So, after the lectures today on plant and soil physics, I want to talk
a little bit about physical limits to life.
So, it's also something that Gaby probably touched on in the first
lecture that discussed the work of Voko, who is obviously a very
important person in this field.
And I just thought I'd start out with a little bit of observations on
life in general, or I mean life as we observe it in nature.
And I like this picture here.
It's a painting of the entry of animals into Noah's Ark.
Obviously, we don't see the entry of the plants into Noah's Ark, but
that's a completely different story why that is so.
But if you just look outside in nature, you could get the impression
that everything is possible.
If you look at these animals here, some are small, some are big, some
have two legs, some have four legs, and they use a variety of
mechanisms for drinking, feeding, doing locomotion, even flying.
So, it seems everything is possible.
I will argue that this is not the case, that physics imposes
fundamental limits, which is driven somewhat by the pressures of
evolution, like we discussed earlier today.
Here are some of the papers we are discussing today.
Recently, we had an important event, a centennial of a book by Darcy
Thompson.
This is a Dutch translation, On Growth and Form.
It's a really important book that I highly recommend.
It's also very elegantly written.
Essentially, he discusses the effects of physics on shape of organisms
and size of organisms.
Essentially, this whole field is about understanding what constrains
biology, and what constrains the mechanisms that are possible and not
possible in nature.
Like the previous lecture, I would like to start with a question.
It's a very simple question.
Why are there no wheels in nature?
We have lots of wheels in technology.
It seems like an obvious thing.
Why don't they exist in nature?
Maybe you can discuss a little bit.
Let's start again.
We had a good discussion why there are no wheels.
People pointed out there are tumbleweeds, which are a wheel-like
strategy.
But a major constraint is that the world is not flat.
If you try with your bicycle to go over a hole like this in your road,
you will have problems.
It's not that wheels are not possible in nature.
It's that they're perhaps not practical.
That's at least one point of view that you could have.
But I want to show one example of a wheel, which is not the
tumbleweed, but which is the dung beetle.
The dung beetle feeds on dung.
It makes these balls made from dung, perfect spherical objects, and
then rolls them over a flat surface.
They basically just try to roll them to wherever there are no other
dung beetles, and then they eat them.
They just try to get them away.
People have been interested in how they navigate.
Does anyone know how they navigate?
Any ideas?
The sun?
Good.
Good suggestion.
What about at night?
How would you navigate?
Yes?
Magnetic field?
That's also a possibility.
Some organisms use that, yes.
In this particular case, it's the Milky Way.
Researchers have shown.
Anyway.
Yes.
Many different strategies are possible, but we need to return to the
question of what drives evolution.
I think we go back again and discuss and see if we can come up with
some good...
Alright.
This is a really broad question that I think has many answers.
One of my personal favorites is, like we also discussed earlier today,
a perspective due to Lotka, which is almost 100 years ago, which is
kind of... there are two ways that you can be good in biology.
One is to be very efficient, very economical, so to be able to use
your resources more efficiently than the organisms you are competing
with.
The other one is just being able to use the most energy, which is a
little bit non-trivial if you think about it.
But it's like if you can use the most energy in the most efficient
way, that's the best thing.
And so it's clear that there are differences between elephants and
mice in their metabolic rate and so on, and they have different
strategies and different objectives.
But you also have this in nations.
If you have the energy use per GDP, so if you take this graph up here,
you have the United States, for instance.
Just after 1900, they used 0.5 tons of oil per $1,000 GDP.
So that's one strategy for growth.
If you want to be number one in the world, which the United States
arguably was, at least in the last century, this is one strategy to
becoming number one.
And you see clearly China is picking up on that idea.
So from the 60s on, roughly, they have spent the most energy per GDP.
So GDP is roughly like how much you...
the production of your country.
So one strategy for nations, at least, is just to use most energy.
So the energy efficiency, I personally think, is a really important
factor in life and in society.
So I want to start by talking about a particular problem, which is the
size of living organisms, just in general.
What could possibly set these limits to size?
And so one fascinating story is that most single-celled organisms, so
bacteria, for instance, tend to be quite small.
So there are no giant bacteria.
There are some algae that are single-celled that can get to centimeter
size, but that's about it.
And so to understand why they are relatively small, I just want to
talk briefly about how they feed.
