Microbe wars

Hannah Woods

17 November 2025

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The word origami comes from the Japanese ori (to fold) and kami (paper). From its origins in Japan, origami has surged in popularity; according to the Guinness World Records, the largest paper crane has a wingspan of 81.94m and the largest display of origami hearts is 3,917,805—even as a mathematician, I wouldn’t want to count all of those!

You may have even tried origami yourself. Perhaps you’ve impressed your friends at dinner with a carefully-folded napkin, or used one of your lecturer’s problem sheets to make a paper plane. However, I can bet one thing you haven’t tried to make your origami from is DNA.

Antimicrobial resistance is the developing resistance of bacteria and microbes to antimicrobial treatments. It is problematic as it could result in antibiotic treatments being less effective. Fortunately, not so long ago in a galaxy not that far away, scientists began experimenting with DNA nanotechnologies for targeted drug delivery. Long strands of DNA can be folded into ‘origami’ with specific shapes suited for direct delivery of the drug to the bacteria.

For example, in one experiment, origami were engineered with wells that carried compounds of an antimicrobial enzyme called lysozyme. These were surrounded by sticky filaments that attached to microbes, ensuring the enzyme was delivered where it was needed. The origami is folded into structures called origami frames.

In this experiment, red E. coli is surrounded by green DNA origami. Image: Ioanna Mela.

DNA is an example of a filament bundle. Filament bundles are abundant in the body and are crucial for both structure and motility. The eyes with which you are reading this article transmit information to the brain through the optic nerve bundle; as you go to reach for a sip of your tea you are using muscles in your arm composed of collagen bundles. Even when you are sitting completely still, neurons in the brain and filaments involved in cell division are busy at work!

Mathematical modelling can play an important role in supporting experimental advancements of this nature, improving our understanding of the dynamics of these structures in vivo. Having said that, biology is complicated and one of the challenges in building a mathematical model is trying our best to keep enough biological detail for realistic simulations, but not too much as to make the model overcomplicated or too computationally costly. This is where an unlikely hero joins our story: energy minimisation.

Bundles within the body are subject to various forces, and under extreme forcing they can deform. For effective drug delivery, it is crucial that the origami frame remains bound to the bacterial membrane; bacteria are less likely to survive direct application of the drug, than if the drug were freely administered into the surrounding fluid. Filament models are one way in which we can build an understanding of how the origami frame responds to forcing, and the use of energy minimisation will help us keep the model under computational wraps.

Join me as we venture through the saga to start your training in filament modelling. Don’t miss the Easter egg on page 45 of the magazine to master making your own origami heart!

Modelling a filament

The first episode in our mission to mathematically model these filaments is to simplify the problem. As with all good models, we make something super simple before adding the complications. One way in which to do this is by considering relevant length scales. The length of a filament is much greater than its radius (with a difference spanning nine orders of magnitude). This gives us a good reason to model it as a one-dimensional rod, which we take to be an open, discrete curve. For computational simplicity we then discretise the curve into $n$ segments, like this:

A discrete rod; point $j$ on rod $i$ is indexed by $ij$.

If we so wish, we could take $n$ to some ridiculously large number (but not quite infinity) and create something that almost resembles a realistic filament. This would, however, come with computational costs. In my own research I’ve taken $n$ to be 100. We further assume the rod is inextensible and elastic; each segment has fixed length and the rod returns to a straight configuration when force is removed. Each segment is also described by two spherical polar angles so we know which direction in 3D space it is pointing in, like this:

For a bundle of $m$ filaments, we therefore have $2mn$ of these angles, describing the shape of the bundle.

The force is strong with this… bundle

Under extreme forcing a filament will deform, and the most common ways this can be done is via bending or twisting (pictured below). For us to describe this mathematically we turn to our good friends in differential geometry.

