Intro to biological physics

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Sep 29, 2014 ... Biological Physics - part of 'Soft Matter and Biological Physics' module. ( PHY3040) ... Biological Physics, by Philip Nelson (Freeman). There are ...
Biological Physics - part of ‘Soft Matter and Biological Physics’ module (PHY3040) Introduction

October 4, 2016

All course notes, problem sets, and answers are/will be on the SurreyLearn ‘Soft Matter and Biological Physics’ module PHY3040. The textbook for this part of the module is: Biological Physics, by Philip Nelson (Freeman) There are copies of this in the library. Note that living organisms are made of soft matter. Our bodies are mostly made up of a liquid (water), polymers (DNA, RNA, proteins) and surfactant-type molecules (cells are surrounded by membranes made of this type of molecule). Thus, in the biological physics part of the course, we will be applying ideas and knowledge from the soft-matter part of the course. For example, in this part of the course we will see how an insect, the pond skater (aka water strider) exploits a soft-matter property: the surface tension of a liquid. Also, in the soft-matter part of the course, you will see examples of soft matter that is or was part of a living organism. In these, the first set of notes, I aim to give a flavour of biological physics, by using two simple examples.

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Introduction to Biological Physics

What is biological physics? It is the application of the basic laws of physics (e.g., statistical mechanics, quantum mechanics, 2nd Law of Thermodynamics, etc) and techniques (e.g., mathematical models like diffusion, soft matter physics, statistical physics, etc) to understanding living things, including ourselves. We all understand that just because we are alive, that does not mean that the laws of physics don’t apply to us. We know that gravity pulls us down, that conservation of momentum applies to us, and that quantum and statistical mechanics are needed to understand the atoms and molecules of which are made. So we can apply our understanding of the laws of physics to understand how living things, including ourselves work. In the London 2012 Olympics, Usain Bolt ran 100 m in 9.63 s. Clearly to do so he needed to accelerate rapidly by generating large forces, then sustain a large power output for the 9.63 s. We can use basic principles in physics to understanding how he did this. We can also use physics to understand disease, for example, cataracts are caused by phase separation in the protein mixtures in the cells of the lens of the eye.

1.1

Background biology

Biology is complex, with many many technical terms. This is a physics course, so I don’t expect you to memorise lots of biology terms, and I will try and keep the number of biology terms to a minimum. But I

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(A)

(B)

Figure 1: Images (from Wikimedia) of: (A) an elephant, and (B) a beetle (species = blister beetle). The elephant’s body is maybe 3 m long, its legs are approximately 40 cm thick, and its mass is few tons. The blister beetle is a relatively large beetle, its body is maybe 4 cm long, its legs are approximately 1 mm thick at most, and its mass is of order 1 g. will use some. I do not expect you to remember more than a handful of key ones for the exam. I will not examine you on biology terms. But you should know basics like have some idea of what DNA is, know that genes are made of DNA, etc. For background biology, you can use Wikipedia or a biology textbook, such as Molecular Biology of the Cell by Alberts et al.. There are many copies of this textbook in the library, and there are versions of this book online. I will keep the number of biology terms you need for the exam to a minimum but with a little bit of background reading into a little of the biology, I think you will get more out of the course.

1.2

Style of the course

In the rest of this set of introductory notes we will consider two simple examples of biological physics in action. The idea is to show how you can use basic physics principles to explain how some aspect of a living organism works, why it is the way it is, and what the laws of physics do and do not allow the living organism to do. The examples will also hopefully show the style of the course. This style will also be reflected in the exam. The style is simple estimates of the order of magnitude of important properties, and how these properties scale with, for example, the size of the organism. Most of the time we will either just estimate the order of magnitude or work to just one significant figure. You can do the same in the problems, and in the exam. We mostly only work to one significant figure of precission because we will be working with simple approximations, that are not accurate to several significant figures of precision, and so keeping those extra figures is useless1 . We work with simple approximations because we want to use simple physics laws and principles to get simple rough estimates, and don’t want to get bogged down doing complex maths calculations where the maths would obscure the basic physics. 1 Note that although for intermediate steps you can keep more figures in the numbers you carry, you should not give your answer with more than about one more significant figure than the accuracy of this answer. For example, if you estimate that a sphere has a radius of roughly 1 m, then giving its volume as (4π/3) × 13 = 4.1887902 m3 is very poor practice. Note that this implies that you should always have a rough idea of the accuracy of every answer you give.

