Microscopes

The microscope is one of the most widely used instruments in science. Microscopy is a huge subject, and I am definitely not an expert. Russ Hobbie and I talk about the microscope only briefly in the 4th edition of Intermediate Physics for Medicine and Biology. In Chapter 14 (Atoms and Light) we give a series of homework problems about lenses. Problem 43 considers the case of an object placed just outside the focal point of a converging lens. The resulting image is real, inverted and magnified (a slide projector, for those of you old enough to remember such things). In Problem 44, the object is just inside the focal point of the lens. The image is virtual, upright, and magnified (a magnifying glass). Then in Problem 45 we put these two lenses together, first a slide projector casting an intermediate image, then a magnifying glass to view that image; a compound microscope. Our discussion is useful, but very simple.

Nowadays, microscopes are extremely complicated, and can do all sorts of wonderful things. Our simple example is nearly obsolete, because almost no one looks through the second lens (the eyepiece) to view the image anymore. Rather, the image produced by the first lens (the objective) is recorded digitally, and one looks at it on a computer screen. I could spend the rest of this blog entry describing the complexities of microscopes, but I want to go in another direction. Can a student build a simple yet modern microscope?

They can, and it makes a marvelous upper-level physics laboratory project. The proof is given by Jennifer Ross of the University of Massachusetts Amherst. In a preprint at her website, Ross describes a microscope project for undergraduates. The abstract reads:

Optics is an important subfield of physics required for instrument design and used in a variety of other disciplines, including materials science, physics, and life sciences such as developmental biology and cell biology. It is important to educate students from a variety of disciplines and backgrounds in the basics of optics in order to train the next generation of interdisciplinary researchers and instrumentalists who will push the boundaries of discovery. In this paper, we present an experimental system developed to teach students in the basics of geometric optics, including ray and wave optics. The students learn these concepts through designing, building, and testing a home-built light microscope made from component parts. We describe the experimental equipment and basic measurements students can perform to learn principles, technique, accuracy, and resolution of measurement. Students find the magnification and test the resolution of the microscope system they build. The system is open and versatile to allow advanced building projects, such as epi-fluorescence, total internal reflection fluorescence, and optical trapping. We have used this equipment in an optics course, an advanced laboratory course, and graduate-level training modules.

This fascinating paper then goes on to describe many aspects of microscope design.

The light source was a white light emitting diode (LED)… We chose inexpensive but small and powerful CMOS cameras to capture images with a USB link to a student’s laptop….The condenser designs of students are the most variable and interesting part of the microscope design. Students in prior years have used one, two, or three lenses to create evenly illuminated light on the sample plane…After creating the condenser, students next have to use an objective to create an image onto the CMOS camera chip.

The equipment is not terribly expensive compared to buying a microscope, but it’s not cheap: each microscope costs about $3000 to build, which means for a team of three students the cost is $1000 per person. But the leaning is tremendous, and Ross suggests that you can scavenge used parts to reduce the cost.

But perhaps even this student-built $3000 microscope is too complicated and expensive for you. Can we go simpler and cheaper? Yes! Consider “foldscope.” The website of foldscope’s inventors says (my italics)

We are a research team at PrakashLab at Stanford University, focused on democratizing science by developing scientific tools that can scale up to match problems in global health and science education. Here we describe Foldscope, a new approach for mass manufacturing of optical microscopes that are printed-and-folded from a single flat sheet of paper, akin to Origami….Although it costs less than a dollar in parts, it can provide over 2,000X magnification with sub-micron resolution (800 nm), weighs less than two nickels (8.8 g), is small enough to fit in a pocket (70 × 20 × 2 mm³), requires no external power, and can survive being dropped from a 3-story building or stepped on by a person. Its minimalistic, scalable design is inherently application-specific instead of general-purpose gearing towards applications in global health, field based citizen science and K12-science education.