So the idea of how they feed is a diffusion process.
So suppose you are bacteria, you're sitting inside the soil that we
heard about earlier, and you have to feed somehow.
And food is far away, it's evenly distributed.
So I have my bacteria at this red dot.
Then I have ample food supply, infinitely far away.
And I'm very good at eating, so I can keep the concentration of my
food zero here.
And I can keep the concentration of food very far away, constant as
well.
Then I can think about how much food can I take up as a function of
how big I am.
So we need to solve this diffusion equation again, that we talked
about this morning.
Concentration can change in time, you can have changes in transport
due to velocity fields, so things being moved around by liquids, and
you can have diffusion.
In this case, we're interested in the case where everything is steady,
so there's no time evolution, in the case where there's no flow.
So we just have to solve this equation, the Laplace operator on the
concentration is equal to zero, with these boundary conditions.
And so if we assume that a bacteria is approximately spherical, we
just get this equation down here.
So let's look at what the solution is.
So the solution to this particular diffusion problem is this profile,
so concentration is C0, infinitely far away, and zero, very close to
the surface of the bacteria.
And basically what we get, you take this concentration and then we
have to basically integrate the flux over the surface of the bacteria,
that's how much it can take up.
So I have my flux, which is the gradient of concentration multiplied
by diffusivity.
And so the diffusive flux here, scales with diffusion coefficients, so
the faster the diffusion coefficient, the more I can eat.
It scales with the concentration, so the more food available, the
better.
And then it scales with the radius of the bacteria, and then the
position that we are at.
So if I evaluate this at the surface of the bacteria, I get this
expression.
So we have some geometric factor, diffusivity, concentration, and then
the radius of the cell.
So it's not the surface area of the cell, it's not the volume of the
cell, but it's just the radius of the cell.
And this is a big problem, because if I double my size, double my
radius, I can take up twice as much food.
But my volume will grow by a factor of 8.
So it's a bad thing to eat by diffusion.
So essentially, I can take up this much, the metabolic rate, let's
suppose that it scales roughly with volume, so you have radius to the
power 3, so you get some diagram like this.
So down here I have cell radius, and up here I have the uptake rate,
and metabolic rate.
So uptake rate, this is blue line, and metabolism is like this.
So there is some critical size, which is set by the physical
parameters, above which you cannot go.
And apparently for bacteria, this corresponds to a few microns in
size.
So if you just want to eat by diffusion, it's not possible to be any
bigger.
Alright, so this is a simple application of the idea that somehow
efficiency, or rate of uptake, is central in determining strategies
used by living organisms.
If you look at the plant again, it's a much more complicated system.
So there's a lot of things going on.
The major thing that I want to discuss today is the action of
photothensis in the leaf, and transporting the product of that
throughout the plant body.
Because if I can do that faster, if I can do that in a cheaper way,
then that will give me an evolutionary advantage over other organisms.
So I just want to recap again.
The plant is this osmotic pump, so it's really just one long cellular
pipe that pushes sugar out of the leaf by osmosis, generating large
pressures.
And the interesting question is, if I am a plant, how should I design
the system?
To be most efficient?
To give me the fastest transport?
To give me the cheapest transport?
That would depend on, say, the size of the plant.
It would depend on the size of this pipe.
It would depend on the parameters like the sugar concentration, how
viscous the sap is, and so on and so forth.
There are many parameters.
It's not clear how they should be chosen to give the most efficient
transport.
So what we do is to make a little model of this to see if we can
elucidate how those different variables enter into the system.
And this is something that started...
Whoa, this is interesting.
I think there's a minor malfunction with the PowerPoint slide.
So it's something that started with a scientist, Horowitz, who was at
Washington State University, who published the first paper on sugar
transport.
So let me just see if there's something here which is not...
It has just taken the wrong...
Anyway, we'll fix that in the version that I upload.
But anyway, you can sort of see the outline of what he was doing over
here.
So he was saying that you have two parallel pipes, one for xylem, one
for phloem, and then you quantify the rate of exchange of liquid
between the two.
So you say that water can go between xylem and phloem, either because
you have a difference in concentration of solutes or a difference in
concentration of sugar.
So that's like I showed here, so water can enter down here because the
water potential or the chemical potential of water is lower inside
than outside.
