Oct 09

The Wason card problem

The Wason card problem is a well-known psychological test that probes how people think about hypothesis testing. The version I use in one of my first-year lectures is shown below. I think the original version used letters and numbers, but I’m a biologist, so obviously I use pictures of dead pets instead of numbers.

  • We know that each of the four cards shown below has a letter on one side and an animal on the other.
  • We suspect that if there is a vowel on one side of a card, then there is a mammal on the other. Which card(s) do we need to turn over to determine whether this hypothesis is false?

It’s well worth doing this for real yourself before reading any further:

Wason cards [CC-BY-SA-3.0 Steve Cook]

Wason cards: an ‘A’, a snake, a cat, and a ‘Q’

Pretty much everyone thinks we should turn ‘A’, and they’re right. If there’s a lizard on the other side, it contradicts the hypothesis “if vowel, then mammal” immediately.

Pretty much everyone thinks we need not turn ‘Q’, and they’re right. Our hypothesis is “if vowel, then mammal”; it says nothing about what a consonant implies, so turning this card is irrelevant.

The sticky bit is that pretty much everyone wants to turn over the cat, but not the snake. In this they are quite wrong.

Having done this test with my fresh-faced biology undergraduates for the past two years, I can confirm that they perform very similarly to the population at large, or at least to the self-selecting and weird subset of the adult population that finds itself being asked deliberately tricky questions by smart-arses. 10-20% want to turn over the snake, but 80%-90% want to turn over the cat.

It’s important to note that the hypothesis is “if vowel, then mammal”, not “if mammal, then vowel”. It doesn’t actually matter one jot what is on the other side of the cat:

  • If it’s the letter ‘Z’, then we have evidence for “if consonant, then sometimes mammal”. This says nothing about the truth of “if vowel, then mammal”.
  • If it’s the letter ‘U’, then we’ve found a card that is consistent with “if vowel, then mammal”, but we already suspected that.

In either case, the thing on the other side of the cat is guaranteed to be either consistent with the hypothesis, or irrelevant to the hypothesis, so there is no point in turning it. The result could never contradict the hypothesis “if vowel, then mammal”.

The snake, on the other hand, must be turned over.

  • If it’s the letter ‘Z’, then we have evidence for “if consonant, then sometimes reptile”. Again, this says nothing about the truth of “if vowel, then mammal”.
  • However, if it’s the letter ‘U’, then we’ve found a card that is inconsistent with “if vowel, then mammal”, and which immediately contradicts the hypothesis.

The reason so many choose the wrong cards can be partly put down to the rather tricksy use of ‘if’: unfortunately, English doesn’t really distinguish between mathematics’ ‘if’ and mathematics’ ‘iff‘ (if-and-only-if).

However, the main reason so many choose the wrong cards is (probably) down to the human inclination to look for data that confirm our hypotheses, rather than actively searching for data that might contradict them. Unless we’re very careful, we humans tend to turn cards, perform experiments, and follow Twitterfolk that stand little or no chance of ever contradicting our beliefs.

We flip the cat, even though what is on the other side cannot change our mind. We read the newspaper, even though what is inside is unlikely to challenge our prejudices. And – shame of shames – we inflict logic puzzles on our students, even though we know the result will almost certainly be consistent with our unworthy suspicions about their deductive abilities. Hypocrisy, thy name is Cook.

Sep 22

Organism of the week #17 – Daughter of Ungoliant

I’d never noticed that a garden spider’s third pair of legs is much shorter than the others before.

Araneus diadematus in web [CC-BY-SA-3.0 Steve Cook]

Araneus diadematus, common European garden spider

Sep 02

Youth of today

One consequence of being an evolutionary dead-end is that I do not get exposed to new cultural touchstones through watching crap children’s telly with offspring I haven’t produced. As an alternative, monitoring the dwindling intersection of history I share with my incoming undergraduates serves the same salutary purpose of reminding me how out-of-touch I am.

The Beloit Mindset is the definitive tool for rubbing the noses of American faculty into time’s unyielding ripsaw, but 2013 is particularly special. This time it’s personal. My incoming first-years were mostly born in 1995 or thereabouts, so – as well as post-dating one of the worst premises for a film by almost a decade – they have also been alive for almost exactly as long as I have been installed at $PLACE_OF_WORK.

I have been at university for as long as my incoming undergrads have been alive. The horror.

