (it can be either a gas or a liquid), the exchanged substance (oxygen, nutrients, waste products, etc.), and the network materials. To our consideration, this model captures the basic characteristics of many biological networks. As mentioned in Section 2, there are many biological distribution networks, like the gills of fish and many other aquatic organisms and the book lungs of land crabs and spiders, that are essentially parallel. Moreover, real biological distribution or collection networks often seem to have properties of both fractal and parallel networks and may be considered as hybrids between the one we model and fractal networks like those of West et al. (1997, 2000). From these considerations, we expect that our results are feasible and comparable with biological data. Recently, Dodds et al. (2001) performed a statistical analysis of the reported data on birds and mammals and concluded that there is no enough evidence supporting the 3=4 allometric relation as a general law. On the contrary, they found that a 2=3 allometric relation is more plausible for birds and small mammals (mass less than 10 kg), but could not reject the 3=4 allometric relation for larger mammals. On the other hand, many reports supporting the 3=4 law in organisms as diverse as arthropoda, amphibians, and reptilians have been published over the years (West and Brown, 2000). Thus, although there is no general agreement concerning the generality of the 3=4 allometric scaling law, it can still be considered as a paradigm for many animal species.
Moreover, one might speculate that, since birds and small mammals are warm blooded animals with a large surface (in proportion with their mass), the original derivation of the 2=3 relation by Rubner (1883) is applicable, while in large mammals and cold blooded animals the distribution-network effects are dominant and thus, they obey an allometric relation with exponent close to 3=4: A by-product of the present work is the allometric scaling law derived for the radius of the oxygen exchanging parallel pipes (see Eq. (22)): rpM 2=9 pM 0:22 : Interestingly, the exchanging pipes of real biological distribution networks also scale obeying allometric laws. For instance, the experimentally observed scaling law for the radius of the mammalian alveolus is rpM 0:13 (Tenney and Remmers, 1963). The comparison with our result is not good, but we believe that this may be due to the different geometries of both systems. In this regard, notice that the exchanging ‘‘pipes’’ of different networks (all of which obey the 3/4 metabolic scaling law) scale with different exponents. For instance, West et al. (1997) consider that the radius of blood vessels is scale invariant in their model of the circulatory system. In conclusion, we present a simple model of an oxygen exchange network which, when its geometry and fluid dynamics are optimized to minimize the power expended in moving the oxygen-transporting fluid, makes the oxygen consuming system obey (under feasible geometric scaling assumptions) a scaling law similar to the experimentally observed biological 3=4 allometric relation. Despite its simplicity, the model bears some resemblance, as discussed above, with many real biological distribution or collection networks. Moreover, it is quite general in the sense that many biological processes rely upon distribution networks like the one modelled here and that the results we obtain depend only on the network geometry and dynamics, and not on the properties of the substances it transports or is made up of. From all these considerations, we believe that this model may contribute to pose the issue of whether the optimization of both static and dynamic properties of biological distribution networks is required to understand the underlying principles that govern the structure and function of real, more complicated systems.
Acknowledgements Thanks are due to F. Angulo-Brown and M.C. Mackey for fruitful discussion. This work was supported by COFAA-IPN; CGPI-IPN, Grant 200606; and by CONACyT, Grant 411300-I333174-E.
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Appendix A. Calculation of the model oxygen exchange rate A.1. One single pipe Consider a single pipe of radius r and length l; which crosses an oxygen reservoir with concentration ms : An oxygen transporting fluid passes through the pipe with a velocity v: The fluid oxygen concentration when it enters the thermal bath (position x ¼ 0) is m0 : Take the differential fluid element located between coordinates x and x þ dx0 : Let its oxygen concentration be mðxÞ: Let a be a constant, proportional to the pipe wall oxygen permeability, such that the transporting-fluid oxygen-uptake rate is given by dJ ¼ ðms mðxÞÞ2par dx0 :
ðA:1Þ
On the other hand, the oxygen concentration increase dm caused by this oxygen uptake can be calculated from dJ dt ¼ pr2 dx0 dm:
ðA:2Þ
By combining Eqs. (A.1) and (A.2) we obtain dm 2pra ¼ ðm mðxÞÞ: ðA:3Þ dt pr2 s If we finally make use of the chain rule dm=dt ¼ v dm=dx (note that v ¼ dx=dt), we have dm 2sð1Þ ¼ ðm mðxÞÞ; dx bð1Þ l s
ðA:4Þ
where sð1Þ ¼ prla is one-half of the pipe surface times a; and bð1Þ ¼ pr2 v is the volumetric flow through the pipe. After integration, the following oxygen concentration profile is obtained, 2sð1Þ mðxÞ ¼ ms ðms m0 Þ exp ð1Þ x ; ðA:5Þ b l where we have employed the boundary condition mð0Þ ¼ m0 : In particular, the oxygen concentration at the pipe end (position x ¼ l) is 2sð1Þ mf ¼ mðlÞ ¼ ms ðms m0 Þ exp ð1Þ : ðA:6Þ b The rate of oxygen exchanged by the whole pipe can be calculated as Z x¼l dJ J ð1Þ ¼ x¼0
¼ 2pra
Z
l
ðms mðxÞÞ dx
0
¼ ðms m0 Þbð1Þ ð1 expð2sð1Þ =bð1Þ ÞÞ:
ðA:7Þ
A.2. n parallel pipes Consider a pipeline of n identical parallel pipes crossing an oxygen reservoir with concentration ms : All
255
