Enzymes do not conform to Brownian motion (2nd Law of Thermodynamics) but to animal-like behavior

By Wolfgang G. Gasser

Future generations will wonder that the majority of the most educated people of the beginning third millennium, despite all the evidence to the contrary, still believe that enzymes construct living cells and macroscopic organisms without perception and purposeful movements*.

In order to better understand what follows look at first at some of the following short videos with animations:

o        What is Brownian motion? (2:58)    Brownian Motion and Random Walks (6:53)

o        Enzyme Action and the Hydrolysis of Sucrose (2:36)    From DNA to protein (2:41)    How Genes are Regulated: Transcription Factors (2:55)    ATP Synthase: The power plant of the cell (3:21)    From DNA to Protein (4:27)   DNA transcription and translation (7:17)    Biology: Cell Structure (7:21)

* The question of perception and purposeful behavior is independent from the question whether enzymes and the technology they use were designed or have been evolving continuously (due to either creative power of Nature or blind chance).


Many enzymes work at defined places in a cell. If we create a model enlarged by factor one million, where enzymes are like little balls, then the volume of the whole cell is about 1000 cubic meters. Imagine concretely this situation: a little ball must come very close to a substrate and the substrate recognition even depends on the correct alignment of the little ball. In addition to that, enzymes often have to pass cell membranes in order to reach their destination. What is the moving force of enzymes? It cannot be electromagnetic attraction or repulsion. So the moving force must primarily depend on random thermal motions, i.e. on Brownian motion.

The voyage of transcription factors to their targets on the DNA can be compared with the voyages of migratory birds and other migratory animals.

Physical laws as described by classical physics or by Quantum Mechanics cannot be responsible for the fact, that living organisms have emerged and survive. The often cited complex dynamic systems as e.g. the appearance of ordered vortices, waves or similar things doesn't affect life and evolution much more than the appearance of solar systems does. And the appearance of crystals (carefully studied by Johannes Kepler) is rather evidence for panpsychism than for reductionism.

One can object that we do not know well enough the chemistry of enzymes in order to conclude that life defies our physics and chemistry theories. There may always be the needed chemical forces responsible for the "apparently" very purposeful movements of enzymes. Yet such an assumption implies that the information for such motions to desired destinations is somehow stored in the amino acid sequence of an enzyme, in addition to the information for folding, substrate specificity and so on, because even similar enzymes can have very different destinations. For instance, a mutation could change a description factor in such a way that the protein would search its usual substrate in a wrong chromosome.


George Acton:

I'm afraid your description of the cellular localization of enzymes misses the point.  The cellular "traffic control" that directs enzymes and other substances to their destination is entirely mechanical.  Not only does the enzyme have the information to assume its shape and form the active site, but it has portions that tag it as destined for different destinations. The transport systems that move the enzymes recognize these "bar codes" and move the enzyme to the appropriate site.  The obvious experiment is to alter that portion of the enzyme without changing the active site.  And the result is an active enzyme located in the wrong place.  And you are correct that a mutation could change the protein so it would wind up in the wrong place, although "search" is more anthropomorphic than most people in science would prefer.

Now it's me who is afraid that the explanation of the purposeful movements of enzymes by an entirely mechanical "traffic control" misses the point. How do you explain such a "transport system"? Is there some kind of currents in the cell, or even some kind of taxis which transport the enzymes to their needed destinations?

With "the information for these motions to desired destinations is somehow stored in the amino acid sequence of an enzyme", I did not mean some kind of "bar code" on the enzyme, which is interpreted by a cellular transport system being responsible for the motions of the enzyme. It seems to me even much more difficult to explain how such a "traffic control" could transport all the enzymes to their very different destinations depending on such a "bar code".

It is obvious that enzymes may consist of several parts or domains, and that one such domain can be responsible for the destination of the whole enzyme. There is no clear transition line between simple proteins and protein complexes.

Your model is correct in size, but the thermal motion at the molecular level is much faster than we would guess from everyday objects.  Some enzymes can catalyze thousands of reactions per second, and the calculations show that random encounters will indeed expose them to that many substrate molecules in a second.

In a model enlarged by factor by 106, little proteins have diameters of few millimeters, e.g. myoglobin with 154 amino acids about 4.5mm x 3.5mm x 2.5mm. The major groove of DNA is then about 2 mm large, and the whole human DNA (both chromosome sets) is about 2000 km long. The major groove is even longer. If the recognition by transcription factors depends on direct contact, then a transcription factor must come very close (maybe 1 mm or 2 mm) to its destination and should even have the correct alignment. A normal living cell with a diameter of about 10 m would consist of 1012 different cubes of 1 mm length.