None of my experiences as a child directly overlap with theirs. They never had the Cold War; they have never not had the World Wide Web. Diana was dead before they had control of their bowels. They’ll never have used a shilling as a 5p piece: in fact, they will never have used a shilling-sized 5p piece, as those were withdrawn in 1993. The new Dr. Who series is the only one they’ll ever have known, and their very conception could have be due to someone leaving an important part of their brain somewhere, somewhere in a field in Hampshire.

Eighteen years before I was born, The Beatles were still The Quarrymen; the Apollo program was a twinkle in JFK’s eye; and being a sexual degenerate would still be illegal in the UK for another decade.

My new students will not remember the issue of New Scientist that carried the Voyager 2 pictures from Neptune on its front cover:

Neptune [public domain, NASA]

Neptune from Voyager 2 (1989) [public domain, NASA]

But having seen those images in 1989, I could not feel quite the same awe at the Cassini images from Saturn that were the talk of the ARPANET when my incoming students had just arrived at their secondary schools:

Saturn [public domain, NASA]

Saturn from Cassini (2006) [public domain, NASA]

And for me, the Blue Marble was already a cliché by the time I was old enough to understand what it was a picture of, no matter how much it had wowed my parents:

Earth [public domain, NASA]

Earth from Apollo 17 (1972) [public domain, NASA]

…Now, that was an obvious set-up for an effortless glide into life-affirming platitude, but frankly, melancholy and curmudgeonliness are far more my style, and I’m not going to give you the satisfaction. So either take heart that the details that separate us are far less than the humanity that unites us across the generations; or just gawp at the pretty pictures until cataract and senility crush mindful sight from you.

You’re all going to die down here.

Aug 09

Organism of the week #16 – From tiny acorns do mighty oaks grow

There’s something very satisfying about growing plants from seed, and none more so than growing a monster from next-to-nothing:

Sequoiadendron giganteum eighteen months old [CC-BY-SA-3.0 Steve Cook]

Sequoiadendron giganteum, eighteen months old

At the moment, this little redwood seedling is just 18 months old. Given 500 years or so, it’s going to get a wee bit bigger:

Sequoia sempervirens in California [CC-BY-2.0 Alex Lomas]

Somewhat more elderly Sequoia sempervirens in Muir Woods

The seeds of redwoods are absolutely titchy, as you can’t see in the picture, below because the idiot who took the photo forgot to put a scale bar on it. Each seed has a mass of about 5 milligrams.

Sequoia sempervirens strobili and seeds [CC-BY-SA-3.0 Steve Cook]

Sequoia sempervirens strobili and seeds

The adult trees are about 70 m in height (h), with a girth at ground-level of about 10 m, hence a radius (r) of 10/2π = 1.6 m. Modelling them thoroughly inaccurately as big cones of water, which has a density (ρ) of 1 tonne per cubic metre, means a typical redwood tree has a mass (M) of about:

M = ⅓ πr² h ρ = ⅓ π × 1.6² × 70 × 1 = 187 t

This is as much as a blue whale, but much larger specimens exist, so this is a conservative estimate of just how astonishingly massive these trees can become.

These trees take about 1000 years to get to this size, which is about 32 billion seconds:

1000 × 365 × 24 × 60 × 60 = 31 536 000 000 s

The trees won’t be growing at a constant rate for their entire life-span, but if we pretend they do (as anything cleverer is beyond my patience), then these trees would put on about 6 milligrams of mass per second:

187 000 000 g / 31 536 000 000 s = 0.006 g s⁻¹

Satisfyingly, this means that if you were to plant one redwood seed, and then spend the next thousand years adding redwood seeds to a pile, one every second, then the tree and the pile would have the same mass by the end of the millennium.

I’m all about the practical gardening tips.

Aug 09

Enzymes provide alternative routes to product with a lower activation energy

An enzyme lowers the activation energy for a reaction

Like a previous post, the problem here is not so much that this idea is flat-out wrong, but that it’s very prone to misinterpretation.

Text-books often state that an enzyme, or any other catalyst, lowers the activation energy of a reaction.

The activation energy for a reaction (written ∆G‡ or Ea in most texts) is supposed to be a measure of how energetically two molecules need to collide with one another to react. If the the molecules collide too slowly, the collision won’t be energetic enough to break bonds, and the molecules will leave the collision unchanged. If they collide into one another fast enough, the collision will be energetic enough for a reaction to occur and for product to form.