of the pipes as well as the flow through each of them have the same dimensions and properties as those specified in Section A.1. Since the pipes are in parallel, the oxygen concentration difference between their ends is the same for all of them. From this assertion and Eq. (A.6) it follows that 2sð1Þ ðA:8Þ mf ¼ ms ðms m0 Þ exp ð1Þ : b On the other hand, the oxygen exchange rate through the whole pipeline, J; is n times the oxygen exchange rate through a single pipe, J ð1Þ : Therefore, it follows from Eq. (A.7) that J ¼ nJ ð1Þ
2sð1Þ ¼ ðms m0 Þnbð1Þ 1 exp ð1Þ : b
ðA:9Þ
Let us define variables s ¼ nsð1Þ and b ¼ nbð1Þ : With these definitions, Eqs. (A.8) and (A.9) become 2s mf ¼ ms ðms m0 Þ exp ðA:10Þ b and
2s J ¼ ðms m0 Þb 1 exp : b
ðA:11Þ
From their definition, s and b can be interpreted as onehalf of the whole pipeline surface times a and as the volumetric flow throw the whole pipeline, respectively. A.3. The oxygen exchanging system Consider the oxygen exchanging system sketched in Fig. 1. It consists of an oxygen consuming system (with steady-state concentration mb ), an oxygen reservoir (with constant concentration me > mb ), a pipeline that crosses and interconnects both of them, and a fluid that circulates through the pipeline and is the active oxygen-exchanging element. The pipeline can be divided into two identical sections. Each section is made up of n parallel pipes, each one having the characteristics of the pipe studied in Section A.1. All of these pipes connect into a single pipe before entering and after leaving the oxygen consuming system and the reservoir. The input of one of these pipeline sections is connected to the output of the other and vice versa. In the steady state, the fluid coming out from the oxygen consuming system and entering the oxygen reservoir will have a constant oxygen concentration mh ; while the fluid coming out from the oxygen reservoir me and entering the oxygen consuming system will have a constant concentration mc ðme > mc > mh > mb Þ:
ARTICLE IN PRESS ! / Journal of Theoretical Biology 223 (2003) 249–257 M. Santillan
256
By applying the results of Section A.2 (Eq. (A.10)) to each of the pipeline sections we have 2s mc ¼ me ðme mh Þ exp ðA:12Þ b and
2s mh ¼ mb ðmb mc Þ exp : b
ðA:13Þ
These equations constitute a complete set of equations for mc and mh which, when solved, give 2s 1 expð2s=bÞ ðA:14Þ mc ¼ me þ mb exp b 1 expð4s=bÞ and
2s 1 expð2s=bÞ mh ¼ mb þ me exp : b 1 expð4s=bÞ
ðA:15Þ
Finally, from these last results and Eq. (A.11), the oxygen exchange rate between the oxygen reservoir and the oxygen consuming system is ½1 expð2s=bÞ2 1 expð4s=bÞ s ¼ ðme mb Þb tanh : b
J ¼ ðme mb Þb
ðA:16Þ
Appendix B. Calculation of the model resistance to flow The resistance to flow of a cylindrical pipe of radius r and length l is given by (Landau and Lifshitz, 1987) 8pnl Rð1Þ ¼ 4 ; ðB:1Þ r with n the fluid viscosity. The resistance to flow of an array of n parallel identical pipes is Rð1Þ 8pnl ¼ 4: nr n
ðB:2Þ
By substitution of Eq. (B.3) into Eq. (B.2), the pipeline resistance in terms of c and n turns out to be RðnÞ ¼
8p3 nl n: c2
ðB:4Þ
On the other hand, the pipeline surface is S ðnÞ ¼ 2prln:
ðB:5Þ
After substitution of Eq. (B.3) into Eq. (B.5) we obtain pffiffiffiffiffiffiffiffiffi S ðnÞ ¼ 2l cpn: ðB:6Þ Solving for n in Eq. (B.6) and substituting into Eq. (B.4) gives the following expression for the pipeline resistance in terms of its cross-section c and its surface SðnÞ : RðnÞ ¼
2p2 n ðnÞ S 2: c3 l
ðB:7Þ
Finally, if we dismiss the resistance of the pipes interconnecting the two pipeline sections of the oxygen exchanging system depicted in Fig. 1, the total system resistance is R ¼ 2RðnÞ ¼
4p2 n ðnÞ2 S : c3 l
ðB:8Þ
Appendix C. Calculation of the optimal sn and bn values For reasons explained in the main text, the optimal sn and bn values we are looking for are those for which
4.4 4.3 4.2 4.1 4
f(β)
RðnÞ ¼
Let c ¼ npr2 be the total pipeline cross-section. From this rffiffiffiffiffiffi c : ðB:3Þ r¼ np
3.9 3.8 3.7 3.6 3.5 3.4
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
β Fig. 4. Plot of the function f ðbÞ ¼ b2 ln½ðb þ oÞ=ðb oÞ with o ¼ 1:
2
ARTICLE IN PRESS ! / Journal of Theoretical Biology 223 (2003) 249–257 M. Santillan
the curves 1 bþo s ¼ b ln 2 bo
ðC:1Þ
and
sffiffiffiffi P1 s¼ wb
ðC:2Þ
are tangent. By combining Eqs. (C.1) and (C.2), this task results to be equivalent to that of looking for the value of P for which the following equation has one single solution ðbn Þ and finding it: sffiffiffiffi P bþo 2 ¼ b ln 2 ðC:3Þ w bo Let us define function f ðbÞ as bþo f ðbÞ ¼ b2 ln : bo
ðC:4Þ
A plot of f ðbÞ (shown if Fig. 4 for o ¼ 1) demonstrates that it is a concave curve p with ffiffiffiffiffiffiffiffiffi a single minimum. Therefore, equation f ðbÞ ¼ 2pffiffiffiffiffiffiffiffi P=wffi (Eq. (C.3)) has one single solution bn only if 2 P=w corresponds to the minimum value of f ðbÞ; being bn the position of such minimum. In conclusion, the optimum bn value we are looking for is that which minimizes function f ðbÞ; given by Eq. (C.4). The minimum of function f ðbÞ can be calculated from df =db ¼ 0: After a little algebra, this leads to the following equation for bn : n b þo bn o ln n : ðC:5Þ ¼ 2 b o b * o2 To solve it we propose bn ¼ eo: After substitution into Eq. (C.5), it becomes eþ1 e ln : ðC:6Þ ¼ 2 e1 e þ1 Eq. (C.6) was solved numerically with the aid of algorithm fzero of Matlab obtaining eC1:2556: From this it follows that bn C1:256 o
ðC:7Þ
and after substituting into Eq. (C.1): sn C1:367 o:
ðC:8Þ
References Angulo-Brown, F., Santill!an, M., Calleja-Quevedo, E., 1995. Thermodynamical optimization in some biochemical reactions. Il Nuovo Cimento 17D, 8790. Angulo-Brown, F., Arias-Hern!andez, L.A., Santill!an, M., 2002. On some connections between first order irreversible thermodynamics and finite-time thermodynamics. Rev. Mex. Fis. 48 suppl. 1, 182–192.
