Many biological and man-made systems rely on transport systems for the distribution of material, for example matter and energy. Material transfer in these systems is determined by the flow rate and the concentration of material. While the most concentrated solutions offer the greatest potential in terms of material transfer, impedance typically increases with concentration, thus making them the most difficult to transport. We develop a general framework for describing systems for which impedance increases with concentration, and consider material flow in four different natural systems: blood flow in vertebrates, sugar transport in vascular plants and two modes of nectar drinking in birds and insects. The model provides a simple method for determining the optimum concentration copt in these systems. The model further suggests that the impedance at the optimum concentration μopt may be expressed in terms of the impedance of the pure (c = 0) carrier medium μ0 as μopt∼2αμ0, where the power α is prescribed by the specific flow constraints, for example constant pressure for blood flow (α = 1) or constant work rate for certain nectar-drinking insects (α = 6). Comparing the model predictions with experimental data from more than 100 animal and plant species, we find that the simple model rationalizes the observed concentrations and impedances. The model provides a universal framework for studying flows impeded by concentration, and yields insight into optimization in engineered systems, such as traffic flow.
Transport systems are ubiquitous in nature and technology. Whether biological such as the vascular systems of plants and animals or engineered such as man-made pipes, roads, electrical grids and the Internet, they serve to move matter, energy or information from one place to another. Owing to the cost of constructing and maintaining redundant channels, it is advantageous for biological transport systems to distribute matter efficiently [1,2]. Oxygen transport in vertebrates [3,4], sugar transport in plants  and drinking strategies of many animals [6,7] are known to be optimized for efficient transport of energy and material. Engineered systems must likewise be cost-effective and able to provide efficient transport under a variety of conditions; for example, considerable resources are spent annually to ease traffic congestion.
In our examination of transport systems, we consider material flow in four different natural systems: blood flow in vertebrates, sugar transport in vascular plants, and two modes of nectar drinking in birds and insects. A common feature of these and other transport systems is that the flow impedance depends on concentration. While the most concentrated solutions offer the greatest potential in terms of material transfer, the increase of impedance with concentration also makes them the most difficult to transport. Additionally, most transport systems are subject to a set of limiting constraints. For example, nectar feeders are typically constrained by a constant work rate, which in turn is a function of flow impedance and hence concentration [7–9]. These transport systems may thus be characterized in terms of an optimization problem subject to appropriate constraints. This approach has been used to rationalize observed concentrations in a wide range of natural systems [4,8–10]. For example, the observed volume fraction of erythrocytes (red blood cells), typically approximately 40–50% in humans, has been shown to maximize oxygen transport [4,10]. With the widespread use of bioinspired design in the development of novel engineered systems, it seems likely that man-made transport systems such as roads or the electrical grid may benefit from improved understanding of natural transport systems.
We here develop a general framework for determining the concentration that maximizes material transfer in transport systems. By drawing on a number of natural examples—either new or drawn from the biology literature—we show how these can be treated within a single framework that provides new insight into the efficiency of transport systems. We compare our model predictions with experimental data from more than 100 animal and plant species collected from the literature. Finally, we show that similar optimization criteria may be applied to engineered systems, and consider traffic flow in the context of our new framework.
2. General formulation
We consider systems in which the material transfer rate (material flow) J can be expressed as the product of a volumetric flow rate (volume flow) Q and a concentration of material c 2.1We express the volume flow as Q = Xf/μ, where X is a constant geometric factor, f quantifies the mechanism driving the flow and μ characterizes the impedance. The material flow in equation (2.1) can be expressed as 2.2where f and μ can depend on c. We now seek the optimal concentration c = copt that maximizes J in equation (2.2) subject to a set of constraints; constant driving force f = f0 or constant work rate W ∼ Qf ∼ f2/μ. We thus consider constraints of the form f ∼ μγ, where γ = 0 corresponds to constant driving force and to constant work rate when f depends on μ. Other values of γ are possible, if there is a direct coupling between the driving force and impedance; for example, for bees that use viscous dipping to drink nectar (see §3.4).
