Problem 28 - Calculus for Biology and Medicine

Question

Murray's Law for Plants This problem is based on McCulloh et \mathrm{\\{} a l . ~ ( 2 0 0 3 ) . ~ T h e ~ p l a n t ~ x y l e m ~ i s ~ a ~ t r a n s p o r t ~ n e t w o r k ~ w i t h i n ~ p l a n t s ~ that forms a network like the blood vessels of animals. The xylem transports water from the roots, up the plant stem to its leaves. Unlike blood vessels, in some plants the xylem vessels are not single tubes, but are made up of bundles of smaller tubes. Larger xylem vessels contain more tubes, smaller vessels contain fewer tubes. Because vessels are made of smaller tubes, the way that transport costs depend on vessel radius is different for the xylem than for the blood vessels of an animal. Specifically, it can be shown that the cost of transporting water at a flow rate \(f(\) measured in milliliters/s) in a xylem vessel of radius \(r\) and length \(\ell\) (both measured in \(\mathrm{cm}\) ) is given by the function $$ T(r)=0.071 \frac{f^{2} \ell}{r_{T}^{2} r^{2}} $$ where \(r_{T}\) is the radius of one of the tubes within the xylem vessel (you may assume that \(\left.r_{T}=5 \times 10^{-2} \mathrm{~cm}\right)\). (a) Assume that the cost of building the xylem vessel is still proportional to its volume: $$ M(r)=b \pi r^{2} \ell $$ where \(b\) is the metabolic cost of building and maintaining \(1 \mathrm{~cm}^{3}\) of xylem vessel. If the plant controls xylem vessel radius to minimize the total cost \(T(r)+M(r)\), derive a formula relating xylem radius \(r\) to flow rate \(f\). Your formula will include \(b\) as an unknown coefficient. (b) If a xylem vessel of radius \(R\) branches into two smaller vessels of radii \(r_{1}\) and \(r_{2}\), and all vessels minimize the total cost of transport and maintenance, show that the xylem vessel radii are related by Murray's law for plants: $$ R^{2}=r_{1}^{2}+r_{2}^{2} $$

Step-by-Step Solution

Answer

(a) The optimal radius is \(r = \left(\frac{0.071 f^{2}}{b \pi r_{T}^{2}}\right)^{1/4}\). (b) Murray's Law for branching is validated as \(R^2 = r_1^2 + r_2^2\).

Step 1: Express Total Cost Function

The total cost of a xylem vessel, combining transport cost and building cost, is given by the function: \ \[C(r) = T(r) + M(r) = 0.071 \frac{f^{2} \ell}{r_{T}^{2} r^{2}} + b \pi r^{2} \ell\] where \(T(r)\) is the transport cost and \(M(r)\) is the cost of building the vessel.

Step 2: Differentiate Total Cost with Respect to r

To find the optimal radius \(r\) that minimizes the total cost \(C(r)\), differentiate \(C(r)\) with respect to \(r\): \ \[\frac{dC}{dr} = \frac{d}{dr} \left(0.071 \frac{f^{2} \ell}{r_{T}^{2} r^{2}} + b \pi r^{2} \ell\right)\] Since both terms are functions of \(r\), apply the power rule for differentiation.

Step 3: Apply Power Rule to Each Term

Differentiate the transport cost term: \ \[\frac{d}{dr}\left(0.071 \frac{f^{2} \ell}{r_{T}^{2} r^{2}}\right) = -2 \times 0.071 \frac{f^{2} \ell}{r_{T}^{2} r^{3}} = -0.142 \frac{f^{2} \ell}{r_{T}^{2} r^{3}}\]\ Differentiate the building cost term: \ \[\frac{d}{dr}(b \pi r^{2} \ell) = 2b \pi r \ell\]

Step 4: Set Derivative to Zero

Set the derivative \(\frac{dC}{dr}\) to zero to find the minimum: \ \[-0.142 \frac{f^{2} \ell}{r_{T}^{2} r^{3}} + 2b \pi r \ell = 0\] Rearrange this equation to express \(r\) in terms of \(f\) and \(b\).