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Banavar, J.R., Martian, A., Rinaldo, A., 1999. Size and form in efficient transportation networks. Nature 399, 130–132. Bishop, C.M., 1999. The maximum oxygen consumption and aerobic scope of birds and mammals: betting to the heart of the matter. Proc. R. Soc. London B 266, 2275–2281. Brown, J.H., Enquist, B.J., West, G.B., 1998. Allometric scaling in biology. Science 281, 751a. Brown, J.H., West, G.B., Enquist, B.J., 2000. Scaling in biology: patterns and processes, causes and consequences. In: West, G.B., Brown, J.H. (Eds.), Scaling in Biology. Oxford University Press, Oxford, pp. 1–24. Brown, J.H., Gupta, V.K., Li, B., Milne, B.T., Restrepo, C., West, G.B., 2002. The fractal nature of nature: power laws, ecological complexity, and biodiversity. Philos. Trans. R. Soc. London B Biol. Sci. 357, 619–626. Changizi, M.A., 2001. Universal scaling laws for hierarchical complexity in languages, organisms, behaviors and other combinatorial systems. J. theor. Biol. 211, 277–295. Dodds, P.S., Rothman, D.H., Weitz, J.S., 2001. Re-examination of the ‘‘3/4-law’’ of metabolism. J. theor. Biol. 209, 9–27. Hoffmann, K.H., Burzler, J.M., Schubert, S., 1997. Endorreversible thermodynamics: a review. J. Non-equilibrium Thermodyn. 22, 311–355. Jr., W.F.W., Liem, K.F., 1994. Functional Anatomy of the Vertebrates: An Evolutionary Perspective. Saunders College Publishing, Fort Worth. Kurz, H., Sandau, K., 1998. Allometric scaling in biology. Science 281, 751a. Landau, L.D., Lifshitz, E.M., 1987. Fluid Dynamics. Pergamon Press, Oxford. Nagy, K.A., Girard, I.A., Brown, T.K., 1999. Energetics of freeranging mammals, reptiles, and birds. Annu. Rev. Nutr. 19, 247–277. . Rubner, M., 1883. Ueber den einfluss der korpergr. osse auf stoffund kraftwechsel. Z. Biol. 19, 535–562. Santill!an, M., 1999. A thermodynamic optimization analysis of a possible relation between the parameters that determine the energetics of muscle contraction in steady state. J. theor. Biol. 199, 105–112. Santill!an, M., Angulo-Brown, F., 1997. A thermodynamic approach to the compromise between power and efficiency in muscle contraction. J. theor. Biol. 189, 391–398. Santill!an, M., Angulo-Brown, F., Arias-Hern!andez, L.A., 1997. Some optimization criteria for biological systems in linear irreversible thermodynamics. Il Nuovo Cimmento 19D, 99–109. Snordgrass, R.E., 1965. A Textbook of Arthropod Anatomy. Hafner, New York. Tenney, S.M., Remmers, D.H., 1963. Comparative quantitative morphology of the mammalian lung: diffusing area. Nature 197, 54–56. Torres, J.L., 2001. Biological power laws and Darwin’s principle. J. theor. Biol. 209, 223–232. West, G.B., Brown, J.H. (Eds.), 2000. Scaling in Biology. Oxford University Press, Oxford. West, G.B., Brown, J.H., Enquist, B.J., 2000. The origin of scaling laws in biology. In: West, G.B., Brown, J.H. (Eds.), Scaling in Biology. Oxford University Press, Oxford, pp. 87–112. West, J.B., Brown, J.H., Enquist, B.J., 1997. A general model for the origin of allometric scaling laws in biology. Science 276, 122–126. West, J.B., Brown, J.H., Enquist, G.J., 1999. The fourth dimension of life: fractal geometry and allometric scaling of organisms. Science 284, 1677–1679. Wiesenfeld, K., 2001. Resource letter: Scl-1: scaling laws. Am. J. Phys. 69, 938–942.
