LSM5194
Limitations of the simple Fickean
law, Complex Diffusion and
Molecular Crowding
Lecture 23
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Does diffusion work in biological systems?
David Goodsell, Triptych “Macrophage engulfing a bacterium”
Here is a paradoxical question: “Does diffusion work in biological systems?”
Two lectures ago we started to study physical chemistry of diffusion explicitly
with the goal to model processes in biological systems. So why would we now
question using classical diffusion theory in application to the biological
systems? Predominantly because of the new body of evidence suggesting
that the organization of real biological cells may be considerably more
complex than the classical diluted aqueous solutions for which the diffusion
theory was developed.
In this lecture we will consider a number of phenomena which point to the
limitations of the simplistic “chemical” approach to understanding of biological
processes in the cell. Namely, we will discuss the dependence of diffusion on
concentration, molecular crowding and its implications for the biochemical
reactions, motility of particles inside the cell and finally we will look at the
validity of the diffusion equation itself in the environment of high density and
large spatial heterogeneity.
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Why intracellular environment is overcrowded
Prokaryotic cell contains (E. coli V = 10-15 L) :
• 2.5 109 Da DNA – 4300 genes – 1,2 transcripts
• 15,000 ribosomes – 52 proteins, ~ 6 MDa, 25
nm, 15% volume, 25% weight
• 1800 RNA pol – 450 kDa
• EF-Tu TF – 100,000
• Beta galactosidase – 3000
Eukaryotic cell contains (V = 2 10-12 L) :
• 5 109 protein molecules of ~10000 types – 75%
of total dry weight
• ~1012 Da DNA – 20,000 genes, 109 bp, 5 pg
D. Goodsell “E. coli”
So what makes cells different from ideal solutions considered in the courses of physical
chemistry? The presence of a great variety of chemical molecules, complexes and structures
whose sizes span many, many orders of magnitude. The cell is a little universe in itself as we
will see in this lecture. Presented on this slide is a look inside the bacterial cell as created for
us by the original scientist and artist David Goodsell. What do bacteria have within? Now we
are very close to be able to answer this question in a more or less comprehensive way. The
largest portion of the volume is occupied by bacterial DNA. Average bacterial genome is
about 3 – 5 thousands genes. This is quite a lot to stuff for the tight volume of the bacterial
cell. To maintain this amount of DNA bacterium needs histones, transriptases, girases,
ligases and other kinds of proteins. Therefore, genome occupies the largest part of the
bacterial volume. To cope with the changing environment and to maintain levels of its
proteins the bacterium needs to keep up its protein production. E. coli cell is known to contain
about 15 thousand ribosomes in the active growth phase. At the intensity of transcription of 12 transcripts per gene, this number is not a luxury but a necessity which only allows to have
1-2 ribosomes per mRNA molecule on average. This is however a huge load on the bacterial
recourses if the consider that each ribosome is a huge complex of 52 ribosomal proteins and
several large RNA molecules of about 25 nm in size. This a single largest population of
protein complexes which takes up about 15% of bacterial volume and whole 25% of the dry
weight. Now we need to add into the pile proteins of RNA polymerase II and other ubiquitous
proteins which are absolutely important for the survival. By the end, the inner volume of the
bacterium can be compared with the interior of a tank or a jet fighter. Everything is important
and no extra space for anything. How is this overload going to affect the cellular transport? In
fact, quite profoundly.
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Steric repulsion and cage effect
Hard sphere abstract model explaining the
concept of excluded volume and steric
repulsion.
Large molecules, due to the effect of excluded
volume, find the cell interior considerably more
crowded than small molecules.
Once inside the compartment, the molecule
cannot simply leave it – cage effect.
Experimental demonstration of cage effect
(Hudder et al., Mol Cel Biol 2003, 23, 9318):
Hamster ovary cells were carefully
permeabilized without destruction of
cytoskeleton and protein release was
measured:
Let us first consider in a classic physical-chemistry style the effect known as
steric repulsion or effect of excluded volume. For simplicity let all the
molecules at first be hard spheres present in high concentration. Interestingly
that this kind of arrangement looks very different for oncoming molecules of
different size. Small molecule will simply see itself in a very complex maze
between the large balls. But since it does not fly ballistically but moves
diffusively, the presence of big spheres will not change anything for a small
molecule. Things are quite different for large molecules of the size
comparable and larger then the obstacles. Due to its own size the new
molecule cannot approach the surface of hard spheres closer than its own
radius – effect known as exluded volume. Given the equal radii of the
obstacles and the new incoming molecule, the excluded volume is 8 times
bigger than the actual volume occupied by the hard spheres! If we now
assume that the obstacle form some kind of spatial structure with cavities, we
oncoming molecule will spend a lot of time to enter the structure but once
inside it will have a very hard time leaving it. This is known as the cage
effect. A perfect illustration of this principle was recently presented in the
elegant experiment by Hudder et al. Using detergent they carefully perforated
the walls of ovarian cells without disrupting the cytoskeleton. The wholes
were made so big to allow passage of protein complexes as big as 800 kDa.