Filaments deform either by bending or twisting

Curvature and twist tell us how a continuous curve is bending and twisting—but our discrete curve is made of straight lines! For this reason, we need discrete analogues. One way to describe curvature is to draw a circle that is the best possible fit at a point on a continuous curve—like the one here:

We then define the curvature $\kappa$ as $\kappa = r^{-1}$. Essentially, curvature is telling us the best ‘circular approximation’ of the curve locally. Again, our problem lies within the fact that our rod is made of straight edges! Not to worry, though; we can still define discrete curvature at vertices and use a `circle of best fit’ to achieve our goal:

Defining twist is a little harder as discrete curves do not have nice properties of continuity and differentiability. What we can do is assign a separate frame of reference, called a material frame, to each segment of the curve. Discrete twist $\gamma_j$ is then the difference in angular rotation of material frames between neighbouring segments. It can be thought of as a `rate of rotation’ of the material frame about its tangent vector. Here’s a diagram:

Energy minimisation

Nature is lazy, in that it will always choose the path of least resistance. If you have ever tried to bend or twist a tube you will have noticed that it resists this motion—this is because it is energetically costly for the tube to be bent or twisted. For this reason we are motivated to factor in energy minimisation to model our bundle.

The way it works is that we first assign a so-called energy function to the bundle, which takes different values depending on the bundle shape, and then minimise this function using calculus techniques. We construct this energy function as a sum of energy `penalties’. Each penalty corresponds to a physical behaviour that is energetically costly, ie bend and twist.

The penalty will be larger the more a bundle exhibits such behaviour, since the actions of bending and twisting the tubes add stress and strain. We should add a further penalty to represent applied loads, and an interaction penalty to prevent unrealistic self-intersection of filaments, so our energy function is $$E = E_{\text{bend}} + E_{\text{twist}} + E_{\text{load}} + E_{\text{interaction}}.$$ A minimum of the energy function, $E$, will be a balance of these energy terms. For example, a straight rod with applied force has higher loading energy and smaller bending energy than a bent rod.

For the remainder of this article, though, we will focus on bending and twisting: these are elastic actions.

“I am your father”

One of the greatest cinematic twists of all time! Although not quite the twist we are talking about here. Let’s take a closer look at constructing the elastic energy penalties.

Bundles in the body are embedded in tissue or surrounded by fluid, and unless something goes very wrong they won’t bend beyond $\pi/2$. It is up to us to ‘tell’ the model that filaments shouldn’t bend this far in sensible parameter regimes. Suppose we define the angle between rods as $\alpha$, like this:

Then one way to stop the filament bending too far is by making the penalty very large when $\alpha\leq\pi/2$. Let’s introduce a bending coefficient $B$ to directly control the resistance of a rod to deformation. We choose a quadratic energy term, $B\kappa^2$, so that the bending energy grows sufficiently fast to penalise extreme deformation:

Sensitivity to bending can be controlled by the bending coefficient $B$, for example, here I make it 10 times larger:

Discrete curvature is defined at each interior vertex. Summing over all interior vertices gives the bending penalty for the rod. In other words, the bend energy depends on the curvature $\kappa$, $$E_{\text{bend}} = B\sum_{j=1}^{n-1} \kappa_j^2,$$ and importantly, we can directly evaluate $\kappa$ from the spherical angles we introduced earlier.

The twist penalty is constructed using a similar argument. We are always looking for minima of the energy functions. A technique widely used in mathematical modelling to do this is via Lagrange multipliers, but we won’t go into detail here. The important point to note is that the twisting energy will automatically be minimised when each segment has a uniform twist, $\gamma$. This simplifies the penalty as we now need just one variable to calculate the penalty, rather than knowledge of a set of $\gamma_j$s, $$E_{\text{twist}} = t_c \sum_{j=1}^n \gamma_j^2 = t_c n \gamma^2,$$ for a twist coefficient $t_c$.

Although we have constructed this twist penalty with safe arguments, it is unclear how we can evaluate it, since unlike $\kappa$, we don’t have a direct way of calculating $\gamma$. We are keeping track of the shapes of the filaments, but two filaments can have the same shape, but a different amount of twist.

Enter a new hero to this tale—the Călugăreanu theorem! This says that the bending (or writhe as Călugăreanu labels it) and twist of a curve always sum to a constant for any given filament. As the filament deforms, this constant remains the same. Mathematically, we can say $$\mathcal{T} + \mathcal{W} = \mathcal{L},$$ where $\mathcal{T} = n \gamma/2 \pi$ is the total twist of a filament, $\mathcal{W}$ is the polar writhe, and the constant $\mathcal{L}$ is called the net winding. To evaluate the twist energy $E_{\text{twist}}$ it is sufficient to rearrange the above equation for $\gamma$: we prescribe $\mathcal{L}$, so we just need to calculate the writhe!