Journal of Theoretical Biology 223 (2003) 249–257
Allometric scaling law in a simple oxygen exchanging network: possible implications on the biological allometric scaling laws Moise! s Santilla! n* ! Depto. de F!ısica, Esc. Sup. de F!ısica y Matematicas, Instituto Polit!ecnico Nacional, Edif. 9, U. P. Zacatenco 07738 M!exico D. F., Mexico Received 29 October 2002; received in revised form 3 February 2003; accepted 26 February 2003
Abstract A simple model of an oxygen exchanging network is presented and studied. This network’s task is to transfer a given oxygen rate from a source to an oxygen consuming system. It consists of a pipeline, that interconnects the oxygen consuming system and the reservoir and of a fluid, the active oxygen transporting element, moving through the pipeline. The network optimal design (total pipeline surface) and dynamics (volumetric flow of the oxygen transporting fluid), which minimize the energy rate expended in moving the fluid, are calculated in terms of the oxygen exchange rate, the pipeline length, and the pipeline cross-section. After the oxygen exchanging network is optimized, the energy converting system is shown to satisfy a 3=4-like allometric scaling law, based upon the assumption that its performance regime is scale invariant as well as on some feasible geometric scaling assumptions. Finally, the possible implications of this result on the allometric scaling properties observed elsewhere in living beings are discussed. r 2003 Elsevier Science Ltd. All rights reserved. Keywords: Allometry; Performance optimization; Biological network
1. Introduction
species, the following relation is observed:
Power laws are ubiquitous in nature. They appear in phenomena as diverse as the magnitude distribution of earthquakes, the fractal geometry of tree branches and roots, the fractal geometry of cardiovascular and respiratory networks, critical thermodynamic states, the self-similarity of biological and non-biological time series, etc. (Wiesenfeld, 2001; Changizi, 2001; Brown et al., 2000). There are some theoretical studies concerning the wide range of biological phenomena where power laws arise (West et al., 1997; Banavar et al, 1999; West et al., 1999; Torres, 2001; Brown et al., 2002). A problem in such direction which has been recently addressed to by several authors (West et al., 1997; Banavar et al., 1999; West et al., 1999, 2000; Kurz and Sandau, 1998; Brown et al, 1998; Nagy et al, 1999; Bishop, 1999; Dodds et al., 2001) is the, experimentally observed, allometric relation between metabolic rate B and mass M: According to this, when we compare the metabolic rate and the mass of individuals of different
BpM x ;
*Tel./fax: +52-57-29-60000. E-mail address: [email protected] (M. Santill!an).
ð1Þ
where the symbol p means proportional to. The value of x was first estimated to be close to 2=3: Later observations of deviations from x ¼ 2=3 eventually lead to its replacement by x ¼ 0:72 0:73; which was then substituted by the simpler approximation x ¼ 3=4 (Dodds et al., 2001, and references therein). However, according to Dodds et al., a statistic analysis of the experimental evidence (on birds and mammals) does not support in all cases the 3=4 B–M allometric law. Conversely, they suggest that for birds and small mammals ðMo10 kgÞ a 2=3 allometric relation is observed, while for larger mammals they could not reject the 3=4 relation. On the other hand, plenty of reports supporting the 3=4 allometric law in species ranging from arthropoda to fish, amphibians, and reptilians have been published over the years (West and Brown, 2000). Thus, although there is no general agreement concerning the generality of the 3=4 allometric relation, it can still be considered as a paradigm for many animal species.
0022-5193/03/$ - see front matter r 2003 Elsevier Science Ltd. All rights reserved. doi:10.1016/S0022-5193(03)00097-3
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The first attempt to explain the B–M allometric law was due to Rubner (1883). He asserted that the rate of heat exchanged with the environment should be proportional to the individuals surface; and since this surface is proportional to M 2=3 ; then BpM 2=3 (The heat exchange rate is assumed to be proportional to B). More recently, West et al. (1997, 2000), studied a fractal-like circulatory network and derived the 3=4 B–M power law by minimizing the network resistance to blood flow. The same allometric law was obtained by Banavar et al. (1999) by looking for the local connection pattern of the circulatory network that minimizes the total blood volume. Finally, West et al. (1999) recovered the same allometric relation by maximizing the scaling of the circulatory network area. The most recent (West et al., 1997, 2000; Banavar et al., 1999; West et al., 1999) of the previously mentioned approaches, for explaining the appearance of biological allometric scaling laws, focus on the study of distribution networks like the circulatory system of reptilians, birds, mammals, etc. All of them emphasize the fractal properties of the system and rely upon the optimization of adequate static properties (the network resistance to flow, the total blood volume, and the scaling of the network circulatory area), ignoring the dynamics of the transporting fluid. The performance of any dynamic system is determined by both its static and dynamic properties. In the case of biological distribution networks, properties like the power required to move the transporting fluid and the efficiency of transport and distribution of nutrients, oxygen, etc. depend on the network geometric characteristics, but also on the fluid velocity through the network pipes. This leads us to wonder whether the allometric scaling properties of biological systems depend only on the distribution network geometry, or they also depend on the fluid dynamics. In order to show that the fluid dynamics can indeed be important to explain the allometric properties of distribution networks we introduce a simple oxygen exchanging network and find that, by minimizing the power expended in moving the transporting fluid (which demands the optimization of both the system design and the transporting fluid volumetric flow) and using some feasible geometric scaling properties, it obeys an allometric scaling law similar to the B–M biological one. Finally, the feasibility of this network as a model of real biological distribution networks and the possible implications of our results on the understanding of the, experimentally observed, biological allometric relations are discussed.
vessel’’ splits into various generations of branches. Examples include the exchanges of respiratory gases between gills or lungs and tissues, exchanges of nutrients between gusts and tissues, and exchanges of metabolic waste products between tissues and kidneys or other excretory organs (Jr. and Liem, 1994; Snordgrass, 1965). All of these systems involve flows of materials and their overall design is quite similar from a geometrical point of view. In the following paragraphs we introduce a simple parallel oxygen-exchanging network, which is intended to account for the basic properties of real biological distribution systems. It is a deliberately oversimplified model. Nevertheless, real biological distribution or collection networks often seem to have properties of both fractal and parallel networks. For example, the gills of fish and many other aquatic organisms and the book lungs of land crabs and spiders are fundamentally parallel structures that are supplied by blood vessels that show a few generations of branching (Jr. and Liem, 1994; Snordgrass, 1965). Moreover, many biological networks may resemble hybrids between the one modelled here and fractal networks like those of West et al. (1997, 2000). Consider the oxygen-exchanging system depicted in Fig. 1. It consists of an oxygen consuming system of mass M; an oxygen reservoir, a pipeline that crosses and interconnects them, and an oxygen transporting fluid that passes through the pipeline with a volumetric flow Q: The oxygen consuming system requires a continuous supply to maintain, in the steady state, a constant oxygen concentration mb : The oxygen reservoir has a constant concentration me : The pipeline is made up of two identical sections, one of them passing through the oxygen consuming system and the other passing through the oxygen reservoir. Every section consists of n identical parallel pipes of length l and radius r; which
µb
Q
µc
µh
µb < µe
Q
µe
2. A simple oxygen exchanging system Many biological processes rely upon exchanges carried out by distribution networks in which a ‘‘main
Fig. 1. Schematic representation exchanging system.