Only very slow release of proteins from the perforated cells was observed as
shown on the figure. When however they added a drug that destroys the
integrity of cytoskeletal components, a drastically different picture was
observed. The loss of proteins was fast and profound.
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Size-dependent viscosity and failure of propagation
1
Using Einstein theory of diffusion, the effect of
crowding can be represented as viscosity
dependent on the radius of the particle R:
D
1 D = 6πµ ( R) R / kT = C (T )µ ( R ) R
R
For large particles R = 6 10-7m viscosity was
found to be 200 Pa sec 105 times viscosity of
water (Bausch et al, Bioph J 1999, v 76, 573)
Rcr
In fact, after some nonlinear range of
viscosity there exist the critical particle size
above which particle diffusion can be
neglected over biologically realistic time.
From the available data (Verkman, TIBS
2002, v 27, 27) critical radius corresponds to
molecular weight of ~ 1 million Da (RNA
polymerase assembled)
kDa
Let us now try to formalize the influence of cage effect on molecular diffusion. If we continue
to use the Einstein’s theory of diffusion, we will come to realization that diffusion coefficient
cannot be simply inversely proportional to the effective radius of the particle (we again
assume approximately spherical molecules). In fact we will have to request that the viscosity
of the medium surrounding the molecule (or molecular complex) depends on the size of the
molecule itself. In this case the cage effect will result in the observation that starting from
some radius the viscosity is not longer constant but starts growing. At some value of Rcr the
viscosity shoots to infinity and the diffusion propagation of the particle is no longer possible
on biologically reasonable time scales. This effect results in enormously different estimates of
cellular viscosity depending on how the authors measured viscosity. Thus biochemist working
with metabolites and small proteins would estimate viscosity of cytoplasm to be only slightly
higher than that of water. On the contrary bioengineers using microne-sized beads driven by
magnetic field found viscosity of the cell to be about 200 Pa s which is five orders of
magnitude higher than water.
An excellent reference material is provided by the figure from Verkaman who plots the ratio of
diffusion coefficients in cell and water. It shows that for majority of molecules including
midsized proteins, the viscosity of cytoplasm is only 4-5 times higher. For large complexes
however there is a clear propagation failure threshold that seems to be reached at the level of
molecular weights of about 1 MDa. This corresponds to a molecule of DNA of roughly 1kbp or
a very large protein complex (e.g., RNA polymerase). It has to be noted that for prokaryotes
this critical size should be several times smaller as their cellular interior is more overcrowded.
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Sticky business of biological diffusion
Diffusion of proteins inside the cell strictly cannot
be described as diffusion of inert spherical
particles through viscous liquid.
Instead, diffusion of macromolecules should be
considered as sequence of diffusion and
binding/unbinding events.
Apparent diffusion constant of PDZ domain
containing proteins was found to be 2-3 times
smaller with the PDZ domain deleted (Haggie et
al, J Biol Chem 2004, V 279, 5494).
With specific binding taken into the consideration,
diffusion coefficient depends on the location in
the cell.
The pure steric effects are by far not the only reasons for macromolecules to
diffuse slower inside the cells than in the water. In reality protein molecules
are forming lots of interaction with other diffusing molecules or with more
stationary parts like cytoskeleton and organelles. In fact most of vitally
important signaling molecules are endowed with a variety of binding cites like
PDZ domains. Constant binding / unbinding along the way results in slowing
down by a factor of 2 to 3 times as was shown in experiments with the cystic
fibrosis transporter and interacting with it molecule EBP50.
In fact to make the recall of the effects of cytoplasmic overcrowding easier,
one can use the following comparison between cellular and highway traffic.
The time that is needed to get from A to B by a car is dependent on the
speed of the car (aqueous viscosity, only 5 times higher than water), the
route taken (the more obstacles, the more detours need to be taken) and the
time spent at the traffic lights (binding to over macromolecules). To add the
effect of the critical size to this picture, we just need to assume that the trucks
with height exceeding the clearance of the bridges simply cannot go
anywhere.
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Molecular motors – solution to the transport problem
All large biological particles of subcritical and
supercritical size are actively transported
through the cytoplasm with molecular motors:
myosin, kinesin, dynein, etc.
Intracellular transport is also utilized by
parasites: bacteria and viruses.
Despite significant diversity in physical
mechanisms of molecular motors, all of
them eventually convert chemical energy
stored in ATP into mechanical work.
For mathematical modeling, see textbook
by Fall et al. Diffusion description is not
applicable in this case.