A Jedi’s guide to net winding

Here we are merely scratching the surface of some deep topological relations between curve behaviours. ‘Net winding’, $\mathcal{L}$, is an invariant of a curve, and remains constant as a curve deforms. It can be thought of as the number of turns in a straight filament. We can control this value of net winding in simulations. Importantly, net winding changes the elastic behaviour of a curve.

The tubes here have the same writhe (shape) but different amounts of twist. This is a result of each tube having a different associated net winding.

Spot the difference! Image: Naina Praveen/Chris Prior.

However, before we hit the hyperdrive to celebrate our victory, the enemy have a secret weapon.

The special relationship we are talking about is a tantrix: a continuous set of tangents along a curve that map out a continuous curve on the sphere of radius 1.

How to construct a tantrix: If you look at the purple knot on the left, you’ll see we’ve drawn a green tangent at a point. We can draw that tangent vector so its tip is on the unit sphere, as in the central image. If we trace the tip of this arrow as we go round the purple knot, we get the tantrix, the yellow curve, on the right. Image: Naina Praveen/Chris Prior.

In the case of a discrete rod, we have a defined set of $n$ tangents, which are then each mapped to a point on the sphere, joining points by shortest paths as we go. With these few adjustments to account for the fact our curve is open and discrete, we can use a relation between the writhe and surface area bounded by the tantrix of a discrete curve, $$\mathcal{W} = \frac{\mathcal{A}}{2 \pi}.$$ With this, we can then evaluate and minimise the energy function. Thankfully, much of the minimisation process can be handled by a computer—at each stage of minimisation, the computer calculates gradients of the energy function and uses these to ‘update’ the parameter set.

The computer runs until the gradients are approximately zero (ie when we have reached a minimum). The details of gradient calculations are quite tedious, but the important thing is we need to be able to calculate $\gamma$. We therefore have our full recipe:

1. Find the tantrix for our filament and calculate the spherical surface area $\mathcal{A}$ that it bounds.
2. Calculate the total twist using $$\mathcal{T} = \mathcal{L}-\frac{\mathcal{A}}{2\pi}.$$
3. Determine the twist, $$\gamma = \frac{2\pi\mathcal{T}}{n}.$$
4. Evaluate the energy gradients to minimise.

Simulations of a hexagonal bundle using different values of net winding: from top to bottom, $\mathcal{L} = 0$, $1.5$ and $2.5$. The bundle is more twisted for higher winding values.

The mathematician strikes back

Kirchhoff rod theory, developed by German physicist Gustav Kirchhoff, models continuous slender bodies. What we have actually done so far is define a discrete approximation of the energy functional used in the continuous case. The discrete approach has the advantage of being very flexible. New, often more realistic, complications can easily be added to the model by adding new penalties to the energy function and minimising as before.

One of the more simple first attempts at adding complications is to connect the ends of filaments in the model by Hookean springs: those where the exerted force is proportional to its displacement from equilibrium. A Hookean energy term penalises stretching and compression of the springs, representing the high energetic cost of stretching the bacterial membrane, and is a relevant energy penalty to include due to the observation that filaments attach the DNA origami to a somewhat `stretchy’ bacterial membrane. If we imagine a setup like this:

then the new energy term is $$E_{\text{membrane}} = \sum_{j=1}^{m-1} k \left(\mathit{\Delta}_j-\mathit{\Delta}_0 \right)^2.$$ Here, $\mathit{\Delta}_0$ is the equidistant distance between endpoints at rest, $\mathit{\Delta}_j$ is the distance between endpoints $\boldsymbol{p}_j$ and $\boldsymbol{p}_{j+1}$, and $k$ is the spring constant. Energy is proportional to the square of extension in accordance with Hooke’s law.

By including further energy penalties we can build up a more complex and realistic model. Mathematical modelling opens the door to exploring complex behaviours of objects in our physical and biological world, complementing experimental efforts. DNA origami is our Jedi master in the fight against antibiotic resistance!

Hannah Woods

Hannah has just started a PhD in mathematical biology and fluid mechanics at the University of Oxford. She enjoys running and ambitious baking. Last Christmas, she took on a gingerbread construction of the Durham University science site with friends. (The very next day, she gave it away.)

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