Although we may have limited knowledge of the exact functional form of the concentration-dependent material flow J(c) in equation (2.2), two general statements can be made. First, we expect J to be proportional to the concentration c at low concentrations and to approach zero with the concentration, i.e. J(c) ∝ c, when c ≪ copt. Second, we are concerned with situations where the system impedance increases with concentration, and J increases monotonically up to a maximum value J(copt). To describe the system, we thus propose the first order governing equation 2.3where J* = J(c)/J(copt) and c* = c/copt is the normalized material flow and concentration, respectively. A and B are parameters that are determined by the boundary conditions: J* (0) = 0, J* (1) = 1 and . This leads to 2.4While equation (2.4) does not reveal the absolute value of the optimum concentration copt, it does contain information concerning the impedance at the optimum concentration μ (copt). By expressing the constraint as f ∝ μγ, we find from equation (2.2) that the normalized flux J* =J(c)/J(copt) can be written as 2.5By using equation (2.4), the normalized impedance μ* = μ(c)/μ(copt) may be expressed as 2.6It follows that the impedance at the optimum concentration is 2.7where the power α = 1/(1 − γ) = log2(μ(copt)/μ(0)) and μ (0) is the impedance at zero concentration.
Equations (2.4), (2.6) and (2.7) provide a general framework for analysing optimization of concentration-impeded material flow in biological and engineered systems. To test the quantitative predictions of the theory, we proceed in §3 by considering a series of biological examples where the flux J can be optimized along the lines outlined above. In §4, we apply our model to traffic flow. Finally, in §5, we consider universal properties of concentration-impeded material transport systems.
3. Biological transport systems
3.1. Nectar drinking from a tube
Perhaps the simplest situation in which we may apply equation (2.2) is drinking from a cylindrical tube. Many insects and birds such as butterflies and hummingbirds  feed on floral nectar, an aqueous solution of sugars, through tubes formed from proboscises or tongues. Quick energy ingestion is advantageous for nectar feeders owing to the threat of predation. While the sweetest nectar offers the greatest energetic rewards, the increase of viscosity with sugar concentration also makes the sweetest nectar the most difficult to transport . An optimal concentration may thus be sought for maximizing energy uptake rate.
Two different suction mechanisms are typically used by nectar feeders: active suction and capillary suction [7–9,11]. Active suction feeders such as butterflies use muscle contraction to suck nectar through their roughly cylindrical proboscises. In the limit of low Reynolds number Hagen–Poiseuille flow, the nectar mass flow rate Js can be expressed as 3.1where a is the radius and l is the length of the proboscis, is the w/w sugar concentration, η is the viscosity (see appendix A), ρ is the density of the nectar solution and Δp is the pressure difference generated by muscular contraction. The manner in which biological constraints determined the dependence of the pressure Δp on nectar viscosity has been treated elsewhere [8,9]. Active suction feeders are typically constrained by constant work rate W = QΔp = πa4/(8ηl)Δp2, so the pressure Δp = (8Wl/(πa4))1/2η1/2 depends on viscosity and, hence, concentration. Comparing the nectar flow rate in equation (3.1) with the general expression in (2.2), we find that the impedance corresponds to the viscosity of the sugar solution μ = η, the concentration to , the geometric factor to X = πa4/(8l), and the driving mechanism to the pressure f = Δp. We can thus express the constraint as f = (8Wl/(πa4))1/2μ1/2 (table 1).
Capillary suction feeders such as hummingbirds use surface tension to draw nectar along their tongue, during repeated cycles of tongue insertion and retraction [7,9]. If the duration of a cyclic motion is the sum of the nectar loading time T and unloading time T0, the average nectar mass flow rate Js can be expressed as where πa2l(T) is the time-dependent nectar volume extracted during the loading. In the loading phase, the volumetric flow rate is given by πa2(dl/dt) = πa4Δp/(8ηl), where Δp = 2σ/a is the capillary pressure. The solution with initial condition l(0) = 0 is given by l(t) = (aσt/(2η))1/2, which depends explicitly on viscosity and hence concentration. The average nectar mass flow rate Js can be expressed as 3.2By comparing the nectar flow rate in equation (3.1) with the general expression in equation (2.2), we find that X = πa3, f = (σηT/(2a(T + T0)2))1/2 and μ = η. If T/(T+T0)2 is assumed to be independent of viscosity , we find a relation between driving force and impedance: f ∝ μ1/2 (table 1).