of
a
simple parallel oxygen
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connect into a single pipe before entering and after leaving the oxygen consuming system and the reservoir. Both pipeline sections connect each other so that the fluid that gains oxygen, after passing through the reservoir, then passes through the oxygen consuming system, loosing it, and so forth. The oxygen exchange between the transporting fluid and both the oxygen consuming system and the source takes place through the parallel pipe walls. Let a be a constant, proportional to the pipe-wall oxygen permeability, such that the oxygen exchange rate ðdJÞ through the wall of a pipe differential element of surface dS is given by dJ ¼ aDm dS; where Dm is the oxygen concentration difference across the pipe wall. In this model, the number ðnÞ of oxygen-exchanging parallel pipes can change, but their length ðlÞ and their total cross-section ðc ¼ npr2 Þ are assumed to be functions of the system mass ðMÞ and thus, they are constant for a given system. In the steady state, this oxygen exchanging system transports oxygen at a constant rate J from the reservoir to the consuming system. This oxygen exchange rate is assumed to be a function of the system mass M too. So, it is also constant for a given system.
251
the pipeline is too high, the oxygen exchange rate results to be proportional to the pipeline oxygen permeability and is independent of the fluid velocity. In other words, in the regime of very high velocities, the oxygen exchange rate depends only on the pipeline wall properties. On the other hand, given that tanhðxÞC1 when xb1; the oxygen exchange rate is proportional to b when sbb: This means that when the fluid velocity is too small, the oxygen exchange rate is proportional to the transporting fluid volumetric flow. That is, in the regime of very low fluid velocities, the oxygen exchange rate is determined by the flow properties. After solving for s in Eq. (5) we get 1 bþo : ð6Þ s ¼ b ln 2 bo
The dynamics of the system sketched in Fig. 1 are studied with detail in Appendix A. In there, the steadystate oxygen exchange rate between the oxygen consuming system and the reservoir is shown to be s J ¼ ðme mb Þb tanh ; ð2Þ b
This last equation permits us to plot the o level curves in the s vs. b space. Such level curves are shown in Fig. 2 for different values of o: Each one of these curves determines an infinite set of ðb; sÞ pairs that lead to the same oxygen exchange rate. Since we are assuming that the oxygen exchange rate ðJ ¼ oðme mb ÞÞ is a function of the system of mass M; it attains a constant value for a given oxygen consuming system. This constant value determines on it own a level curve in the s vs. b space, like those of Fig. 2. Every point in such curve represents one of the infinite number of possible combinations of s (total surface of the oxygen exchanging system) and b (oxygen transporting fluid volumetric flow) to exchange the oxygen rate determined by o: A natural choice would be to pick up, of all those ðb; sÞ pairs, the one that requires the minimum energy expenditure to move the fluid through the pipeline. The resistance to laminar flow of the present work’s oxygen-exchanging network (Fig. 1) is analysed in Appendix B and is given by
where s and b are respectively defined as
R¼
3. Optimization of the oxygen exchanging system
s ¼ pr ln a
ð3Þ
ð7Þ
where n is the fluid viscosity,
and b ¼ pr2 nv:
4p2 n 2 S ; c3 l
ð4Þ
From their definition, s is nothing else but one half of the pipeline surface times a; while b is the pipeline volumetric flow, Q: For the sake of decreasing the number of variables we have to dealt with, let us introduce a new variable o ¼ J=ðTb Te Þ; which is proportional to the oxygen exchange rate. From this definition and Eq. (2) it follows that s o ¼ b tanh : ð5Þ b Since tanhðxÞCx when x51; we have that oCs when s5b: This means that when the fluid velocity through
c ¼ npr2
ð8Þ
is the total cross-section of the n parallel pipes passing through the oxygen consuming system and the reservoir, and S ¼ n2prl
ð9Þ
is their total surface. The resistance of the network interconnecting pipes is not accounted for in Eq. (7). The reason for this omission is that our main goal is to find the optimal number and geometry of the oxygen-exchanging parallel pipes that minimize the power expended in moving the fluid. Thus, the inclusion of the interconnecting-pipe resistance would not alter the final results
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252 5
ω = 0.5 ω = 1.0 ω = 1.5 ω = 2.0
4
Direction of increasing ω
σ
3
2
1
0
0
1
2
3
4
5
β Fig. 2. Plots of the o level curves in the s vs. b space for different values of o (look at the main text for the definition of variables o; s; and b). These plots asymptotically approach to the b ¼ o and s ¼ o lines. The direction of increasing o is shown close to the level-curve bending points.
5
lines of constant P line of constant ω
4
σ
3
Direction of increasing P 2
σ* 1
0
0
1
β*
2
β
3
4
5
Fig. 3. In this graphic the P level curves are plotted with solid lines in the s vs. b space for different values of P: The o ¼ 1 level curve is also plotted (dashed line). See the main text for the definition of variables P; s; b; and o: The direction of increasing P is shown close to the P level-curve symmetry axis. The point ðbn ; sn Þ where the o level curve is tangentially touched by one of the P level curves is also indicated.