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Introductory example problem 1: Why do elephants have fat legs?

Animals vary widely in size, some are too small to be seen with naked human eye, while others, like elephants, are huge. We are quite a big animal, in the sense that there are many more animals smaller than us, than there are animals larger than we are. The size of an animal matters. Large and small animals are different in a number of ways. Here we will see one way basic physics forces small and large animals to be different. Look at Fig. 1. It shows a large animal, the elephant, and a small animal, the beetle – and deliberately they are shown at close to the same size on the page. An elephant is roughly 2 m high at the shoulder and with mass of maybe 3,000 kg. The beetle is maybe 2 cm high at the shoulder, and has a mass of around 1 g. The two images are very different scales of course and it is only on the page that the two bodies of the two animals are roughly the same size. But note that when their bodies are shown at about the same lengths, the thicknesses of the elephant’s and the beetle’s legs are very different. In proportion to its body, the legs of the elephant are much thicker. This is generally true, the legs of large animals such as elephants, hippos, rhinos, large dinosaurs, etc, are thicker in relation to their body than are the legs of small animals, such as shrews, beetles, mice, etc. Here we are interested in understanding why this is.

2.1

Bones support the weight of animals’ bodies

The weight of animals like mice, elephants and ourselves is born by bones. When an elephant or we are standing, our weight acts as a compression force on our bones like the femurs2 in our legs. Bone is quite a complex material, and it does vary a bit from one type of bone to another, but roughly speaking, bones can be broken by stresses (i.e., force per unit cross-sectional area of the bone) of the order of tens of MPa. To give a good margin of safety let us say that when I am standing the stress in each femur should be no more 1 MPa. Of course, there should be a large margin between my weight and the force needed to break my bones. An animal needs bones that are large enough that when standing the stress on these bones is no more than around 1 MPa maximum stress while standing = 106 Pa (1) This stress is just the force of gravity on the animal divided by the cross-sectional area of the bones.

2.2

Are humans’ bones big enough and strong enough to support our weight?

My mass is around 80 kg, so the force of gravity is around 800 N, taking g = 10 m s−2 . And the stress per femur should be about 400 N/cross-section of each femur. Let us say that the cross-section of one of my femurs is 10 cm2 . Then the stress when I am standing is approximately 400/(10 × 10−4 ) = 400,000 Pa, or about half a MPa. So I am OK. But what about larger animals? Let’s think about an animal that is h m high. As the animal size h increases, the mass of the animal scales as its volume, which scales as h3 , while the cross-sectional area of its femurs scales as an area, as h2 . Thus as h3 is a faster increase than h2 , at some point the stress becomes too large for bone to support.

2.3

Scaling of leg thickness with animal size, for cubic animals

To get a rough idea how we expect the thickness of animals’ legs to scale with their size h, let us consider an animal that is roughly cubic. Of course animals are all sorts of shapes, but never perfect cubes, but we just want an estimate, so this is a sensible simplifying assumption. 2

The large bones in our thighs

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Figure 2: Artist’s impression (from Wikimedia) of a candidate to be the largest land animal that has ever lived, the Argentinosaurus dinosaur. Our best guess is that they could reach masses of around 60 to 100 tons and be around 30 m long. Recently (2014) it has been suggested that another dinosaur, Dreadnoughtus schrani, may be at least as big if not bigger. This is an active field of research and both animals are known only from incomplete fossilised skeletons. The mass of an animal is its mass density times its volume. All animals have a mass density close to that of water, because we are all mainly water. This mass density 1000 kg m−3 . So the mass of an animal h high is then about 1000h3 kg. With g = 10 m s−2 , this gives a force of gravity on an animal, i.e., its weight, weight of cubic animal ∼ 104 h3 N