For both active and capillary suction, we find that f ∝ μ1/2 (i.e. ) and the optimal concentration can thus be found by maximizing c/μ1/2. For nectar sugar solutions, we thus predict that w/w and (i.e. α = 2 in (2.7)). This is in good agreement with experimental data on 16 butterfly and hummingbird species (figure 1a and table 2), where optimal concentrations in the range 30–45% are reported.
3.2. Blood flow in vertebrates
Another biological flow problem that can be analysed within our framework is oxygen and nutrient transport within the cardiovascular system of vertebrate animals. Here, red blood cells transport oxygen between the lungs and distal parts of the organism. The cells are suspended in blood plasma, which primarily consists of water . Red blood cells typically measure 10 µm in diameter , and the blood's bulk viscosity increases with the haematocrit , the volume concentration of red blood cells (see appendix B). While blood with the highest haematocrit is the most oxygen rich, the increase of viscosity with the haematocrit also makes such blood the most difficult to transport. Accordingly, an optimal haematocrit may be sought for maximizing oxygen transport.
In the limit of low Reynolds number Hagen–Poiseuille flow, the red blood cell volume flow rate Jr in vessels larger than 1 mm can be expressed as 3.3where a and l are the radius and length of the blood vessel, η is the blood viscosity (see appendix B) and Δp is the pressure difference generated by the heart. The diastolic pressure is of the order of 10 kPa for most animals , but the pressure difference associated with flow in large vessels is small; most of the pressure drop in blood flow occurs in vessels with size comparable to red blood cells. The dependence of blood pressure on the haematocrit is, however, negligible , i.e. Δp does not depend on . Although blood viscosity η generally depends on the shear rate, this dependence is weak for typical blood conditions, specifically in vessels with diameters larger than 1 mm and shear rates greater than 50 s−1 . Comparing the blood flow rate in equation (3.3) with the general expression in equation (2.2), we find that , μ = η, X = πa4/(8l) and f = Δp. We express the constraint as that of constant pressure f = f0 = Δp (table 1).
For blood flow, we thus find that Xf ∝ μ0 (i.e. γ = 0) and the optimum concentration can be found by maximizing c/μ. We thus predict that v/v and (i.e. α = 1, in (2.7)), in good agreement with experimental data from 57 species observed throughout the animal kingdom (figure 1b and table 2). We attribute the significant variation in the observed concentrations in part to the complex interactions between red blood cells and flow in smaller vessels, which we do not consider in our model. An important feature of the red blood cell volume flow rate Jr is also that it varies by less than 20 per cent over the range of concentrations from c = 20–60%, suggesting that concentrations in this interval are acceptable given additional biological constraints. For example, we note that, for some diving mammals (e.g. Weddell seals and whales), oxygen storage in the blood may also be an important factor, resulting in a higher haematocrit (up to 63%; ). It is also likely that lack of thermoregulation may explain why poikilothermic animals (e.g. the rainbow trout) have a lower haematocrit value (23%) than the average, probably as a result of thermally induced variations in blood viscosity .
3.3. Sugar transport in plants
Plants, like animals, rely on vascular systems for distribution of energy and nutrients. Energy distribution in plants takes place in the phloem vascular system. Here, an aqueous solution of sugars, amino acids, proteins, ions and signalling molecules flows through a series of narrow elongated cylindrical cells, known as sieve tube elements, that lie end-to-end, forming a microfluidic distribution system spanning the entire length of the plant. The flow is driven by differences in chemical potential between distal parts of the plant . While phloem sap with high sugar concentration has the greatest potential for energy transfer, the increase of viscosity with sugar concentration makes it the most difficult to transport. Accordingly, an optimal concentration may again be sought for maximizing energy flow.
Assuming low Reynolds number Hagen–Poiseuille flow, the phloem sugar mass flow rate Jp can be expressed as 3.4where a is the radius of the phloem sieve tube (a ≃ 10 µm), l is the length of the plant, is the sugar concentration, η is the phloem sap viscosity and Δp is the pressure difference driving the flow. By comparing the sugar flow rate (3.4) with the general expression in (2.2), we find that , , and f = Δp. We express the constraint as that of constant pressure f = f0 = Δ p (table 1).