appreciably since it shall only play the role of an additive constant. The power P needed to move a flow o through a pipeline with resistance R is
By solving for s in Eq. (11) we obtain sffiffiffiffi P1 : s¼ wb
P ¼ Ro2 :
With the aid of Eq. (13) we can plot the level curves for P in the s vs. b space. These plots (which are hyperbolas whose symmetry axis is the s ¼ b line) are shown in Fig. 3 for different values of P; along with the o ¼ 1 level curve. As seen in there, every one of the P level curves either crosses the o level curve at two different points, is tangent to it, or does not touch it at all. The P level curves that do not touch the level curve for o correspond to values of power ðPÞ expended in moving the oxygen transporting fluid with which it is impossible to transfer the oxygen rate determined by o:
ð10Þ
Then, from Eqs. (7) and (10), and the definition of s (Eqs. (3)), it follows that P ¼ ws2 b2
ð11Þ
with w defined as w¼
16p2 n 1 : a2 lc3
ð12Þ
ð13Þ
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The P value of the P-level curve that is tangent to the o level curve, is the minimum power we can expend in moving the oxygen transporting fluid in order to transfer such oxygen rate. From Eq. (11), this minimum power expenditure is given by Pmin ¼ ws * 2 b * 2 ;
ð14Þ
where bn and sn are the coordinates of the tangent point of the P and o level curves. For larger power expenditures there exist two possible combinations of s and b values (those corresponding to the points where the P and o level curves cross each other) to transfer the demanded oxygen rate. One of them corresponds to a high b and a low s values, while the other corresponds to a low b and a high s values. The optimal bn and sn values are calculated in terms of o and w in Appendix C rendering: bn C1:256o
ð15Þ
and sn C1:367o:
ð16Þ
Notice that neither bn nor sn depend on w; they only depend on o: This, in turn, means that the optimal bn and sn are independent of the network geometric characteristics ðl; cÞ; as well as of its material properties ða; nÞ; see Eq. (12). From the results of the present section we can conclude that, when we have an oxygen consuming system of mass M; which maintains a constant oxygen concentration mb by means of a continuous oxygen supply at rate J ¼ oðme mb Þ from a reservoir me ; and the oxygen is exchanged by a network as depicted in Fig. 1, we have to optimize both the pipeline surface (variable s) and the fluid velocity (variable b) in order to minimize the power expended in moving the fluid through the pipeline. The optimal sn and bn values are given by Eqs. (15) and (16). The minimum power expenditure is on its own given by Eq. (14). Recall that through the minimization process, the pipeline length ðlÞ and total cross-section ðcÞ; as well as the oxygen exchange rate ðJÞ; were kept constant, since they are assumed to be functions of the system of mass ðMÞ: The precise form of these functions will be discussed in the following section.
253
the power expended in moving the fluid is minimized, given the rate of oxygen exchange ðJðMÞ ¼ oðMÞðme mb ÞÞ; the pipeline total cross-section ðcðMÞÞ; and the pipeline length lðMÞ; then from Eqs. (12), (14), (15), and (16), the power expenditure is related to o; l; and c by o4 P p 3: ð17Þ lc The proportionality constant implicit in Eq. (17) involves only fluid and pipeline wall properties: The fluid viscosity ðnÞ and pipeline wall oxygen permeability ðaÞ: The performance regime of biological and nonbiological energy converting systems is determined by the ratio of quantities such as the energy input rate, the power output, the rate of heat exchanged with the environment, etc. (Angulo-Brown et al., 1995; Santilla! n et al, 1997; Angulo-Brown et al., 2002; Hoffmann et al., 1997). Some authors suggest that there might be some optimal operation regimes which are attained by biological systems (muscle contraction, ATP synthesis by anaerobic and aerobic glycolysis, etc.) of living organisms of different sizes (Angulo-Brown et al., 1995; Santilla! n et al., 1997; Santilla! n and Angulo-Brown, 1997; Santilla! n, 1999). They also propose an explanation for this fact from evolutive considerations. Based on the above considerations, we assume a scale-invariant performance regime for our system, which in terms of variables P and o implies that, as the system mass changes, ð18Þ
PpopB;
with B the oxygen consuming system ‘‘metabolic’’ rate and M the system mass. Then, from Eqs. (17) and (18) Bpðlc3 Þ1=3 :
ð19Þ
If we assume that the system grows up proportionally in all directions and that the ratio of the system to the pipeline cross-sections is scale invariant, as well as the system mass density, that is, lpM 1=3 ;
cpM 2=3 ;
ð20Þ
then, the following B–M allometric relation is obtained from Eqs. (19) and (20): B pM 7=9 pM 0:78 :
4. Scaling properties derived from a scale-invariant performance regime If we have an oxygen system optimized in the way described in Section 3, i.e.: The pipeline surface (variable s) and the volumetric flow (variable o) are set up so that
ð21Þ
Finally, from Eqs. (3), (4), (8), (15), (16), (18), (20), and (21), it is possible to derive, after a little algebra, the following scaling relations for variables r; n; and v: rpM 2=9 ;
npM 2=9 ;
vpM 1=9 :
ð22Þ
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5. Concluding remarks We have studied a simple oxygen exchanging network, whose task is to transfer oxygen from a source (with a constant concentration me ) to an oxygen consuming system (with steady-state concentration mb ). Under the supposition that the oxygen exchange rate J ¼ oðme mb Þ; the cross-section available for the pipeline passing through the system c; and the pipeline length l are functions of the system mass (and thus fixed for a given system), it was shown that both the pipeline total surface and the transporting fluid volumetric flow have to be optimized, according to Eqs. (15) and (16), in order to minimize the power expended in moving the fluid through the pipeline. Then, we found that, if the oxygen consuming system performance regime is scale invariant then, the ‘‘metabolic’’ rate of the optimized system has to scale up according to the allometric relation given by Eq. (21), which is quite similar to the experimentally observed biological 3=4 allometric scaling law. The model here presented is deliberately oversimplified in order to be able to optimize its performance regime analytically. Nevertheless, despite its simplicity it is quite general in the sense that the results we obtain are independent of the nature of the transporting fluid (it can be either a gas or a liquid), the exchanged substance (oxygen, nutrients, waste products, etc.), and the network materials. To our consideration, this model captures the basic characteristics of many biological networks. As mentioned in Section 2, there are many biological distribution networks, like the gills of fish and many other aquatic organisms and the book lungs of land crabs and spiders, that are essentially parallel. Moreover, real biological distribution or collection networks often seem to have properties of both fractal and parallel networks and may be considered as hybrids between the one we model and fractal networks like those of West et al. (1997, 2000). From these considerations, we expect that our results are feasible and comparable with biological data. Recently, Dodds et al. (2001) performed a statistical analysis of the reported data on birds and mammals and concluded that there is no enough evidence supporting the 3=4 allometric relation as a general law. On the contrary, they found that a 2=3 allometric relation is more plausible for birds and small mammals (mass less than 10 kg), but could not reject the 3=4 allometric relation for larger mammals. On the other hand, many reports supporting the 3=4 law in organisms as diverse as arthropoda, amphibians, and reptilians have been published over the years (West and Brown, 2000). Thus, although there is no general agreement concerning the generality of the 3=4 allometric scaling law, it can still be considered as a paradigm for many animal species.