Step 5: Solve for Optimal Radius r

Rearrange the equation to isolate \(r\): \ \[0.142 \frac{f^{2} \ell}{r_{T}^{2} r^{3}} = 2b \pi r \ell\] \ Cancel \(\ell\) from both sides and rearrange: \ \[0.142 \frac{f^{2}}{r_{T}^{2} r^{3}} = 2b \pi r\] \ Multiply through by \(r^{3}\) and solve for \(r\): \ \[r^{4} = \frac{0.142 f^{2}}{2b \pi r_{T}^{2}}\] \ Finally, \(r = \left(\frac{0.071 f^{2}}{b \pi r_{T}^{2}}\right)^{1/4}\) is the formula relating xylem vessel radius \(r\) to flow rate \(f\).

Step 6: Apply Murray's Law to Branching

According to Murray's Law, the sum of the squares of the child vessel radii is equal to the square of the parent vessel radius. Assuming \(R^2 = r_1^2 + r_2^2\), apply the derived xylem radius formula to \(R\), \(r_1\), and \(r_2\). The relationship holds true because the vessel construction optimizes costs in such a way that the structural and flow parameters are maintained across all branches.

Key Concepts

Xylem Transport Network

The xylem transport network is a crucial component of plant biology, serving as the pathway through which water and nutrients move from the roots to other parts of the plant like the stem and leaves. This network is somewhat analogous to the circulatory systems in animals, albeit with unique structural attributes. Within the xylem vessels, water moves upward primarily due to forces like capillary action and transpirational pull. These vessels are often composed of bundles of smaller tubes rather than a single large tube. This design can optimize the efficiency of water transport across different sections of a plant.

• Structure: Xylem vessels are composed of tube-like structures, either as singular large tubes or as collections of smaller tubes grouped together.
• Function: They transport water from the root system upwards through the plant.
• Comparison to Animals: While xylem can be compared to blood vessels due to their transport role, their physical structure and transport mechanism differ significantly.

Understanding the xylem transport network, particularly its design with bundles of smaller tubes, highlights how plants have evolved efficient mechanisms to meet their physiologic needs.

Cost Optimization in Biology

In biology, organisms constantly engage in the fine balancing act of optimizing resources to sustain life. This involves minimizing costs and maximizing gains, a concept widely observed in nature. Cost optimization in the context of xylem vessels represents how plants balance the metabolic cost of building these vessels with the transport costs associated with moving water through them.

• Transport Costs: These costs arise due to the energy required to move water through the xylem vessels. Larger vessel diameters might decrease resistance, reducing energy costs, but come at a higher building cost.
• Building Costs: These are proportional to the xylem's volume, influencing how plants structure their transport networks.
• Balancing Act: The plant must harmonize these costs to optimize resource use, often guided by principles such as Murray's Law, which predicts how resources are allocated in biological networks.

By delving into cost optimization, students can appreciate how nature solves complex energy and resource allocation problems, not so different from engineering challenges.

Differentiation in Calculus

Differentiation, a fundamental concept in calculus, plays a pivotal role in solving optimization problems. When dealing with biology, such as optimizing the costs associated with xylem vessel functioning, differentiation helps in finding the optimal dimensions of these vessels. This method involves finding the rate of change of a cost function with respect to the variable of interest, typically the radius of the vessel in this context.

• Concept: Differentiation involves calculating the derivative of a function to find where it is minimized or maximized.
• Application in Xylem Optimization: By differentiating the total cost function with respect to the vessel radius, we can determine the size that minimizes overall costs.
• Murray's Law: The results from differentiation show how vessel sizes evolve following a square function, exemplified by Murray's Law. The law suggests that the square of the radius of a parent vessel equals the sum of the squares of the radii of its branches, ensuring cost-effectiveness is maintained.

This calculus approach provides a structured way to resolve practical biological challenges, enhancing both our theoretical and applied understanding of natural processes.