For sugar transport in plants, we thus find that Xf ∝ μ0 (i.e. γ = 0), and the optimum concentration can thus be found by maximizing c/μ. We find that w/w and (i.e. α = 1, in (2.7)), in good agreement with experimental data (figure 1c and table 2). While sugar concentrations observed in plants generally span a wide range, this analysis provides a rationale for the observation that plants that use active sugar loading (data shown in figure 1c) typically have a higher sugar concentration than plants that use passive loading . Active loaders expend metabolic energy to increase the sugar concentration in the phloem . The process is driven by membrane transporters and sugar polymerization and occurs against a sugar concentration gradient. However, in passive loading species, sugars move into the phloem without the use of metabolic energy by travelling down a concentration gradient from sites of carbohydrate synthesis and/or storage to the phloem . We also note that plants with the highest sugar concentrations are crop plants, for example potato (50%) and maize (40%), suggesting that selection for high crop yield tends to lead to increased sugar concentration in the phloem sap .
3.4. Drinking by viscous dipping
So far, we have limited our attention to transport in closed channels. However, it is straightforward to extend the problem to situations where free surfaces are involved. Most bees whose tongues are solid rather than hollow use a drinking style termed ‘viscous dipping’ in which the fluid is entrained by the tongue surface. The average nectar volume entrained can be expressed by Q ∼ 2πaeu, where a is the tongue radius, e is the thickness of the nectar layer on the tongue and u is the tongue extraction speed. Based on Landau–Levich–Derjaguin theory when the Reynolds number Re ≪ 1 and Bond number Bo ≪ 1, the nectar film thickness is given by e ∼ aCa2/3, where Ca = ηu/σ ≪ 1 is the ratio of viscous to capillary forces . Because the fluid is entrained on the tongue by viscous forces, we define the driving force and geometric factor as f = ηue/a2 and X = 2πa3. The movement of the tongue in the fluid requires power W ∼ ηu2l to overcome the viscous drag, where l is the immersed tongue length. Assuming a constant work rate W for a given creature leads to the constraint on velocity u ∼ (W/(ηl))1/2, which in turn leads to f ∝ μ 5/6 (; table 1).
For viscous dipping, we find that f ∝ μ 5/6 (i.e. ) and the optimum concentration can thus be found by maximizing c/μ1/6. We find that copt = 57% w/w and μ (copt) = 64 μ0 (i.e. α = 6, cf. equation (2.7)). This is in reasonable agreement with experimental data on six bees species (figure 1d and table 2), where optimal concentrations in the range 50–60 % are found. This may explain why the nectar concentration of flowers pollinated by bees is generally higher than that of those pollinated by tube-feeding butterflies and hummingbirds .
4. Applications to engineered transport systems: traffic flow
We have thus far seen many qualitative similarities between different biological flows. Although the detailed physiological and physical mechanisms are different, provided increased concentration leads to greater impedance, we can rationalize the optimal concentrations. An interesting question naturally arises. In which engineered systems might one expect to observe similar phenomena? It appears likely that most efficient communication and transport systems will exhibit similar features. Nevertheless, we limit our discussion to traffic congestion on highways.
A measure of the efficiency of a given section of road is the vehicle flow Jv, the number of vehicles passing a given point per unit time [23–26]. Designers of road networks strive to maximize the vehicle flow that can be expressed as Jv = ρv, where v is the speed of the individual vehicle and ρ is the number of vehicles per unit length of roadway. Generally, car speed v = v(ρ) is a decreasing function of density ρ. At very low densities, where inter-vehicle interaction is negligible, however, the speed approaches the speed limit vmax and the vehicle flux is proportional to density Jv ≃ ρvmax. At higher vehicle densities, interaction between adjacent cars leads to flow impedance and a significant reduction in the speed of individual vehicles, causing congestion and a net decrease in the flux Jv. The vehicle interactions initially take the form of synchronized flow, a form of congested traffic in which each driver attempts to maintain a safe distance from the neighbouring cars. As the density increases, wide moving jams form, that is, stop-and-go traffic in which the vehicle flux approaches zero . From these considerations, one anticipates an optimal vehicle density ρopt that maximizes the vehicle flux.