From Schliwa & Woehlke, Nature 2003, v 422, 759
On the other hand, cells need to transport lots of different things exceeding
critical size across and in and out. How do they solve this problem? Recently
it has become increasing clear that the majority of large components are
being actively transported along the microtubules by molecular motors. The
topic of molecular motors is fascinating in itself but for our purposes we only
need to know that they are used for directed non-diffusive transport of large
cellular particles. Until now, motors present significant challenge for the
biophysicists trying to explain their mode of action. There exist several
designs of motors and surprisingly so it appears that the nature reinvented
these molecular devices from scratch rather than reusing the already existing
components. However, the absolute majority are similar in their major
principle of action. The chemical energy of ATP is utilized to convert some
protein part into a loaded configuration which while relaxing to its ground
state, performs the useful work of carrying the cargo. Principal difference
between this mode of motion and the thermal diffusion is that in case of
molecular motors there exists a clear direction as opposed to purely random
Brownian motion driven by thermal fluctuations. There is a number of texts
presenting mathematical description of the molecular motors themselves,
however the description of transport mediated by molecular motors is still a
mostly unsolved problem.
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Implications of molecular crowding for chemical kinetics
In non dilute solutions, concentrations of
species should be replaced by activies ai
where activity coefficients are
γ i ≡ ai / ci = vtot / vi
ln γ i =< g i > / kT
γi
The apparent equilibrium constants (e.g.,
for binding) observed in the cell are
related to thermodynamic constants K0
K12 = c2 / c12 = K120 (γ 21 / γ 2 )
As a result, equilibrium constants in the
cell for typical conditions may be 10 – 100
different from those measured in dilute
solutions (Minton, J Biol Chem 2001 v
276, 10577).
Until now we only discussed the effects of overcrowding on cellular transport.
For completeness, we should consider the influence of crowding on reaction
kinetics. Physical chemistry teaches us that in non-ideal solutions (that is with
non vanishing concentrations of molecules) we need to replace
concentrations with activities. Activity coefficient gamma has a physical
meaning of the average free energy of non-specific interaction between
species i and the rest of species. For very dilute solutions, gamma is close to
one and it is possible to use concentrations directly. As the concentration
grows, gamma also grows and in dense solutions it may exceed actual
concentration by the factor of 100! This has everything to do with the effect of
the excluded volume or steric repulsion. The consequence of this
phenomenon for the chemical reactions is that all the kinetic constants will
effectively depend on the intracellular concentrations. Therefore, chemical
kinetics measured in vitro may be significantly different from actual behavior
in vivo. This needs to be taken into the consideration when building
mathematical models for real biological processes using experimental data
from in vitro conditions.
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Diffusion equation for high-density media
r
x
The derivation of the diffusion equation
was based on the empirical Fick’s law
J ( x, t ) = − D
∂
c (x, t )
∂x
which assumes simple proportionality of
particle flux to the gradient of
concentration at the point x. At high
densities and degree of inhomogeneity
we should instead consider (see Murray):
r r
J ( x, t ) = F [∇c ( x + r , t )]
From some arcane considerations of
symmetry and isotropy (again Murray):
J = − D1∇c + ∇D2 (∇ 2 c)
R
This results in the transformation of the
reaction-diffusion equation into :
r
r
∂ c ( x, t )
∂  ∂ r

= f (c, x , t ) +  D1 c( x , t )  −
∂t
∂x  ∂x



∂2 r
 D2 2 c ( x, t ) 
∂
x


Where diffusion coefficients can be
functions of concentration, space and
time.
−
∂2
∂x 2
Finally we are going to question we diffusion equation itself! But what can be
different in the crowded conditions from the point of analytical form of
diffusion equation. Please recall that the derivation of the diffusion equation
consists of two parts and is based on two different laws. The equation itself is
a form of conservation of mass and therefore the general form of it stays the
same for all systems. However, in the course of derivation we used an
empiric law which connects the diffusion flow with the concentration field.
Fick’s law declares simply that the flow J is proportional to the gradient of the
concentration. This statement is so intuitive that its validity is usually not
questioned. However, Fick’s law was tested on systems with very smooth
variation of concentration field. Inside the cells the heterogeneity of molecule
distribution can be very high (e.g., due to inhomogeneous cage effect) and
the Fick’s law may require some amendments.
James Murray, based on early works of Hans Othmer, suggests that in this
situation higher order terms should be included into the consideration. That
brings to the form with conventional Laplassian as well as the fourth
derivative with two diffusion coefficients. This potentially interesting result
however has not been extensively explored until now.
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What to take home
• The cellular interior is highly overcrowded by the presence of molecules whose
sizes vary over many orders of magnitude
• Diffusion of molecules inside living cells is affected by fluid-phase viscosity, specific
and non-specific binding and steric repulsion. This makes diffusion coefficient to be
dependent on overall concentration of species, radius of the particle, chemical
properties of the molecule and location in the cell
• For small to medium molecules (from ions to signaling proteins) diffusion coefficient
in the cytoplasm is only 3 – 5 times lower than in the water
• There exist a critical size of particles that can propagate through the cell by means
of diffusion
• All large protein complexes, DNA molecules, secretory vesicles, organelles and
endocytosed bodies are actively transported in the cell by molecular motors
• Biochemical activity, protein folding and signal transduction are strongly influenced
by the molecular crowding/ steric repulsion / cage effects.
• Given high concentrations and inhomogeneity of spatial distribution of molecules in
the cell, Fickean diffusion equation may need to be replaced by more complicated
form.
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