It can also exit the tube if the pressure inside becomes sufficiently
large.
So that's what's happening up here.
This tube is closed, so up at this end of the pipe, water will
actually be flowing out.
So Horowitz was the first to formulate sort of the conservation law
for this system.
Now unfortunately, these are relatively complicated coupled partial
differential equations in space and time.
So finding these sort of closed form solutions is a really painful
exercise.
So a lot of progress has been made by using more sort of elaborate
models of plants that are more phenomenological.
So in the sense that you say, oh I have a crown, it's consuming so and
so much.
I have roots supplying so and so much water, I'm doing so and so much
photosynthesis.
Now the problem here lies in the number of parameters.
So these tables that I show here describe the number of parameters
needed to describe this model.
And this goes on.
So there are many parameters needed.
So it's difficult to formulate a good hypothesis that one can test in
an experiment with this type of model.
And this problem is best formulated by Thorp, Laquan and Minchin,
writing in Functional Plant Biology a few years ago.
They have their own model that they describe and they themselves
describe their own model in this way.
The proposed model of flow-on-flow is moderately complex with several
detailed mechanisms described by parameters of unknown value.
Complex models involving a large number of unknown parameters are an
accepted part of theoretical physics and play an important role in the
development of new ideas.
For instance, string theory.
So this is the level of complexity that we are allegedly at.
But I hope that we can do a little bit better today.
We don't need string theory to describe the flow-on.
So what I suggest is to do this sort of hydraulic resistance modeling.
So the idea is to treat the plant as a hydraulic circuit.
So up here in the leaf we have a resistance because water needs to
enter the flow-on pipe.
Along the stem I have resistance because the liquid needs to flow
along or inside the cylindrical pipe.
And down in the root you have the equivalent of the leaf.
Water or sugar sap needs to exit the flow-on pipe again.
Now some of the sap will not go all the way.
So some of it will go only halfway.
But the average transport system is still set by the size of the
plant.
So what we can do is to say the resistance in the leaf is basically
inversely proportional to the surface area that I have available.
So if I have a big pipe, a long leaf, I have a lot of surface area to
do the sugar loading.
And it becomes very easy for liquid to flow in and push out of the
leaf.
So that's radius R, the length of the pipe and the leaf.
And Lp is the permeability of the membrane.
So basically the bigger all these numbers here are, the easier it
becomes to flow in.
So the resistance to flow in the leaf becomes very low.
Similarly in the stem, I have this dependence on viscosity and length
of transport.
And then the radius to the power 4, this Poiseuille-Hugon-Poiseuille
law that we saw earlier today.
So if I put these two together, I say that the total flow is described
by the usual transport equation.
I get a flow speed which looks like this.
So it depends on all the physical parameters.
So permeability of membrane, radius, length of plant, length of leaf,
length of plant and viscosity.
And it's just proportional to the pressure drop.
So this formula has some interesting properties.
If you think of it, well, what would happen if I made my leaf larger?
What would happen if I made my plant shorter?
What would happen if I made my pipe bigger or smaller?
So the most interesting property is if we look in the function of cell
size.
And the cell size here is not a monotonic function.
So essentially if the cell size is very small, you have a lot of
resistance in the stem to flow.
So viscous resistance.
So the flow speed goes down.
If I make the cell very large, then I have to have a very big inflow
of liquid in the leaf to get a high speed.
So it means there is some optimum choice of conduit size for which the
speed is as fast as possible.
And so it turns out that it's basically the radius to power 3
proportional to permeability of membrane, viscosity, leaf length, and
plant height.
And so when I optimize here, I take the speed to make that as large as
possible.
You could say, well, don't you want to maximize the flow rate, for
instance.
But the point is here that here I allow for comparing two small pipes
and one big pipe, which would give the largest flow rate.
And by looking at the velocity, I just maximize the flux.
So for whatever area I allocate to vascular transport, this is the
relevant quantity.
Because a larger individual pipe will have a larger flow rate than a
small individual pipe.
But there's room for more small pipes than big pipes.
So now the question is, do plants know about this?
And it turns out they do.
So here I plot the radius, which we measured on these flown tubes.
And here I plot the length of the leaf multiplied by the size of the
plant.
So these parameters, size of the leaf, height of the plant.
And you see that they fall more or less along this line.