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This force is distributed across the total cross-sectional area of the bones in the legs. Let us assume that the animal has 4 legs, and that each leg is w across and so has a cross-sectional area 2 w . If the bones occupy 10% of the cross-section of leg – the remainder is needed for muscles and nerves etcm, it cannot be solid bone – then the cross-sectional area of bone in a single leg is 0.1w2 . As the mass is 10000h3 N, then the stress on the bone is Stress =

3 weight 104 h3 5h = ∼ 0.3 × 10 Pa total bone cross-section area 4 × 0.1 × w2 w2

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If we equate this to the maximum stress bone can safely support, 106 Pa, and rearrange, we get w2 ∼

h3 30

w ∼ 0.2h3/2

or

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for the minimum led thickness needed to support the weight. To get the thickness of the legs relative to the thickness of the animal’s body we use the dimensionless ratio w/h, which is relative thickness of leg to body = 2.3.1

w ∼ 0.2h1/2 h

N.B. h in metres

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Predictions of our simple model

Now that we have a simple model that predicts how thick legs should be, let us see if its predictions are correct. For a small animal, like a beetle h = 1 cm = 0.01 m tall, then the legs can be very spindly. So, for a beetle w/h = 0.02, it has legs only 2% as wide as its body. But for larger animals like us, not only are our legs thicker in absolute terms, they are also thicker relative to our bodies. For us h = 1 m, and so the model predicts w/h = 0.2 and w = 20 cm. This is a reasonable estimate. It should be borne in mind that this is only a rough estimate, prediction is that our legs should be somwhere in the range of tens of cms, which is correct.

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Figure 3: Image (from Wikimedia) of a water strider, standing on water. Water striders, also called pond skaters, are common in ponds in the UK and all over the world. The image above was taken in India. The size of the body of a water strider is roughly 1 cm long. The mass of a water strider is about 0.01 g. Then for still larger animals, the equation predicts that for an animal 25 m high, its legs are as wide as its body, and so that this is the limit. Animals can’t be bigger than this. Again this is a reasonable estimate. The largest land animal3 ever to have lived is believed to be Argentinosaurus, a dinosaur that was around 40 m nose-to-tail and had a mass of about 110 tons. An artist’s impression of an Argentinosaurus is shown in Fig. 2. Note that the Argentinosaurus is a lot longer that it is high. Our assumption of a cubic animal is clearly a bit dodgey. We could do a more complicated calculation taking into account the animal’s shape to get a more accurate answer. But this would not change our basic conclusion, which is that bigger animals need thicker and thicker bones, relative to the thickness of their body, and above a certain size they can’t be any thicker. This sets the size of the largest land animals that can walk the Earth.

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Introductory example problem 2: Why I can’t walk on water

Neither you nor I can walk on water, but a type of insect called pond skaters or water striders can and do - hence the names water strider and pond skater. Here we will try and understand why they can walk (and stand) on water and we can’t. Figure 3 shows a picture of a water strider standing on water. The body of a water strider is about 1 cm long, and you can see that its legs are a bit longer than the body. Note that there are elongated dimples where each of the 4 legs touch the surface - each of the 4 legs has the whole lower part (I guess the equivalent of our shin) in contact with the surface of the water. There the weight of the insect deforms the surface of the water forming a dimple-like depression. A water strider does not fall into the water because its weight is supported by the surface tension of water. Surface tension was introduced in the first year course Properties of Matter, and is covered in the soft-matter 3 Animals in the sea can be larger as water supports their weight, and they are larger. The blue whale is believed to be the largest animal by mass to have ever lived and can be up to 180 tons

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part of this course, so here we just give the surface tension of water, and briefly review the properties of a surface tension.