For example, if you put some hydrogen peroxide into a beaker, and observe how quickly it breaks down to oxygen and water at different temperatures, it is possible to calculate that the activation energy for this uncatalysed decomposition is about 75 kJ mol⁻¹. This reaction is very slow at room temperature.

However, in the presence of the enzyme catalase (minced liver or blood is a cheap source), the reaction is almost explosively fast, and the calculated activation energy drops all the way down to about 25 kJ mol⁻¹.

Hydrogen peroxide decomposition [CC-BY-SA-3.0 Steve Cook]

Hydrogen peroxide decomposition

When a reaction’s activation energy is low, then a greater proportion of the molecules zipping around in the beaker collide with enough energy to break down. The reason for this is usually shown using a diagram of the sort below (although generally with much more vaguely labelled axes):

Maxwell Boltzmann distribution [CC-BY-SA-3.0 Steve Cook]

Maxwell Boltzmann distribution: the area under the curve and to the right of a threshold speed gives the proportion of molecules moving fast enough to react

This diagram shows the speed distribution of molecules. Technically it’s only applicable to gases composed of unrealistically helpful particles, but the results are qualitatively applicable to beakers of peroxide.

The y-axis shows how probable it is that a molecule is travelling at some given speed, and is therefore a measure of how likely it is that a collision in the beaker will have some particular energy. From the graph, you can see that the majority of the molecules are travelling somewhere between 250 and 1000  m s⁻¹. The peak is centred on 500 m s⁻¹ (in old money, this is about 1000 mph!)

For the uncatalysed reaction, the activation energy is large (75 kJ mol⁻¹), so the number of molecules moving fast enough to react when they collide is small. This is represented by the purple shaded area on the graph, which is related to the proportion of molecules in the beaker that could collide fast enough to react.

In the presence of the catalase enzyme, we saw that the apparent activation energy is much smaller (25 kJ mol⁻¹). The speed of collision doesn’t need to be so high to get the molecules to react, and the proportion of molecules that can react is therefore larger. This is represented by the sum of the shaded purple area plus the extra red area.

So, it does appear that in the presence of a catalyst, the reaction really does have a lower activation energy. The catalysed reaction runs faster and the reaction reaches equilibrium more quickly.

But…

Why would the addition of a catalyst lower the activation energy of the reaction?

The short answer is that it doesn’t, because when you add the enzyme, the peroxide is no longer reacting on its own to make oxygen and water, it is reacting with the enzyme. The enzyme becomes a fundamental part of the reaction, and the enzyme-catalysed reaction is completely different from the uncatalysed reaction. The reaction is no longer:

Peroxide → unstable ‘transition state’ → oxygen + water

but (at least):

Peroxide + enzyme → enzyme/peroxide complex →

unstable ‘transition state’ →

enzyme/product complex → oxygen + water

An enzyme doesn’t lower the activation energy for a reaction, because the reactions with and without the enzyme are very different things.

A better way of thinking about how enzymes catalyse reactions is that…

An enzyme provides an alternative route to products that has a lower activation energy

Quite why an enzyme-catalysed reaction will have a lower activation energy than an equivalent one that does not involve an enzyme is a matter of some interesting debate, and I’ll leave that to some other time.

Aug 09

The wisdom of crowds

I tried this about five years ago, with more or less the inverse of these results. The nerdy grammar fascist bit of me is pleasantly surprised by the improvement, but poor Karl Albert still has some way to go…

 

Jul 28

Organism of the week #15 – Try not to think about the poo

Apologies for the long gap between postings; it’s been exam season at work, and I’ve not had much time to blog.

The garden was buzzing with green-bottles yesterday, thanks to a combination of warm weather and overflowing brown bins. These flies tend to get a bit of a bad wrap, what with the spitting onto dog shit, sucking up the slurry, and then tramping it all over our food.

However.

If you can somehow put out of your mind the poo, they’re really very pretty:

Lucilia sericata perky [CC-By-SA-3.0 Steve Cook]

Lucilia sericata – a perky little green-bottle

If the metallic green iridescence of a green-bottle’s arse were on the wing of a butterfly or the feather of a peacock, you wouldn’t be able to stop yourself blathering on about the soul-lifting delight of it all, would you?