So there are some outliers.
These black dots are herbaceous plants.
They work pretty well.
There are some angiosperm trees.
They open circles here.
They also work pretty, fairly well.
There's the gymnosperm trees that have needles, these square boxes.
And then the gymnosperm trees that have scales are these guys down
here.
So for scale, I don't know if you know what a scale on a plant looks
like.
But it's a little bit difficult to define what you mean by the length
of the leaf.
Because it's like a little overlapping long leaf.
So here we took the smallest length and so that's why it's down here.
So for whatever reason, this guy up here, which has the biggest pipe,
that's one of these sort of cucumber plants or cucurbits, which also
make these giant pumpkins that I showed earlier.
So it's always an outlier.
Whenever we do any of this stuff, it has the biggest pipes, the
biggest fruits, whatever.
I don't know.
It's just something strange is going on there.
But one interesting thing we notice here is that these pipes are on
the order of maybe 10 microns.
They are small pipes.
And there is some variation as you look along the stem of a plant, but
not a lot.
So most of them are basically confined down here.
And it's an interesting question why that is exactly.
Because if you were able to make bigger pipes, then you could have,
according to this theory at least, larger trees, you could have larger
leaves, you name it.
About the physical limits to size of plant leaves.
And there are many obvious limits to plant size and leaf size.
And many of them have to do with mechanics.
So there is a limit to how tall you can make a skyscraper building or
the structure of a bridge.
And these are all fundamentally physical laws.
And many, many people have thought about the size of leaves.
And I hope that we have some new perspective on this, which is related
to transport and not mechanics.
But I want to give a really quick review of saying who was the first
person to actually think about this.
And it's not Leonardo da Vinci.
It's much earlier.
So the first person, at least to my knowledge, is this guy,
Theophrastus, who wrote a book, Inquiry into Plants, which is a
beautiful book, very well written, highly recommended.
And it also contains some minor works on odors and weather signs.
So back then you could do pretty broad research on your stuff.
And this book is inspired by... so Theophrastus was a student of
Aristotle.
And so Aristotle wrote the book on animals.
So then that was taken, so he had to do then the book on plants.
So you have to go one step down.
He's also most famous, perhaps, for his book on human characters.
So he has a nice, highly recommendable book on that.
Anyway, so he writes about... so I don't know if any of you know the
Greek, but I put the English translation just in case.
And you can go back and read this in detail later on.
So I actually have one of my students who is Greek, check the
translation of this particular piece, and it's okay.
But he remarks that there are many differences between leaves in their
size, their number, their shape, their width, their roughness, and the
way that they are arranged on the plant.
And then he has a statement about all the differences, and then he
says, these are all the differences there are.
So, problem solved.
So now the question is, if we just look at the size, what is the
spectrum that we actually have?
And so that's what I try to make in this plot.
So we collect the data from some 2,000 species of trees.
And I just plot it here as a function of how large the total organism
is.
So here you have the height of the plant, height of the tree.
In this case, these are all ankyosperm trees, so flowering plant
trees.
And then I plot the diameter of the leaf.
So diameter is just the longest distance you can measure on any given
leaf.
So the spectrum looks like this.
So there are lots of trees that are down here between 10-20 meters
that have leaves that range from 1 centimeter up to about 1 meter.
So some palm trees have leaves that are about 1 meter in size.
But as you go up in height of the plant, the spectrum becomes more and
more narrow.
And this is not just because there are fewer species.
Because obviously up here, if you want to make statements about 100
meter tall trees, you have one species.
So you have a problem in making general statements.
But this trend is not only due to the fact that there are fewer
individuals up here.
So the clear observation we have here is that there are some forbidden
regions.
So it's basically not allowed to be very tall and have very small
leaves.
It's not allowed either to be very tall and have very large leaves.
And so I wanted to understand the reason why this is.
And I mean, this observation is something that... I mean, we have 20
,000 species here, which is...
I would think it's a fair representation of plants in general.
But obviously there's maybe 1 million angiosperm species.
So this is a tiny fraction.
So some people more recently looked at sort of a more global spectrum.
So there's a paper in Nature from last year, a plant form and
function.
And they have, let's see, plant height for almost 25,000 species.
And so they also have leaf size, they have some seed mass, and so on
and so forth.