3.1

Surface tension of water

The surface tension of water is approximately Γ = 0.07 N m−1 , or 70 mN m−1 , at 20◦ C. Thus a surface tension is a force or tension per unit length. N.B. As energy has dimensions of force times a length, the units of energy/area are the same as force/length, i.e., N/m in SI units is the same as J/m2 , and so a surface tension is also an energy per unit area. A surface tension of 70 mN/m for water, means that if we have a water surface that is 1 m wide along the y axis and we try and stretch the surface along the x axis then surface tension resists this stretching with a force of Γ N, also along the x axis. If the liquid surface is 2 m wide along the y axis and we try and stretch it along the x axis, the force is 2Γ N, and so on. To give you an idea of how big that is note that 1 N is about the weight of an apple, so a water surface of width about 15 m exerts a force large enough to suspend an apple against the force of gravity. So in that sense the force is not that large.

3.2

Why a water strider can walk on water and we can’t

Now, we can go back to seeing how the water strider can walk on water. If you look at the water strider in Fig. 3, you can see it has 4 long legs, each a bit longer than its body, and this body is about 1 cm long. For each leg, all the lower part of the leg is in contact with the water - this is what causes the 4 elongated dimples in the surface of the water. So, if we work to one significant figure, we have of order 1 cm length of the surface of water in contact with, and being stretched by, the water strider’s legs, and water’s surface tension is of order 1 mN/cm. Thus the maximum force the water’s surface can exert is of order 1 mN. To support the water strider against the pull of gravity, this force should be larger than the force of gravity, which is mg, for a water strider of mass m. So the requirement is that surface tension force > force of gravity

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So, the surface tension force is 10−3 N, and gravity is mg = 0.01 × 10−3 × 10, for an insect of mass 0.01 grams. So, we have 10−3 N > 0.01 × 10−3 × 10 N 10−3 > 10−4

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so for an insect of 10−5 kg, the maximum force surface tension can exert is larger than gravity, and so the insect can stand on the surface of water, without falling through. However, the force surface tension exerts increases only linearly with the size of the organism, and this is a problem as mass increases as the volume, which increases as the cube of the size. To see the problem let us consider a human trying to walk on water. My feet are roughly 10 cm long, and so the maximum surface tension force the surface of water can exert on one of my feet is about 10 mN. I am about 80 kg which gives a weight of 800 N. Obviously 800 N  20 mN, so of course I can’t walk on water, I go straight through the surface. Indeed, it is easy to see that water striders are not far from the upper limit on the size of organisms that support themselves on the surface of water using water’s surface tension. If we scale a water strider up by a factor of 10 in all directions, i.e., 10 times as long, as wide and as tall, then its mass will scale up by the same factor as its volume. This factor is 103 , so the mass of the ten-times-bigger water strider is 0.01 × 103 = 10 g, and so its weight is 0.1 N or 100 mN. Its legs are now 10 cm and so can support a weight that is only 10 times as big as the real water strider, 10 mN.

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The ten-times-bigger water strider is too heavy, its weight of 100 mN is too large to be supported by the force of surface tension, 10 mN. On water on Earth, the water strider is about as big an animal as can walk on water, according to the laws of physics.

Figure 4: Image (from Wikimedia) of a coin (Japanese 1 Yen) on water. The coin is supported by the surface tension of water, just as the water strider’s weight is. Note that the coin is made of aluminium which has a density greater than that of water, so it is not floating, aluminium objects are too dense to float on water. The 1 Yen coin is 2 cm across and has a mass of 1 g. You can show, repeating the steps above we used for the water strider, that the coin is just about light enough to be supported by the surface tension of water.