It is necessary to have a concrete imagination of the proportions between cells, enzymes, molecules and so on. Therefore I have introduced the enlarged model where 1 mm corresponds to 1 nanometer. The 'diameters' of enzymes are then in the order of a few millimeters and the 'diameter' of a water-molecule is about 0.3 mm (there is room for 33.3 water molecules in 1 cubic millimeter).

Ian Musgrave:

As Gavin Tabor pointed out, in aqueous solution there are around 10^14 interactions/sec between molecules, and the volume involved is nanoliters, which is miniscule compared to the free diffusion paths of these molecules, so it is not particularly difficult for a substrate to meet its enzyme in the right orientation just by randomly bouncing around in the cell.

One nanoliter gives a cube with a side length of 0.1 mm, and in our enlarged model this corresponds to 100 m! You even claim that such lengths are minuscule compared to the free diffusion paths of the enzymes. Are you confusing kinetic theory of gases with diffusion in aqueous solutions?

Einstein calculated in a 1905 paper [p. 18] that a particle with a diameter of 0.001 mm (the size of a bacterium) would result in an average motion of 0.0008 mm in a second and of 0.006 mm (less than the length of a normal cell) in a minute (at a temperature of 17C). The average movements per time of enzymes are certainly longer because they are much smaller. The bigger the particles, the slower are random thermal motions. The reason is simple: random collisions with many surrounding molecules can cancel each other out and the remaining change in momentum does not increase proportionally to the mass of the moving particle.

In our enlarged model, a normal living cell with a diameter of about 10 m consists of 10^12 different cubes of 1 mm length. So it is rather difficult for a substrate to meet its enzyme in the right orientation "just by randomly bouncing around in the cell". Don't forget, most of the 10^14 random collisions primarily with water molecules have no effect at all! Every square millimeter of the enzyme surface corresponds to around 10 water molecules.

That "enzymes often have to pass cell membranes in order to reach their destination" happens very rarely, and then the enzymes are usually secreted via a specific membrane targeting peptide.

Only at most 5 percent of the hundreds of different mitochondrial proteins are coded by mitochondrial DNA. The proteins even have to pass a double membrane in order to reach their destination. According to the very convincing endosymbiont theory, at least most of these proteins (or their ancestors) were coded once by the mitochondrial DNA itself.

How do you explain the fact that the proteins could find their destinations in the mitochondrion even after the transfer of the genetic code into the nucleus?

What apparent purposeful motion of proteins?

For instance: motions of transcription factors or of ribosomal proteins. There are many other examples: Look for instance at the many enzymes involved in DNA replication. If there were only random thermal motions, only an extremely small percentage of the enzymes would work (by chance).

A transcription factor can cross the cell in a nanosecond, the observed rates of transcription factor initiation is entirely compatible with the enzymes just randomly bouncing around in the cell.

If transcription factors generally reach their destinations some micrometers away in a nanosecond, than materialist reductionism is dead!


Ian Musgrave:

Good heavens, just look at your own calculations, a bacteria sized particle travels 800 micrometers a second, that's roughly 8 times the diameter of an average mammalian cell! And that's at 17 degrees, whereas mammalian cells are at 37C! Protein sized particles travel much faster as you say. Your own calculations show you are wrong. Now add in diffusion down a chemical gradient, and you are way out of field.

I don't understand: according to Einstein's formula a bacteria sized particle (1 μm diameter) travels 0.8 μm and not 800 micrometers in a second. The mean path is proportional to the square root of both the absolute temperature and the time, and inversely proportional to the square root of the diameter (of a spherical particle). The mean path of a spherical enzyme with a diameter of 10 nm is roughly 8 μm in one second and roughly 8 nm in a microsecond*. For a mean path of 1 nanometer (about three times the length of a water molecule) 10 ns are needed! The mean path of a little protein such as trypsin in 10 ns is in the order of 2 nm.

Why is a 100-fold increase in time needed for 10-fold increase in the mean path? Because the movements are purely random, and the particle will come back close to the starting point over and over again.

Imagine very concretely the many water molecules (0.3 mm in the enlarged model) colliding with the enzyme (e.g. a diameter of some millimeters). Think about the effect of such a collision on a protein whose mass is thousands of times bigger than the mass of the water molecule.

According to you, enzyme reactions can't take place at all, even in a test tube. However, they take place in test tubes at rates that are entirely consistent with substrates and enzymes meeting randomly in a diffusion-limited way (with the exception of enzymes with lipid soluble substrates). This is all elementary enzyme kinetics.