And if you cover all these parameters together, you can get plots like
this.
You look at how various things vary.
But so roughly speaking, this axis here along this direction is the
leaf area.
And this axis here is the height.
So you see sort of a similar constraint here.
That the height, for a given height, you can have several different
leaf areas, but it's a tightly constrained volume that it has.
So basically the same thing as what we saw here.
So now the question is, what are these guys really doing?
So essentially, what is a leaf?
A leaf is an organ for capturing solar energy.
So in that case, the bigger the surface area of the leaf, the better.
Then you can say, do I want two small leaves or one big leaf?
And there's definitely some economics of scale there.
But the other function that the plant has to be able to do is to drain
the sugar out.
The sugar sitting inside the leaf is just useless for the organism, so
it needs to be able to pump it out.
And you can see here, there's the mid-vein here and second outer veins
going out.
So they have an elaborate channel network for extracting the sugar.
So earlier today, we looked at how fast you can pump sugar out.
So we did that basically by saying, so I have one conduit inside the
leaf.
So it starts somewhere inside the leaf and then it goes into the stem.
And then it flows to the root.
So, rough approximation.
It gave us this expression for the speed.
So it depends on the size of the conduit, the size of the leaf and the
height of the plant in some particular way described by this equation.
And now, it's important to emphasize, if I look at all these plants,
just a little while ago I showed you that the radius of the sieve
tube, the radius of the pipe, that can vary.
The problem is that in trees, all of these trees basically have the
same conduit size.
So the maximum size, which is diameter of roughly 50 microns, that's
what all of these guys have.
So radius in all of these is constant.
So in this equation down here, I can really only play with the size of
my leaf.
That's the only thing I have to play with.
Let's try to see what happens when I try to find how big should I make
my leaf.
Because if I just look at this equation up here, I can say the energy
flux coming out of the leaf, so the per area of esculature
proportional to speed, shear concentration.
So I just get some equation for that here.
And if I plot that as a function of just leaf size, I get a curve
which looks like this.
So in the beginning, when my leaf is relatively small, there's a lot
to be gained in terms of how much energy you can pump out.
So if I double the area, I can easily double the export capacity, no
problem.
However, as I increase, I keep increasing the size of my leaf, the
flow rate going out saturates.
And the reason is that as I make my leaf bigger and bigger, I have
less and less resistance going into the pipe system.
But my stem is constant, so now that becomes the limiting factor.
So this upper limit here is just set by the constraint of having to go
flow through the stem all the way.
The good question is then, when to stop?
When is enough?
So for this particular case, I think I chose a 20 meter tree, and here
if I have a 40 centimeter leaf, I can export so much, like 90% of the
maximum.
If I have an 80 centimeter leaf, I can export maybe 95.
So by doubling the size of my leaf, I can only export a few percent
more.
So you could say I have to stop when I reached some particular
fraction of the maximum output.
So tau here is a small number, so I want to reach 90-95%, whatever.
And that sets the size of leaf at which I should stop.
So I don't want to make a bigger leaf than this maximum size, which is
set by some fraction of the maximum output.
And so it gives some equation for the maximum leaf size, which depends
on the parameters like this.
In particular, it's inversely proportional to height.
So the bigger my plant is, the smaller my largest leaf should be.
That's the upper limit.
If I want to make a lower limit to leaf size, I can say, well, what is
the purpose of having a vascular system actually?
So the purpose is to move material more efficiently than other ways.
The obvious other way here is to do diffusion.
So if I have one cell here and one there, they can communicate by
diffusion.
But it's not feasible to do diffusion over tens of meters.
So the minimum requirement I have is that the ratio of how quickly...
Oh!
Of how quickly I can do flow to how quickly I can do diffusion, it has
to be bigger than one.
Otherwise, it would not make sense to make a vasculature in the first
place.
We call this a Peclet number.
So the ratio of how quickly you do advection, you go with the flow to
how quickly you diffuse.
Peclet number has to be bigger than one in this case.
This also sets then a lower limit, because as you can see, the speed
increases initially.
So I don't want to have a leaf that's too small.
Then I would not be able to go faster than diffusion, essentially.
So there's no point in having a vasculature in the first place.
So basically, that sets an equation for my smallest leaf size here.
And also, actually a maximum size to the plant.