Details are described in a preprint available at http://arxiv.org/abs/1403.1211. Also, listen to Manu Prakash give a TED talk about foldscope. The goal is to provide “a microscope for every child.” I think Prakash and his team means EVERY child (as in every single child in the whole wide world).

The Airy Disk

I hate to find errors in the 4th edition of Intermediate Physics for Medicine and Biology. When we do find any, Russ Hobbie and I let our readers know through an errata, published on the book’s website. Last week, I found another error, and it’s a particularly annoying one. First, let me tell you the error, and then I’ll fill in the backstory.

In the errata, you will now find this entry:

Page 338: In Chapter 12, Problem 10. The final equation, a Bessel function integral, should be

Error found 6-10-14.

In the 4th edition, we left out the leading factor of “u” on the right-hand-side. Why does this bother me so much? In part, because Problem 10 is about a famous and important calculation. Chapter 12 is about imaging, and Problem 10 asks the reader to calculate the two-dimensional Fourier transform of the “top hat,” function equal to 1 for r less than a (a circular disk), and zero otherwise. This Fourier transform is, to within a constant factor, equal to J₁(u)/u, where J₁ is a Bessel function and u = ka, with k being the magnitude of the spatial frequency. This function is known as the “Airy pattern” or “Airy disk.” The picture below shows what the Airy disk looks like when plotted versus spatial frequencies k_x and k_y:

The Airy disk.

A picture of the square of this function is shown in Fig. 12.1 of IPMB. If you make a smaller, so the “top hat” is narrower, then in frequency space the Airy disk spreads out. Conversely, if you make a larger, so the “top hat” is wider, then in frequency space the Airy disk is more localized. The Bessel function oscillates, passing through zero many times. Qualitatively, J₁(u)/u looks similar to the more familiar sinc function, sin(ka)/ka. (The sinc function appears in the Fourier transform of a rectangular “top hat” function).

The Airy disk plays a particularly important role in diffraction, a topic only marginally discussed in IPMB. Interestingly, diffraction isn’t important enough in our book even to make the index. We do mention it briefly in Chapter 13

One property of waves is that diffraction limits our ability to produce an image. Only objects larger than or approximately equal to the wavelength can be imaged effectively. This property is what limits light microscopes (using electromagnetic waves to form an image) to resolutions equal to about the wavelength of visible light, 500 nm.

We don’t talk at all about Fourier optics in IPMB. When light passes through an aperture, the image formed by Fraunhofer diffraction is the Fourier transform of the aperture function. So, for instance, when light passes through the objective lens of a microscope (or some other aperture in the optical path), the aperture function is the top hat function: all the light passes through at radii less than the radius of the lens, and no light passes through at larger radii. So the image formed by the lens of a point object (to the extent that the assumptions underlying Fraunhofer diffraction apply) is the Airy disk. Instead of a point image, you get a little blur.

Suppose you are trying to image two point objects. After diffraction, the image is two Airy disks. Can you resolve them as two separate objects? It depends on the extent of the overlap of the little blurs. Typically one uses the Rayleigh criterion to answer this question. If the two Airy disks are separated by at least the distance from the center of one Airy disk to its first zero, then the two objects are considered resolved. This is, admittedly, an arbitrary definition, but is entirely reasonable and provides a quantitative meaning to the vague term “resolved.” Thus, the imaging resolution of a microscope is determined by the zeros of the J₁ Bessel function, which I find pretty neat. (I love Bessel functions).

So, you see, when I realized our homework problem had a typo and it meant the student would calculate the Airy disk incorrectly, my heart sunk. To any students who got fooled by this problem, I apologize. Mea culpa. It makes me all the more determined to keep errors out of the upcoming 5th edition, which Russ and I are working on feverishly.

On the lighter side, when I run into scientists I am not familiar with, I often look them up in Asimov’s Biographical Encyclopedia of Science and Technology. When I looked up George Biddell Airy (1801–1892), Astronomer Royal of the Greenwich Observatory, I was shocked. Asimov writes “he was a conceited, envious, small-minded man and ran the observatory like a petty tyrant.” Oh Myyy!