Simple approximations using dimensional analysis A surface tension of 70 mN m−1 means, by definition, that if you take a strip of water surface of length 1 m then along that length there is a total force of 70 mN acting so as to shrink the area of the water. Now as we see in Fig 3 each of the water strider’s four legs deforms the water surface into a dimple the shape of a roughly oval ellipse. In other words the surface of the water is deformed in quite a complex way. So to calculate the exact force the deformed water surface exerts on the leg of the water strider we would have to solve a quite nasty partial differential equation. We don’t want to do this. So we are going to guess that the water surface in contact with a long thin leg of length 1 cm = 0.01 m cannot exert a force more than around 0.7 mN. Basically we are saying that a water surface is characterised by the ability to exert a force of around 70 mN for each metre of its length that is deformed. By around 70 mN we mean to within an order of magnitude of 70 mN. In practice, guesses of this sort works out around 95% of the time, but it is good to be clear that it is a bit of a guess. Making educated guesses like this is a valid and very useful thing to do in physics, but you should bear in mind that it is an approximate guess. Another way of putting this is to say the surface tension sets the scale of forces per unit length, so we expect any force per unit length to be of order 70 mN m−1 . And if the surface of water is deformed over a length l then the order of magnitude of the associated force is 70l mN. This is just an estimate in the sense that the exact force will depend on details. If we compare the dimple on the water surface around a long thin water strider leg in Fig. 3 with the dimple around a circular coin in Fig. 3 we see that they have different shapes, and so the forces will be a bit different. But the forces exerted by the water surface on a long thin leg of length say 1 cm and a circular coin of diameter 1 cm are expected to differ by a factor of less than 10. Maybe one is 2 times the other but one will not be 20 times the other.

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Question: What are living organisms made of ? Answer: Soft Matter

Living organisms are mostly made of soft matter, and our cells are essentially entirely made from soft matter. The composition of our cells is roughly speaking: 1. 50% water. All the processes that run cells occur in water. 2. 20% protein. Proteins are a type of polymer. 3. 10% lipids. These include surfactant like molecules that form the membranes round cells, as well as fats and oils used for energy storage. 4. 1% RNA. Our proteins are coded for by our genes, which are made from DNA, but RNA molecules are made as an intermediate step. RNA is also used in the nanoscale molecular machines called ribosomes that actually make proteins, i.e., that string amino acids (the monomers here) into polymers. 5. 0.1% DNA. Your genes are physically made from DNA, in the form of 46 huge DNA molecules called chromosomes plus smaller DNA molecules in your mitochondria (which are part of the energy metabolism of cells). The total length of these 46 molecules is about 1.5 m. If you’re short, the total length of your DNA is longer than you are. plus salts (sodium, potassium, chloride, etc), some small molecules, and another type of polymer called polysaccharides. Polysaccharides are polymers made by stringing sugar molecules together. This composition varies a bit, the % water in your body will drop if you are dehydrated, the lipid (in the form of fat) % will go up if you eat a lot of pies. So, apart from water, the main ingredient in our cells is polymers, of 4 types: proteins, RNA, DNA, and polysaccharides. Our cells are held together mainly by a protein called collagen, a form of which, gelatin, is also what holds jelly together. Plants are a bit different, they use polysaccharides to provide mechanical strength. The main polysaccharide is cellulose. You are probably wearing some as well as cotton is largely cellulose. The surfactants are mostly the lipids that form the membranes around and inside cells. Our bodies are mostly composed of cells but we also have bones, these are composite material made up of a crystalline mineral called hydroxyapatite, and a matrix of polymers, mostly collagen. The crystalline mineral hydroxyapatite is a crystalline form of calcium phosphate.

4.1

Soft matter behaviour

We are made of soft matter and so, as you would expect, living organisms show a wide range of typical soft matter behaviour. Also, food which of course is just bits of dead (and often heated) living organisms, also shows typical soft matter behaviour. In the soft matter part of the course, you will look at viscous liquids, diffusion, elasticity, polymers, self-assembly, etc. Our bodies are full of all these, and without them life could not exist. Examples are: 1. Viscous liquids. Many biological liquids are viscous, e.g., honey, some yoghurts, etc. Many have complex flow behaviour, e.g., blood is shear thinning, which means if you let it settle it becomes more viscous but when it flows (as of course it is doing right now in your arteries, veins and capillaries) its viscosity decreases. 2. Diffusion. As we will see in this part of the course, diffusion is how oxygen and other molecules move around your cells. 3. Elasticity. Rubber is a natural product and is elastic. Your skin is elastic due to the polymers like collagen that hold it together. Jellies are soft solids held together by collagen (aka gelatin).