Moreover, one might speculate that, since birds and small mammals are warm blooded animals with a large surface (in proportion with their mass), the original derivation of the 2=3 relation by Rubner (1883) is applicable, while in large mammals and cold blooded animals the distribution-network effects are dominant and thus, they obey an allometric relation with exponent close to 3=4: A by-product of the present work is the allometric scaling law derived for the radius of the oxygen exchanging parallel pipes (see Eq. (22)): rpM 2=9 pM 0:22 : Interestingly, the exchanging pipes of real biological distribution networks also scale obeying allometric laws. For instance, the experimentally observed scaling law for the radius of the mammalian alveolus is rpM 0:13 (Tenney and Remmers, 1963). The comparison with our result is not good, but we believe that this may be due to the different geometries of both systems. In this regard, notice that the exchanging ‘‘pipes’’ of different networks (all of which obey the 3/4 metabolic scaling law) scale with different exponents. For instance, West et al. (1997) consider that the radius of blood vessels is scale invariant in their model of the circulatory system. In conclusion, we present a simple model of an oxygen exchange network which, when its geometry and fluid dynamics are optimized to minimize the power expended in moving the oxygen-transporting fluid, makes the oxygen consuming system obey (under feasible geometric scaling assumptions) a scaling law similar to the experimentally observed biological 3=4 allometric relation. Despite its simplicity, the model bears some resemblance, as discussed above, with many real biological distribution or collection networks. Moreover, it is quite general in the sense that many biological processes rely upon distribution networks like the one modelled here and that the results we obtain depend only on the network geometry and dynamics, and not on the properties of the substances it transports or is made up of. From all these considerations, we believe that this model may contribute to pose the issue of whether the optimization of both static and dynamic properties of biological distribution networks is required to understand the underlying principles that govern the structure and function of real, more complicated systems.
Acknowledgements Thanks are due to F. Angulo-Brown and M.C. Mackey for fruitful discussion. This work was supported by COFAA-IPN; CGPI-IPN, Grant 200606; and by CONACyT, Grant 411300-I333174-E.
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Appendix A. Calculation of the model oxygen exchange rate A.1. One single pipe Consider a single pipe of radius r and length l; which crosses an oxygen reservoir with concentration ms : An oxygen transporting fluid passes through the pipe with a velocity v: The fluid oxygen concentration when it enters the thermal bath (position x ¼ 0) is m0 : Take the differential fluid element located between coordinates x and x þ dx0 : Let its oxygen concentration be mðxÞ: Let a be a constant, proportional to the pipe wall oxygen permeability, such that the transporting-fluid oxygen-uptake rate is given by dJ ¼ ðms mðxÞÞ2par dx0 :
ðA:1Þ
On the other hand, the oxygen concentration increase dm caused by this oxygen uptake can be calculated from dJ dt ¼ pr2 dx0 dm:
ðA:2Þ
By combining Eqs. (A.1) and (A.2) we obtain dm 2pra ¼ ðm mðxÞÞ: ðA:3Þ dt pr2 s If we finally make use of the chain rule dm=dt ¼ v dm=dx (note that v ¼ dx=dt), we have dm 2sð1Þ ¼ ðm mðxÞÞ; dx bð1Þ l s
ðA:4Þ
where sð1Þ ¼ prla is one-half of the pipe surface times a; and bð1Þ ¼ pr2 v is the volumetric flow through the pipe. After integration, the following oxygen concentration profile is obtained, 2sð1Þ mðxÞ ¼ ms ðms m0 Þ exp ð1Þ x ; ðA:5Þ b l where we have employed the boundary condition mð0Þ ¼ m0 : In particular, the oxygen concentration at the pipe end (position x ¼ l) is 2sð1Þ mf ¼ mðlÞ ¼ ms ðms m0 Þ exp ð1Þ : ðA:6Þ b The rate of oxygen exchanged by the whole pipe can be calculated as Z x¼l dJ J ð1Þ ¼ x¼0
¼ 2pra
Z
l
ðms mðxÞÞ dx
0
¼ ðms m0 Þbð1Þ ð1 expð2sð1Þ =bð1Þ ÞÞ:
ðA:7Þ
A.2. n parallel pipes Consider a pipeline of n identical parallel pipes crossing an oxygen reservoir with concentration ms : All
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of the pipes as well as the flow through each of them have the same dimensions and properties as those specified in Section A.1. Since the pipes are in parallel, the oxygen concentration difference between their ends is the same for all of them. From this assertion and Eq. (A.6) it follows that 2sð1Þ ðA:8Þ mf ¼ ms ðms m0 Þ exp ð1Þ : b On the other hand, the oxygen exchange rate through the whole pipeline, J; is n times the oxygen exchange rate through a single pipe, J ð1Þ : Therefore, it follows from Eq. (A.7) that J ¼ nJ ð1Þ
2sð1Þ ¼ ðms m0 Þnbð1Þ 1 exp ð1Þ : b
ðA:9Þ
