Optimal concentrations in transport systems

1. Introduction

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 [5] 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 = c_opt 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 ∼ μ^γ, 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 ≪ c_opt. Second, we are concerned with situations where the system impedance increases with concentration, and J increases monotonically up to a maximum value J(c_opt). To describe the system, we thus propose the first order governing equation 2.3where J* = J(c)/J(c_opt) and c* = c/c_opt 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 c_opt, it does contain information concerning the impedance at the optimum concentration μ (c_opt). By expressing the constraint as f ∝ μ^γ, we find from equation (2.2) that the normalized flux J* =J(c)/J(c_opt) can be written as 2.5By using equation (2.4), the normalized impedance μ* = μ(c)/μ(c_opt) 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(c_opt)/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 [7] 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 [11]. 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 J_s 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))^1/2η^1/2 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))^1/2μ^1/2 (table 1).

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Table 1.

Parameters describing the material flow J = Qc = Xfc/μ (see equation (2.2)) for each of the systems considered. See appendices A and B for details on the viscosity η of blood, nectar and phloem sap (BHN, Bando, Hasebe and Nakayama).

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 J_s 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 J_s 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 [12], we find a relation between driving force and impedance: f ∝ μ^1/2 (table 1).

For both active and capillary suction, we find that f ∝ μ^1/2 (i.e. ) and the optimal concentration can thus be found by maximizing c/μ^1/2. 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.

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Table 2.

Comparison between theoretical predictions (T) and experimental observations (E) of the optimum concentration c_opt, the optimum viscosity μ_opt and the exponent α. Concentration units are % w/w for nectar drinking and sugar transport in plants, % v/v for blood flow and % vehicle density/max. vehicle density for traffic flow. The experimental data are available in the electronic supplementary material.

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Figure 1.

Optimal concentrations in biological transport systems. (a) Drinking from a tube. Histogram showing the distribution of observed sugar concentrations that maximize nectar uptake for 16 bird and insect species that use muscular contractions or surface tension to feed through cylindrical tubes [9,13]. Normalized sugar mass flow J_s/J_s,max (solid line, equations (3.1) and (3.2)) and nectar viscosity μ/μ₀ (dashed line, data from [14]) are plotted as a function of nectar sugar concentration . Mass flow is predicted to be maximum when c̄_opt = 35%, in good agreement with the observed average nectar concentration (37%). (b) Blood flow. Histogram showing the distribution of observed red blood cell concentrations (haematocrit) from 57 vertebrate species [4]. Normalized oxygen flow J_r/J_r,max (solid line, equation (3.3)) and blood viscosity μ/μ₀ (dashed line, see appendix B) are plotted as a function of haematocrit . Flow is predicted to be maximum when c̃_opt = 39%, in good agreement with the observed average haematocrit (40%). (c) Sugar transport in plants. Histogram showing the distribution of observed sugar concentrations from 28 plant species that use active sugar loading [15]. Normalized sugar flow J_p/J_p,max (solid line, equation (3.4)) and sap viscosity μ/μ₀ (dashed line, data from [14]) are plotted as a function of phloem sugar concentration . Mass flow is predicted to be at a maximum when c̄_opt = 24%, in good agreement with the observed average sugar concentration (22%). (d) Nectar drinking by viscous dipping. Histogram showing the distribution of observed sugar concentrations that maximize nectar uptake for six insect species that use viscous dipping [9,13]. Normalized sugar mass flow J_v/J_v,max (solid line) and nectar viscosity μ/μ₀ (dashed line, data from [14]) are plotted as a function of nectar sugar concentration . Mass flow is predicted to be at a maximum when c̄_opt = 57%, in good agreement with the observed average nectar concentration (55%). In (a)–(d), the numbers given above the bins (coloured bars) indicate the percentage of species in the bin. The experimental data are available in the electronic supplementary material.

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 J_v, 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 J_v = 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 v_max and the vehicle flux is proportional to density J_v ≃ ρv_max. 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 J_v. 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 [23]. From these considerations, one anticipates an optimal vehicle density ρ_opt that maximizes the vehicle flux.

To estimate ρ_opt, 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 [27], 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(ρ_max/ρ), where ρ_max is the density at which traffic stops owing to congestion. The vehicle flow rate J_v = 𝒞ρ ln(ρ_max/ρ) is at a maximum when ρ = ρ_opt = ρ_max/e, and the constant 𝒞 = v(ρ_opt) is the vehicle speed at the optimal concentration. Because vehicles typically occupy 7.5 m in a totally congested flow [23], we estimate that ρ_max ≃ 133 vehicles per kilometre. Greenberg's model overestimates the optimal density, predicting ρ_opt = ρ_max/e ∼ 50 vehicles per kilometre, whereas the true value is known to be ∼ 20 vehicles per kilometre. Nevertheless, the vehicle flow rate J_v is qualitatively consistent with empirical traffic data (figure 2). The data are plotted as a function of vehicle concentration c = ρ/ρ_max in figure 2 along with Greenberg's flow rate J_v, deduced using ρ_max = 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/ρ_max → 0) = v_max 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) [28] suggested a traffic model in which the vehicle speed depends on the distance from the car in front, Δx. This leads to v = v_maxtanh(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) = v_maxtanh[(1/ρ − L)/s], in which case the flow rate J_v = vρ is optimized when ρ = 0.21 and ρ_max = 28 vehicles per kilometre. With v_max = 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).

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Figure 2.

Optimal vehicle concentration for maximizing traffic flow. Grey dots show measured vehicle flow rate J_v plotted as a function of vehicle concentration c = ρ/ρ_opt, where ρ_opt = 133 vehicles per kilometre. The flow rate is normalized by 1483 vehicles per hour, which corresponds to J_v(ρ_opt) = J_v,max in Bando, Hasebe and Nakayama's (BHN) model [28]. Histograms show the states occupied by the system in the morning (green, 06.00–08.00) and evening (blue, 16.00–18.00) rush-hour traffic. The data were collected by the Minnesota Department of Transportation from a sensor on the westbound direction of I-94 (Minneapolis, MN, USA) on Fridays (7, 14, 21, 28) in September 2012 [29]. The predicted vehicle transport rate J_v/J_v,max (thick solid black line, BHN model; thin solid red line, Greenberg's model) and traffic impedance μ/μ₀ (dashed line, BHN model) are plotted as a function of vehicle concentration c. The experimental data are available in the electronic supplementary material.

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 J_v. 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 μ_opt = 2^αμ₀, 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. η_opt = 2η₀, in good agreement with observed values (table 2).

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Figure 3.

Universal properties of biological and engineered flows. (a) Normalized flow rate J* = J(c)/J(c_opt) plotted as a function of normalized concentration c* = c/c_opt. The solid thick black line shows the prediction of equation (2.4). (b) Normalized impedance μ* = μ(c)/μ(c_opt) plotted as a function of normalized concentration c*. The solid and dashed thick black lines show the predictions of equation (2.6). The inset indicates the dependence of (μ*)^1−γ on c*.

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 μ_opt = 2^αμ₀ 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; [30]), and the macro-alga Chara where streaming distributes the content of the cell cytosol (observed viscosity: ∼3η₀ [31]). 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 v_max, 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 μ_opt = 2 μ₀. This is in rough accord with the BHN model that yields μ_opt = 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.