To estimate ρopt, we require v(ρ), which can be either found empirically or deduced from vehicle interaction models. One of the simplest models that leads to a reasonable expression for v(ρ) was proposed by Greenberg , who treated traffic flow as a one-dimensional flow of an ideal compressible gas. He assumed (i) that the local speed is a function of density only v = v(ρ(x,t)), (ii) that vehicles are conserved , (iii) that vehicle flow satisfies the Euler equation and (iv) that traffic ‘pressure’ is proportional to density p = 𝒞2ρ. This leads to the relation v(ρ) = 𝒞 ln(ρmax/ρ), where ρmax is the density at which traffic stops owing to congestion. The vehicle flow rate Jv = 𝒞ρ ln(ρmax/ρ) is at a maximum when ρ = ρopt = ρmax/e, and the constant 𝒞 = v(ρopt) is the vehicle speed at the optimal concentration. Because vehicles typically occupy 7.5 m in a totally congested flow , we estimate that ρmax ≃ 133 vehicles per kilometre. Greenberg's model overestimates the optimal density, predicting ρopt = ρmax/e ∼ 50 vehicles per kilometre, whereas the true value is known to be ∼ 20 vehicles per kilometre. Nevertheless, the vehicle flow rate Jv is qualitatively consistent with empirical traffic data (figure 2). The data are plotted as a function of vehicle concentration c = ρ/ρmax in figure 2 along with Greenberg's flow rate Jv, deduced using ρmax = 133 vehicles per kilometre.
A shortcoming of Greenberg's theoretical model is that the vehicle speed v diverges when the car density is very low. To ensure that v(ρ/ρmax → 0) = vmax and to account for other aspects of traffic flows, numerous other models have been proposed [23–26,28]. For example, Bando, Hasebe and Nakayama (BHN)  suggested a traffic model in which the vehicle speed depends on the distance from the car in front, Δx. This leads to v = vmaxtanh(Δx/s), where s is a fixed length scale determined by the road conditions. With a minimum vehicle distance L = 7.5 m, we can express the density in terms of Δx as ρ = 1/(L+Δx). This leads to v(ρ) = vmaxtanh[(1/ρ − L)/s], in which case the flow rate Jv = vρ is optimized when ρ = 0.21 and ρmax = 28 vehicles per kilometre. With vmax = 120 km h−1 and s = 60 m, the BHN model provides a better quantitative fit to the empirical data than does Greenberg's model (figure 2).
Comparing the Greenberg and BHN models of traffic flow with the formulation introduced in equation (2.2), we see that traffic flow can be treated in the same general framework where 4.1for Greenberg's model, and 4.2for the BHN model. In both cases, N is the number of lanes.
Comparing traffic flow with the biological transport problems considered above, we find that the normalized flux and impedance curves follow the same pattern (figure 2). While traffic flow can be treated in the same framework as biological flows, it is important to note that the congested highway (figure 2, data recorded from 16.00 to 18.00) is very far from being optimized. This is presumably the result of two main effects. First, the individual vehicle operator attempts to minimize his or her own travel time, which does not necessarily optimize the overall vehicle flow Jv. Second, traffic flows are intrinsically time dependent, which leads to the formation of travelling density waves and shocks [23–26].
5. Universal properties of transport systems
To compare characteristics of the particular biological and man-made transport systems considered in §3 and §4 with the general formulation in equations (2.4) and (2.6), normalized material flow and impedance curves are plotted in figure 3. Despite the complex dependence of impedance on concentration (see appendices A and B), both the material flow J* and impedance μ* are adequately approximated by the simple forms given in equations (2.4) and (2.6). From (2.6), it follows that the impedance at the optimum concentration is μopt = 2αμ0, where μ0 is the impedance of the pure carrier medium (with c = 0) and the power α is determined by the flow constraints. In the cases of vascular transport in plants and animals, the power α = 1, because there is no coupling between the constant driving pressure (f) or the vascular geometry (X) and the impedance (μ). This suggests that the optimum in material flow should occur when the blood or phloem sap is twice as viscous as water, i.e. ηopt = 2η0, in good agreement with observed values (table 2).