Physics Research & Education: The Complex Intersection of Biology and Physics

This morning, I am heading home after a productive week at a Gordon Research Conference about “Physics Research and Education: The Complex Intersection of Biology and Physics.” I wish I could tell you more about it, but Gordon Conferences have this policy…

To encourage open communication, each member of a Conference agrees that any information presented at a Gordon Research Conference, whether in a formal talk, poster session, or discussion, is a private communication from the individual making the contribution and is presented with the restriction that such information is not for public use….

So, there is little I can say, other than to point you to the meeting schedule published on the GRC website. I suspect that future blog entries will be influenced by what I learned this week, but I will only write about items that have also been published elsewhere.

 I can say a bit about Gordon Conferences in general. The GRC website states

The Gordon Research Conferences were initiated by Dr. Neil E. Gordon, of the Johns Hopkins University, who recognized in the late 1920s the difficulty in establishing good, direct communication between scientists, whether working in the same subject area or in interdisciplinary research. The Gordon Research Conferences promote discussions and the free exchange of ideas at the research frontiers of the biological, chemical and physical sciences. Scientists with common professional interests come together for a full week of intense discussion and examination of the most advanced aspects of their field. These Conferences provide a valuable means of disseminating information and ideas in a way that cannot be achieved through the usual channels of communication—publications and presentations at large scientific meetings.

Before this, the only Gordon Conference I ever attended was one at which I was the trailing spouse. My wife studied the interaction of lasers with tissue in graduate school, and she attended a Gordon Conference on that topic in the 1980s; I tagged along. I don’t remember that conference being as intense as this one, but maybe that’s because I’m getting older.

The conference was at Mount Holyoke College, a small liberal arts college in South Hadley, Massachusetts, about 90 minutes west of Boston. It is a lovely venue, and we were treated well. I hadn’t lived in a dormitory since college, but I managed to get used to it.

For those of you interested in education at the intersection of physics and biology—a topic of interest for readers of the 4th edition of Intermediate Physics for Medicine and Biology—I suggest you take a look at the recent special issue of the American Journal of Physics about “Research and Education at the Crossroads of Biology and Physics,” discussed in this blog before. In addition, see the website set up based on the “Conference on Introductory Physics for the Life Sciences,” held March 14–16, 2014 in Arlington, Virginia. I’ve also discussed the movement to improve introductory physics classes for students in the life sciences previously in this blog here, here, here, and here.

Now, I need to run so I can catch my plane….

Plant Physics

Perhaps the 4th edition of Intermediate Physics for Medicine and Biology should have a different title. It really should be Intermediate Physics for Medicine and Zoology. Russ Hobbie and I talk a lot about the physics of animals, but not much about plants. There is little botany in our book. This is not completely true. Homework Problem 34 in Chapter 1 (Mechanics) analyzes the ascent of sap in trees, and we briefly mention photosynthesis in Chapter 3 (Systems of Many Particles). I suppose our discussion of Robert Brown’s observation of the random motion of pollen particles counts as botany, but just barely. Chapter 8 (Biomagnetism) is surprisingly rich in plant examples, with both magnetotactic and biomagnetic signals from algae. But on the whole, our book talks about the physics of animals, and especially humans. I mean, really, who cares about plants?

Plant Physics, by
Karl Niklas and Hans-Christof Spatz.