On the contrary: According to reductionist Darwinism, enzyme reactions can't take place at all. The main principle of this modern word view is the hypothesis that biology can be reduced to chemistry and chemistry to physics. But if we explain chemistry by physics, we should do it in a careful and consistent way. It makes no sense to put forward some formulas which are in agreement with the experiments, and to simply assume or claim that they are consistent with our physical theories.


How do you explain the fact that the proteins could find even after the transfer of the genetic code to the nucleus their destinations within the mitochondrion?

Ian Musgrave:

Specific targeting peptide sequences. These can be tailored to particular phospholipid ratios in different membranes, or bind to a peptide receptor.

How probable is it that hundreds of mitochondrial genes received during or after their transfer into the nucleus by random mutations (blind chance) exactly such specific sequences which direct their corresponding enzymes back to the mitochondria?

* Corrected/changed from "is roughly 1 μm in one second and roughly 10 nm in a microsecond"


gavin tabor:

A given molecule will interact with a new molecule every 10^-14 s. If you like, the 'pack of cards' of each molecule is being reshuffled every 10^-14 secs.

The velocity of molecules in the air is around 10^2 or 10^3 m/s. This gives a distance of 10^-11 m or 10^-12 m every 10^-14 second. (Einstein's formula for Brownian molecular motions gives for 10^-14 s similar results for water molecules, but for a factor x increase in distance, a factor x^2 increase in time is necessary.)

The length of simple molecules is around 10^-9 meter, this is around two orders of magnitude bigger than the distance molecules move in 10^-14 s. How is it possible that a "given molecule will interact with a new molecule every 10^-14 s", if the given molecule can only move a small fraction of the length of a molecule?


Wade Hines:

In my estimation, every macroscopic observation regarding the 2nd law of thermodynamics is derivable from statistical mechanics.

I hope you agree with me that the theory of Brownian motion is related to the Second Law because it is based on the same principles. The development of macroscopic life depends on the capacity of innumerable enzymes to efficiently move to different sites in living cells. Movements of particles depending on random thermal motions are described by the theory of Brownian motion.

According to Einstein's formula of his 1905 paper on this subject, a little enzyme (diameter of 1.6 nm resp. of 50 water-molecule diameters) moves a mean path of about 2 nanometers in 10 nanoseconds.

Mean path  needed time

2 nm       10 ns

20 nm      1000 ns

200 nm     0.1 ms*

2 μm       10 ms*

20 μm      1000 ms*

A typical diameter of a living cell consists of around 10 μm, and there are around 10^12 = 1'000'000'000'000 cubes of a side length of 1 nanometer in it [and space for 33 water molecules in each cubic nanometer].

If the recognition of the correct DNA position by transcription factor depended on direct contact of an active site of the transcription factor with the DNA, then this active site would have to come very close to the corresponding DNA position, maybe even closer than 1 nanometer. A nanometer is three times the length of a water molecule, and the major groove of DNA is about 2 nanometers large.

The fact that enzymes move very efficiently in living cells clearly refutes the second law, and your statement "every macroscopic observation regarding the 2nd law of thermodynamics is derivable from statistical mechanics" is simply wrong.

* corrected from "second" to "millisecond"


Ian Musgrave:

You have to consider a number of factors in calculating how far a molecule can move across a cell, charge and non-sphericity of some molecules for example.

Why should being not spherical lead to a substantial increase in the mean path per unit of time? Only in case of a voltage gradient, charges could have substantial effects over distances longer than the molecule sizes. Such a voltage gradient however would affect all charged molecules.

Also, the collision radius is 4 times the particle radius, partly due to water cage effects.

If that's true, then it is very strong evidence for my claim that enzymes are able to perceive their surroundings (see The Nature of Life).

Imagine a spherical enzyme with a radius of 2 nm, a volume of around 34 cubic nanometers and a surface of around 50 square nm. The active site may have a surface of 4 square nanometers. If the collision radius is 4 times the particle radius, we must conclude that the active site can perceive a substrate over a distance of at least 6 nm. But these 6 nanometers are filled with other molecules such as water. The diameter of a water molecule is around 0.3 nanometers and there is room for around 33 water molecules in one square nanometer.

If our enzyme were only surrounded by water, than within such a 'collision radius' there would be as many as 70 thousand water molecules.

The number of encounters between two particles a and b (say an enzyme and its substrate, where the substrate is smaller than the enzyme) per second per milliliter of solution is given by

   4π (Da + Db) (ra + rb) na ∙ nb

Where D is the diffusion constant of molecules a and b, r is the radius and n the number of molecules of a and b.