Lucilia sericata on Catalpa leaf [CC-BY-SA-3.0 Steve Cook]

Green-bottle on a Catalpa

Look into its puppy-dog eyes and tell me your heart doesn’t burst from cuteness overload:

Lucilia sericata face on [CC-BY-SA-3.0 Steve Cook]

Lucilia sericata says “I wub you”

May 26

Organism of the week #14 – Turtles all the way down

Cassiopea, which looks rather like a spelling mistake (but isn’t), also looks a bit like an anemone (but isn’t):

Cassiopea sp. [CC-BY-SA-3.0 Steve Cook]

Cassiopea ‘upside-down jellyfish’ in the aquarium beneath the Palm House at Kew Gardens

It’s actually a jellyfish, but one that spends most of its time living upside-down (relative to its relatives). Unlike its close relatives, it gets much of its energy from sunbathing rather than from fishing.

Inside the cells of the jellyfish live tiny photosynthetic algae called dinoflagellates. These algae produce food for the jellyfish by photosynthesis, and presumably benefit from the protection afforded by the jellyfish’s stings.

Beneficial relationships like this between two different organisms are called ‘mutualistic symbioses‘. Mutualistic symbiosis is very common in biology. Many large organisms act as hosts for smaller organisms, and relationships based on photosynthesis are some of the commonest:

Ramalina farinacea [CC-BY-SA-3.0 Steve Cook]

Ramalina farinacea, like all lichens is an intimate mutualistic symbiosis between a fungus and an photosynthetic alga. Like the upside-down jellyfish, the algae here (arguably) benefit from the fungus’s protection, and the fungus benefits from the sugars made by the algae.

The marvellous thing about the jellyfish/dinoflagellate symbiosis is that the dinoflagellate itself is the product of a mutualistic symbiosis. Inside the cell of a dinoflagellate there is a sub-structure that actually performs the photosynthesis. This structure itself is the remains of a different kind of alga that has taken up residence inside the dinoflagellate.

Although they are not closely related to dinoflagellates, the cells of brown algae have a very similar set-up. Brown algae are perhaps more familiar to you than dinoflagellates, as they include most of the larger seaweeds like wracks, kelps and rockweeds:

Fucus vesiculosus [CC-BY-SA-3.0 Steve Cook]

Fucus vesiculosus or bladder-wrack is a common brown alga found around the UK coast

The photosynthetic structures of dinoflagellates and brown algae are (usually) the remains of red algal cells. In the case of the brown algae, these were taken up in the dim and distant past and are now permanent residents. The many species of dinoflagellate are flightier, and appear to have gained, lost, and re-acquired the photosynthetic helpers by gobbling up other algae on many occasions during their evolutionary history.

Kryptoperidinium [CC-By-SA-3.0 Steve Cook]

The dinoflagellate Kryptoperidinium has not just one, but two different kinds of photosynthetic structures (‘plastids’). The yellow thing the the left is the remains of a red alga, and acts as an eye-spot. The much larger plastid to the right contains its own nucleus (grey wibbly thing) mitochondria (pink things) and internal plastids (yellow). This plastid is the remains of another algal cell, in this case a diatom , which itself has a structure similar to the brown algal cell shown in the next diagram

But it doesn’t even stop there! If you peel open a red algal cell, you’ll find that the photosynthetic structures are again restricted to small sub-compartments of the cell. These structures look very similar to certain sorts of photosynthetic bacteria called cyanobacteria, and indeed that is precisely what they are: the dwindled remains of photosynthetic bacteria captured by the ancestors of red algae about a billion years ago.

This ancestor of the red algae also happens to be the ancestor of all green algae and land plants too, which means inside every butterwort and cock-of-the-gods there are also trillions of these bacterial wraiths, photosynthesising away.

Endosymbiotic matryoshka [CC-BY-SA-3.0 Steve Cook]

Symbiotic matryoshka

So an upside-down jellyfish is like a Russian matryoshka doll.

  1. The jellyfish contains dinoflagellate algae, which make food for the jellyfish.
  2. But the dinoflagellate contains the remains of a red alga, which is the source of this food.
  3. But even the red alga is only able to make food because it contains the remains of the real player here…
  4. …which is the remains of a photosynthetic bacterium.

The upside-down jellyfish is a four-level deep symbiosis of bacterial cells living within red-algal cells living within dinoflagellate cells living within jellyfish cells. In comparison to this, living upside down is the least of its marvels.

If anyone knows of a five-deep endosymbiosis I’d be delighted to hear about it…

May 03

Organism of the week #13 – Unlucky for some

I’m a sucker for things that are both poisonous and pretty. Datura metel definitely meets both criteria.

Datura metel (flower) [CC-BY Alex Lomas]

Datura metel (flower)

These photos were taken in southern Spain, where this purple variety of the plant seemed to have naturalised itself out from someone’s garden .