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Figure 5: The protein shell (capsid) of the cowpea mosaic virus. The protein shell is made of 120 protein molecules of two types, the two types are shown (these are false colours) as green (dark in B&W) and yellow (light in B&W). It is approximately 30 nm across. Virus capsids can form spontaneously, they self-assemble without any external mechanism to put them together. Inside the virus are the genes of the virus plus a couple more proteins it needs. For this virus, the genes are encoded in RNA not DNA as our genes are. Note that the capsid is highly symmetric, it has icosahedral symmetry. If it looks somehow vaguely familiar this is because footballs also have icosahedral symmetry. I don’t expect you to remember the details of this virus in an exam. 4. Self-assembly. Cells are complex machines with an amazing array of self-assembled structures. But some of the most beautiful structures that can form via self-assembly are the protein shells, called capsids, of viruses. See Fig. 5. Note that the capsid is highly symmetric, it has what is called icosahedral symmetry. Many, but not all viruses, have this symmetry. The membranes that surround cells, and that keep the stuff that needs to be inside, inside, and the stuff that needs to be outside, outside, also form via selfassembly. Surfactant-like molecules (types of lipids) spontaneously form two back-to-back layers. These two back-to-back layers form the basis of our cell membranes. Another example of self-assembly is what is called protein folding. This is where a protein, which is a polymer, spontaneously self-assembles into a particular structure, see Fig. 6, for the structure of the protein hemoglobin. Proteins in these structures can then self-assemble further, for example in hemoglobin they self-assemble into clusters of four hemoglobin molecules. These are just a tiny selection from the enormous number of ways in which living organisms, from us to viruses, use soft matter physics to live. Self-assembly in living organisms is molecules (mostly proteins, but also some RNA) spontaneously forming an ordered structure, because this structure is the equilibrium state of the molecules. In other words, selfassembly is driven by thermodynamics. Examples of what self-assembly can do in biology are shown in Figs. 5 and 6.

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Figure 6: Schematic showing hemoglobin: the protein in our red blood cells that carries oxygen round in our blood. Image from Wikimedia. Our hemoglobin is actually formed of four protein molecules, two α hemoblobin proteins plus two β hemoglobins. The α and β hemoglobins are shown in red and gold. Each hemoglobin has a heme group, shown in green. It is the iron-containing heme groups that actually bind oxygen molecules. Protein molecules are polymers that can spontaneously self-assemble into a complex structure called the native state of the protein. One of the elements of protein structure that is very abundant in hemoglobin is a helical structure called an α helix. These are shown schematically in the figure as the ribbons bent into helices. Note that these helices are shown purely schematically, each contains many amino acids and each amino acids contains many atoms so we can’t show every atom as that would be a mess. In hemoglobin, four molecules, each in their native state, self-assembles into the roughly square complex we see above. I don’t expect you to remember the details of this protein in an exam. Importance of Soft Matter Physics in Biology Some diseases directly involve soft matter properties. For example, cataracts are caused by a liquid/liquid phase separation occurring in the mixture of proteins inside the cells of the lens of people’s eyes. By liquid/liquid phase separation, I mean a liquid separating into two coexisting phases, like droplets of olive oil in vinegar. Cataracts are when the lens of a person’s eye goes from being clear (which you need to be able to see of course) to being milky white in appearance. Due to phase separation, droplets of one mixture of proteins form in the cells, and these scatter light, just as the droplets of fat scatter light in milk. Also, because living organisms are made of soft matter, and food is basically dead living organisms, you use soft matter physics when you cook. For example, if you cook a tough piece of meat for a long time to make it tender you are destroying protein (mainly collagen) structures with heat, and by destroying these structures you weaken the meat, making it softer. In summary: ideas from soft matter physics are used everywhere from cooking to research into diseases such as cataracts, cancer, muscular dystrophy, etc.

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