Guess what? Some people care very much about plants! Karl Niklas and Hanns-Christof Spatz have written a book titled Plant Physics. What is it about? In many ways, it is IPMB redone with only plant examples. Their preface states

This book has two interweaving themes—one that emphasizes plant biology and another that emphasizes physics. For this reason, we have called it Plant Physics. The basic thesis of our book is simple: plants cannot be fully understood without examining how physical forces and processes influence their growth, development, reproduction, and evolution….This book explores…many…insights that emerge when plants are studied with the aid of physics, mathematics, engineering, and chemistry. Much of this exploration dwells on the discipline known as solid mechanics because this has been the focus of much botanical research. However, Plant Physics is not a book about plant solid mechanics. It treats a wider range of phenomena that traditionally fall under the purview of physics, including fluid mechanics, electrophysiology, and optics. It also outlines the physics of physiological processes such as photosynthesis, phloem loading, and stomatal opening and closing.

The chapter titles in Plant Physics overlap with topics in IPMB, such as Chapter 4 (The Mechanical Behavior of Materials), Chapter 6 (Fluid Mechanics), and Chapter 7 (Plant Electrophysiology). I found the mathematical level of the book to be somewhat lower than IPMB, and probably closer to Denny’s Air and Water. (Interestingly, they did not cite Air and Water in their Section 2.3, Living in Water Versus Air, but they do cite another of Denny’s books, Biology and the Mechanics of the Wave-Swept Environment.) The differences between air and water plays a key role in plant life: “It is very possible that the colonization of land by plant life was propelled by the benefits of exchanging a blue and often turbid liquid for an essentially transparent mixture of gasses.” The book discusses diffusion, the Reynold’s number, chemical potential, Poiseuille flow, and light absorption. Chapter 3 is devoted to Plant Water Relations, and contains an example that serves as a model for how physics can play a role in biology. The opening and closing of stomata (“guard cells”) in leaves involves diffusion, osmotic pressure, feedback, mechanics, and optics. Fluid flow through both the xylem (transporting water from the roots to the leaves) and phloem (transporting photosynthetically produced molecules from the leaves to the rest of the plant) are discussed. Biomechanics plays a larger role in Plant Physics than in IPMB, and at the start of Chapter 4 the authors explain why.

The major premise of this book is that organisms cannot violate the fundamental laws of physics. A corollary to this premise is that organisms have evolved and adapted to mechanical forces in a manner consistent with the limits set by the mechanical properties of the materials out of which they are constructed…We see no better expression of these assertions that when we examine how the physical properties of different plant materials influence the mechanical behavior of plants.

Russ and I discuss Poisson’s ratio in a homework problem in Chapter 1. Niklas and Spatz give a nice example of how a large Poisson’s ratio can arise when a cylindrical cell has inextensible fibers in its cell wall that follow a spiral pattern. 

Values [of the Poisson’s ratio] can be very different [from isotropic materials] for composite biological materials such as most tissues, for which Poisson’s ratios greater than 1.0 can be found. A calculation presented in box 4.2 shows that in a sclerenchyma cell, in which practically inextensible cellulose microfibers provide the strengthening material in the cell wall, the Poisson’s ratio strongly depends on the microfibrillar angle; that is, the angle between fibers and the longitudinal axis of the cell.

Given my interest in bioelectric phenomena, I was especially curious about the chapter on Plant Electrophysiology (Chapter 7). The authors derive the Nernst-Planck equation, and the Goldman equation for the transmembrane potential. Interestingly, plants contain potassium and calcium ion channels, but no sodium channels. Many plants have cells that fire action potentials, but the role of the sodium channel for excitation is replaced by a calcium-dependent chloride channel. These are slowly propagating waves; Niklas and Spatz report conduction velocities of less than 0.1 m/s, compared to propagation in a large myelinated human axon, which can reach up to 100 m/s. Patch clamp recordings are more difficult in plant than in animal cells (plants have a cell wall in addition to a cell membrane). Particularly interesting to me were the gravisensitive currents in Lepidium sativum roots. The distribution of current is determined by the orientation of the root in a gravitational field.

Botanists need physics just as much as zoologists do. Plants are just one more path leading from physics to biology.

For those wanting to learn more, my colleague at Oakland University, Steffan Puwal, plans to offer a course in Plant Physics in the winter 2015 semester.