Let us define variables s ¼ nsð1Þ and b ¼ nbð1Þ : With these definitions, Eqs. (A.8) and (A.9) become 2s mf ¼ ms ðms m0 Þ exp ðA:10Þ b and
2s J ¼ ðms m0 Þb 1 exp : b
ðA:11Þ
From their definition, s and b can be interpreted as onehalf of the whole pipeline surface times a and as the volumetric flow throw the whole pipeline, respectively. A.3. The oxygen exchanging system Consider the oxygen exchanging system sketched in Fig. 1. It consists of an oxygen consuming system (with steady-state concentration mb ), an oxygen reservoir (with constant concentration me > mb ), a pipeline that crosses and interconnects both of them, and a fluid that circulates through the pipeline and is the active oxygen-exchanging element. The pipeline can be divided into two identical sections. Each section is made up of n parallel pipes, each one having the characteristics of the pipe studied in Section A.1. All of these pipes connect into a single pipe before entering and after leaving the oxygen consuming system and the reservoir. The input of one of these pipeline sections is connected to the output of the other and vice versa. In the steady state, the fluid coming out from the oxygen consuming system and entering the oxygen reservoir will have a constant oxygen concentration mh ; while the fluid coming out from the oxygen reservoir me and entering the oxygen consuming system will have a constant concentration mc ðme > mc > mh > mb Þ:
ARTICLE IN PRESS ! / Journal of Theoretical Biology 223 (2003) 249–257 M. Santillan
256
By applying the results of Section A.2 (Eq. (A.10)) to each of the pipeline sections we have 2s mc ¼ me ðme mh Þ exp ðA:12Þ b and
2s mh ¼ mb ðmb mc Þ exp : b
ðA:13Þ
These equations constitute a complete set of equations for mc and mh which, when solved, give 2s 1 expð2s=bÞ ðA:14Þ mc ¼ me þ mb exp b 1 expð4s=bÞ and
2s 1 expð2s=bÞ mh ¼ mb þ me exp : b 1 expð4s=bÞ
ðA:15Þ
Finally, from these last results and Eq. (A.11), the oxygen exchange rate between the oxygen reservoir and the oxygen consuming system is ½1 expð2s=bÞ2 1 expð4s=bÞ s ¼ ðme mb Þb tanh : b
J ¼ ðme mb Þb
ðA:16Þ
Appendix B. Calculation of the model resistance to flow The resistance to flow of a cylindrical pipe of radius r and length l is given by (Landau and Lifshitz, 1987) 8pnl Rð1Þ ¼ 4 ; ðB:1Þ r with n the fluid viscosity. The resistance to flow of an array of n parallel identical pipes is Rð1Þ 8pnl ¼ 4: nr n
ðB:2Þ
By substitution of Eq. (B.3) into Eq. (B.2), the pipeline resistance in terms of c and n turns out to be RðnÞ ¼
8p3 nl n: c2
ðB:4Þ
On the other hand, the pipeline surface is S ðnÞ ¼ 2prln:
ðB:5Þ
After substitution of Eq. (B.3) into Eq. (B.5) we obtain pffiffiffiffiffiffiffiffiffi S ðnÞ ¼ 2l cpn: ðB:6Þ Solving for n in Eq. (B.6) and substituting into Eq. (B.4) gives the following expression for the pipeline resistance in terms of its cross-section c and its surface SðnÞ : RðnÞ ¼
2p2 n ðnÞ S 2: c3 l
ðB:7Þ
Finally, if we dismiss the resistance of the pipes interconnecting the two pipeline sections of the oxygen exchanging system depicted in Fig. 1, the total system resistance is R ¼ 2RðnÞ ¼
4p2 n ðnÞ2 S : c3 l
ðB:8Þ
Appendix C. Calculation of the optimal sn and bn values For reasons explained in the main text, the optimal sn and bn values we are looking for are those for which
4.4 4.3 4.2 4.1 4
f(β)
RðnÞ ¼
Let c ¼ npr2 be the total pipeline cross-section. From this rffiffiffiffiffiffi c : ðB:3Þ r¼ np
3.9 3.8 3.7 3.6 3.5 3.4
1
1.1
1.2
1.3
1.4
1.5
1.6
1.7
1.8
1.9
β Fig. 4. Plot of the function f ðbÞ ¼ b2 ln½ðb þ oÞ=ðb oÞ with o ¼ 1:
2
ARTICLE IN PRESS ! / Journal of Theoretical Biology 223 (2003) 249–257 M. Santillan
the curves 1 bþo s ¼ b ln 2 bo
ðC:1Þ
and
sffiffiffiffi P1 s¼ wb
ðC:2Þ
are tangent. By combining Eqs. (C.1) and (C.2), this task results to be equivalent to that of looking for the value of P for which the following equation has one single solution ðbn Þ and finding it: sffiffiffiffi P bþo 2 ¼ b ln 2 ðC:3Þ w bo Let us define function f ðbÞ as bþo f ðbÞ ¼ b2 ln : bo
ðC:4Þ
A plot of f ðbÞ (shown if Fig. 4 for o ¼ 1) demonstrates that it is a concave curve p with ffiffiffiffiffiffiffiffiffi a single minimum. Therefore, equation f ðbÞ ¼ 2pffiffiffiffiffiffiffiffi P=wffi (Eq. (C.3)) has one single solution bn only if 2 P=w corresponds to the minimum value of f ðbÞ; being bn the position of such minimum. In conclusion, the optimum bn value we are looking for is that which minimizes function f ðbÞ; given by Eq. (C.4). The minimum of function f ðbÞ can be calculated from df =db ¼ 0: After a little algebra, this leads to the following equation for bn : n b þo bn o ln n : ðC:5Þ ¼ 2 b o b * o2 To solve it we propose bn ¼ eo: After substitution into Eq. (C.5), it becomes eþ1 e ln : ðC:6Þ ¼ 2 e1 e þ1 Eq. (C.6) was solved numerically with the aid of algorithm fzero of Matlab obtaining eC1:2556: From this it follows that bn C1:256 o
ðC:7Þ
and after substituting into Eq. (C.1): sn C1:367 o:
ðC:8Þ
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