Datura metel [CC-BY Alex Lomas]

Ominously looming over the Spanish countryside

Datura metel is closely related to Atropa belladonna, the deadly nightshade, and – like it – the plant contains high concentrations of atropine and scopolamine. If eaten, these chemicals clog up receptors in the nervous system, the normal job of which is to cause the body to “rest and digest“. By blocking these receptors, high doses of atropine or scopolamine cause a “flight or fight” response: increased heart rate, dry mouth, reduced gut movement, and dilated pupils. The last of these effects is the source of the belladonna (‘beautiful lady’) of Atropa belladonna: juice from deadly nightshade was used cosmetically during the middle ages to dilate the pupils alluringly – and, one suspects – alarmingly.

Atropine and scopolamine are still used medically, but in standardised and pure form rather than as the random rag-bag of assorted poisons used by mediaeval herbalists. Atropine is used to speed up dangerously slow heart beats, and as a treatment for organophosphate poisoning, where it counteracts the excessive “rest and digest” symptoms caused by nerve agents and accidental consumption of some insecticides. Atropine is no longer used to dilate pupils for eye examinations as it is too long-lasting in its effects (days to weeks!) Scopolamine is mostly used in the treatment of motion sickness: it reduces nausea by reducing motion of the gut.

Scopolamine [CC-BY-SA-3.0 Steve Cook]

Scopolamine

The fruit of the Datura is called a thorn-apple, but I would certainly not recommend baking them into pies. In Datura stramonium, the fruit varies markedly in its shape if it carries an extra copy of one of its chromosomes. Extra chromosomes (up to and including a whole extra set, or four, or even eight!) are quite well tolerated by plants, although they can lead to fertility issues. However, in humans, most (but not all) changes to chromosome number are lethal. Quite why plants are more tolerant than humans of this sort of large-scale chromosomal change is not well understood. On the other hand, doubling of chromosome number has occurred at least twice in the evolutionary history of human beings, and our own chromosome 2 is the product of fusion of two smaller chromosomes (which are retained in our nearest relatives), so these events – whilst rare – are not unheard of in mammals.

Datura metel (fruit) [CC-BY Alex Lomas]

Thorn-apple

May 01

The Michaelis-Menten model is not applicable to most enzymes in a cell

Enzymes in cells can be modelled using the Michaelis-Menten model

Enzymes can be, and often are, modelled by the Michaelis-Menten (well, Briggs-Haldane) model:

v = vmax · [S] / (KM + [S])

Where:

  • v is the velocity of the enzyme, which is the rate at which product accumulates
  • vmax is the maximum velocity of the enzyme (i.e. the velocity to which it tends as substrate concentration increases towards infinity)
  • [S] is the concentration of substrate
  • KM is the Michaelis constant for the enzyme, which is the concentration of substrate needed to make the enzyme run at half its maximum velocity

This model works out very well when you mix together an enzyme and its substrate in a test-tube and measure the initial rate of product accumulation. Here are some data from a practical I run with my first-year biology students, showing a nice hyperbolic curve as predicted by the model:

Michaelis-Menten kinetics [CC-BY-SA-3.0 Steve Cook]

Wheat-germ acid phosphatase shows Michaelis-Menten kinetics when acting on para-nitrophenol phosphate, with a vmax of about 17 µM min−1 and a KM of about 0.5 mM.

However, the Michaelis-Menten model relies on a number of assumptions. During the Veneration and Memorisation of the Derivation, these assumptions may be dashed past by the lecturer doing the mathturbation, and may be ignored by the students trying to grasp the point of the modelling. This is problematic, as not understanding the assumptions leads to the Michaelis-Menten being applied thoughtlessly in situations in which it does not apply.

Which turns out to be almost all situations.

The assumptions are:

  1. The only relevant chemical species in the system are the enzyme (E), the substrate (S), the enzyme-substrate complex (ES‡), and the product (P).
  2. The only relevant chemical reactions occurring in the system are the reversible association of enzyme and substrate to form the enzyme-substrate complex; and the irreversible breakdown of the enzyme-substrate complex to product.
  3. The concentration of the enzyme-substrate complex is constant.

For the Michaelis-Menten model to be applicable to an enzyme in a cell, then these assumptions need to hold.

Are the only chemicals in a cell E, S, P and ES‡?