The only concrete consequence from this formula is that the number of encounters is proportional to the number of molecules of a and b. The diffusion constant and the radius of a molecule are not independent. So it is not possible to test the formula by changing the radius or the diffusion constant independently.

It is clear that it is always possible to attribute such diffusion constants to the molecules that that the formula is valid. But if these diffusion constants cannot be derived from fundamental principles, then we have only an ad-hoc-formula.

Now the collisions are inelastic, because of water molecules forming a "cage" around the enzyme and its substrate, so you can get 100-200 collisions per encounter!  Given the rotational rate of an enzyme is mighty fast, the enzyme can present many orientations to the substrate molecule during these collisions, and that is ignoring charge effects.

Imagine concretely the situation taking into account the corresponding sizes and distances of the involved molecules. How and why do such water "cages" form? There are innumerous different substrate sites in a living cell. Does an enzyme form water cages with all potential substrate sites? And what happens if after several hundreds of collisions in the cage no matching substrate has been found by the active site?

And what happens to the angular momentum of the enzyme ("the rotational rate of an enzyme is mighty fast") in the moment it presents the correct orientation to the substrate molecule?

From the French-Metzler formula (derived from the Stokes Einstein equation) a single enzyme can sweep through roughly 20% of a standard cells volume (or all of the cell's nucleus) in one second.

Where does the Stokes-Einstein equation come from? Stokes died in 1903 and Einstein hardly participated in creating a formula totally inconsistent with his own work on Brownian motion.



Ian Musgrave states that a transcription factor can cross a cell (around 10 micrometer) in nanoseconds,

Ian Musgrave:

Which is the value you get correctly applying the Stokes-Einstein equation and Markov chaining. It's actually closer to a microsecond.

Laurence A. Moran:

Ian, I'm having trouble seeing how you did your calculation. The Einstein equation is,

      x^2 = 2∙D∙t

where x^2 is the mean square displacement, t is time (sec), and D is the diffusion coefficient. For a typical transcription factor the diffusion coefficient would be about 10-6 (cm squared/sec). If we want to know the time for diffusion over 10 micrometers,

        x^2       (10 x 10^-4 cm)^2

   t = -----  =  -------------------  = 0.5 sec

        2 D            2 (10^-6)

This is a great deal slower than the nanosecond range you were suggesting. Are you using a different diffusion coefficient? Perhaps you have taken into account some micro-viscosity effects that I'm not aware of? What do you mean by "Markov chaining?

Here Einstein's formula [1]:

x^2 = t ∙ (R∙T/N) / (3π∙k∙P)

where x is the mean square displacement of the x-coordinate, t is time, P the radius of the suspended particles, R the gas constant, T absolute temperature, N Avogadro's number and k viscosity.

In the case of water and suspended particles with a 1 micrometer diameter, Einstein calculates a mean square x-displacement of 0.8 micrometer in a second [2]. Einstein also shows that the formula is consistent with the diffusion of saccharose (sucrose) in water, if saccharose particles have a diameter of around one nanometer [3].

If we deal only with the order of magnitude we can say that the mean displacement of a sucrose sized particle is around 10 or 100 μm and the displacement of protein sized particles around 10 micrometers in a second.

That is in agreement with Larry's calculation. The discrepancy between Larry's and Ian's values lies in the order of 10^6 or even more:

-        Ian states that a transcription factor can cross a cell (around 10 micrometer) "in nanoseconds", "actually closer to a microsecond".

-        Larry states that the time for diffusion over 10 micrometers is 0.5 sec.

Einstein's formula takes into account viscosity. But because water has yet a low viscosity and the mean displacement is proportional to the inverse of the square root of viscosity, viscosity cannot account for the huge discrepancy.

In the same way, I think that micro-viscosity could rather be responsible for a lower mean path than for a higher.

It is generally acknowledged that enzyme-substrate-interactions are normally based on the lock-and-key-principle. If we take into account that an average eukaryotic cell consists of 10^12 cubic nanometers and that the diameter of a water molecule (relevant to van der Waal's forces) is around 0.3 nanometers, we must conclude that enzymes are just randomly bouncing around most time.

One should really create a computer-simulation of the enzyme-substrate-recognition based only on fundamental physical and chemical principles.

A computer simulation based on ad-hoc-hypotheses would be almost worthless. Here an example of such an ad-hoc-hypothesis:

The recognition radius is four times the enzyme radius. If the substrate is by chance within this radius, enzyme and substrate meet and change their orientations to each other until the active site and the substrate eventually get together in the right way.

[1] "The Collected Papers of A. Einstein", Volume 2, P.U.P. 1989, p. 212

[2] p. 234 and p. 500

[3] p. 501

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