In transport systems that are constrained, for example by constant work rate, α will generally be greater than unity, because of the coupling between flow and impedance. The impedance at the optimum concentration μopt = 2αμ0 can thus be significantly greater than that of the carrier medium. This is most clearly seen in the case of viscous dipping (§3.4), where the observed nectar viscosity is up to 50 times greater than that of water, roughly consistent with the value (26 = 64) predicted by our simple model (table 2).
These observations suggest that this general framework may also provide the rationale for the viscosities found in other biological transport systems where efficient transport is favoured. Examples of systems with constant forcing include mammals that drink whole milk (observed viscosity: η ∼ 2η0; ), and the macro-alga Chara where streaming distributes the content of the cell cytosol (observed viscosity: ∼3η0 ). Although detailed studies of these systems are left for future consideration, we note that both are roughly consistent with the predictions of our general theory with α = 1.
Comparing traffic flow with the biological transport problems considered, we find that the normalized flux and impedance curves follow the same pattern (figure 3 and table 2). Since the speed limit vmax, which is fixed on a given road section, corresponds to the flow-driving mechanism in the BHN model, traffic flow is analogous to vascular transport in animals and plants that operate at constant pressure. Our model thus indicates that the flow constraint does not couple to impedance, Xf ∝ μ0 (γ = 0, α = 1), and hence that the optimal impedance is μopt = 2 μ0. This is in rough accord with the BHN model that yields μopt = 1.9 μ0.
6. Discussion and conclusion
We have seen many qualitative and quantitative similarities between different natural and engineered transport systems. Although the detailed transport mechanisms are different, key common features have allowed us to develop a general framework. Provided impedance increases with concentration, our model provides a means of rationalizing the optimal concentrations. Collecting data from more than 100 plant and animal species, we have observed that optimization of material flow appears to be a universal feature of biological transport systems. This deduction provides a rationale for the observation that the simple model introduced in §2 collapses flow and impedance curves for all the systems considered (figure 3), suggesting a universal component to all natural transport systems.
Finally, we have shown that an interesting analogy can be made between biological systems and self-driven systems such as traffic flows. Here, we find that the impedance analogy is still valid, but that the system is far from optimized owing to conflicting interests between individuals and the collective. The consideration of other man-made transport systems, such as the electrical grid or the Internet, is left for future consideration.
The authors wish thank Maciej Zwieniecki, Ruben Rosales, Jessica Savage, Nick Carroll, Kenneth Ho and David Weitz. This work was supported by the NSF (grant nos 1021779 and DMS-0907955) and the Materials Research Science and Engineering Center (MRSEC; grant no. DMR-0820484) at Harvard University.
Appendix A. Viscosity and density of nectar and phloem sap
Phloem sap and flower nectar consist of an aqueous solution of sugars, amino acids, proteins and other nutrients. Sugars, of which sucrose, fructose and glucose are the most abundant types, constitute about 90 per cent of the total solute mass . To approximate the viscosity η and density ρ of phloem sap and nectar, we thus used data from sucrose solutions of concentration obtained from Hainsworth . Least squares fits to sucrose data yield the approximate expressions for viscosity and and density . We note that viscosity and density data from other sugar types (glucose and fructose) are well approximated by the fit, suggesting that the major determinant of viscosity is the mass fraction , and not the type of sugar.
Appendix B. Viscosity of blood
Vertebrate blood is composed of blood cells suspended in blood plasma, a liquid that consists mostly of water. The viscosity of blood η depends primarily on the volume concentration (haematocrit) of red blood cells, and on temperature [4,19]. As demonstrated by Saitô  and Stark & Schuster , blood viscosity is well described by the function , which for blood vessels with diameters larger than 1 mm is consistent with empirical data with less than 5 per cent error for 0 < ([34,35]; electronic supplementary material, table S6).
- Received February 12, 2013.
- Accepted March 25, 2013.
- © 2013 The Author(s) Published by the Royal Society. All rights reserved.