The answer is obviously no. Cells contain thousands of metabolites. To imagine that none of them have any relevant effect on the enzyme under consideration is patent nonsense, particularly when half of the rest of the enzymology course is about enzyme inhibitors, and enzyme regulation by small molecules.

In a test-tube, we can provide a pristine environment of E, S, P and ES‡, but a test-tube of four ingredients isn’t a terribly realistic model of a cell. It’s also not terribly realistic to expect P to sit back and do nothing…

Are enzyme-catalysed reactions in the cell irreversible?

The answer depends on which reaction you’re talking about, but it is certainly true that many of reactions in the cell are very nearly at equilibrium under normal cellular conditions. For example, in glycolysis:

Glucose → Glucose phosphate ⇄ Fructose phosphate → Fructose bisphosphate ⇄

Triose phosphate ⇄ Bisphosphoglycerate ⇄ Phosphoglycerate ⇄ Phosphoenolpyruvate → Pyruvate

…all but three of the glycolytic reactions (those shown with a → rather than a ⇄) are usually close to equilibrium in the cell. At equilibrium, the forward and reverse rates of reaction are equal, so for these enzymes, the Michaelis-Menten model is inappropriate because assumption #2 is completely broken. The backwards reaction is appreciable because the concentration of product is relatively high. So for many enzymes in the cell, the answer to the question above is a resounding “no”.

You can model these enzymes with a (substantially more complex) reversible version of Michaelis-Menten model, but at, or close to, equilibrium, these enzymes are most assuredly not causing products to accumulate in the way implied by a naïve glance at the graph above. Enzymes acting on near-equilibrium reactions are also exquisitely sensitive to changes in substrate and product concentrations. This is something else that is not obvious from the Michaelis-Menten graph, but which is vitally important, as we shall see below.

In a test-tube, we can provide a super-duper pristine environment of just E and S, which will give us more-or-less Michaelis-Menten kinetics after a little while (once the concentration of ES‡ has settled down) and then for a little while (until P accumulates a bit and the reverse reaction becomes appreciable), but this is even less realistic than the four-ingredient tube above!

What of the remaining three reactions of glycolysis? It turns out these enzymes have to violate the assumptions of the Michaelis-Menten model, and for a very good reason.

If enzymes that catalyse irreversible reactions in the cell were to exhibit Michaelis-Menten kinetics, what would their response to substrate look like at low substrate concentrations?

Substrate concentrations in cells are typically lower than, but somewhere in the vicinity of, KM. When [S] is lower than KM, the Michaelis-Menten can be approximated as:

v ≈ (vmax/KM) · [S]

To double the velocity of such a reaction, you need to double the concentration of substrate: if you look at the graph above, you can see it’s more-or-less a straight line between 0 and 0.5 mM.

Requiring the concentration of a chemical to double in order to get an appreciable increase in the rate of the reaction that consumes that chemical is often an extremely bad idea. Your blood normally contains glucose at a concentration between 4 and 6 mM. If your blood glucose needs to double to 10 mM before the enzymes in your body begin to deal with it properly, then you are probably diabetic.

Enzymes acting at irreversible reactions have usually evolved a completely different response to substrate that is most certainly not modelled by the Michaelis-Menten equation:

Sigmoidal kinetics [CC-BY-SA-3.0 Steve Cook]

Enzymes acting at non-equilibrium reactions usually show sigmoidal kinetics, with a very sharp response to substrate: doubling the substrate concentration from 0.5 to 1 mM here leads to a six-fold increase in the velocity

These enzymes are usually multimeric and can co-operatively bind multiple substrate molecules, so there are ES‡, ESS‡, ESSS‡, etc., complexes in the system too. They are also often sensitive to feedback inhibition from other small molecules, so there are plenty of other relevant chemicals you need to take account of when modelling their behaviour. These enzymes violate assumption #1 in the most thorough fashion, and have a sharp sigmoidal (S-shaped) response to substrate.

The Michaelis-Menten is a very good model of the activity of many enzymes in a test tube full of substrate. Its parameters vmax and KM are genuinely useful bits of basic information to obtain about an enzyme, and they are easy to estimate in this simple system. However, Michaelis-Menten kinetics are inapplicable for most enzymes within a cell because most enzymes act on reactions that are close to equilibrium, and they are unsuitable for most of the other enzymes within a cell, because enzymes acting at far-from equilibrium reactions cannot afford to behave in such a lackadaisical fashion. Be really careful when applying toy models to real cells. Be really careful that you know the assumptions of any model you are using.

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