Because if I demand that my Peclet number be one or ten or something
like that, then what the equation tells me is that this is never
possible if your plant is larger than what this expression tells you.
So again, we can go and ask the plant if it knows about this.
So that's what we'll do.
And these lines here, this is the minimum size, this is the maximum
size up here.
And basically, the upper limit is described by this equation, one over
height up here.
And it actually stops, this line describes 95% or 90% of the maximum
output.
So they seem to stop making bigger leaves when you reach 90% of the
maximum output.
This line down here, this range corresponds to Peclet numbers in the
range one to ten.
So if you are down here, like these, I have no good explanation.
They should not exist.
Principle is not allowed, but maybe they have tricks.
So this lower range here is described by...
And these two lines nicely converge in the range 100 to 110 meters,
which is the size limit to trees that we currently observe on Earth.
Okay?
There's a question.
So, the question is, how about the maximum height?
How did I get it?
So, the maximum height is...
I mean, if you just look at these two curves, you'll see there's a
maximum leaf size and there's a minimum leaf size, and they meet
somewhere.
Meaning that to the right of this point, I can no longer make a plant.
Because any leaf that I can make will not be big enough.
The problem with using this type of data to answer the question, why
are trees no taller than 100 meters, is that there are very few trees
that are taller than 100 meters.
So this data set can be used to describe the size of leaves, but I
don't believe that we can use it to determine the maximum height of
trees.
Because there's only one species up here.
So it's very difficult to generalize.
If I had a whole bunch and they all stopped there, you could make
statistical arguments about it.
Any more questions about size?
Not right now.
Okay.
So I want to finish off by talking a little bit about the sap that's
inside the phloem.
And so a very obvious question today is, can we engineer plants that
just make more sugar?
It seems like an obvious idea.
Why not just modify maize or corn or wheat, just to do more
photosynthesis?
What more do we want?
And people are basically saying, oh, we hack the photosynthesis, we do
a faster rubisco, so some of these proteins that are involved in this
process, and you can do more photosynthesis.
But I will argue that this is not the ideal world.
It's not one in which you make individual plants produce or do more
photosynthesis, because there are some essential bottlenecks in phloem
transport.
And so basically, again, we have to start with a leaf and look at this
main architecture and say, okay, let's try to see.
Okay, I want to get a sample of the phloem sap.
Okay, and as we looked at before, what do we use?
We use these aphids.
You saw the movie with the cutting.
I can play it again if you want.
I don't think we have the sound this time.
And this is like the state-of-the-art of how sap is collected from
plants.
There are some that bleed, so some that you can make incisions in to
extract the sap, but essentially, this is what is done.
Okay, so it's a complicated process.
Now, on some plants, you can do something slightly different, which
is, I show this example here, which is a chamber developed by the
Knoblauch Lab at Washington State University, where you can grow
plants between a coverslip and a fine mesh.
So you have soil in the background, then fine mesh, then plant root.
I don't know if you can see it right here.
Plant root.
And this over here is then a movie that Michael Knoblauch took where
you basically deposit fluorescein, a modified fluorescein, on the
leaf.
And then it's able to enter the phloem channel, so it's membrane
permeable.
And then you basically see this.
So you see sugar entering the vasculature.
I don't know how clearly it shows up, but...
Initially it's just in the leaf, then it goes down here.
You can use this for many, many interesting things.
You can use it to measure the speed of translocation.
In the break we discussed how that's otherwise done, for instance,
with radioactive labeling.
That's a very commonly used technique to measure how quickly something
moves inside plants.
You feed it radioactive carbon, see how it's allocated.
Here you can just look for a dye.
It's slightly simpler.
It relies on the fact that the root is transparent in this case.
Now an interesting thing is that this can actually be used to measure
the sugar concentration inside.
So to do that, we have this root which is filled with the dye.
That's shown on here, all the green stuff is the dye.
And basically what we do is that we apply a laser.
So I turned on the laser at this one point.
I run it for 20 seconds, and then that means that since liquid is
moving from left to right, whatever passes my laser, I will break the
fluorescent bond.
So this is what happens.
If you saw it, you see this pulse moving over here.
You can play it again.
There it goes.
It moves from left to right.
So it turns out that we can use this type of data to actually extract
information about the diffusivity of the dye.
And the diffusivity is linked to the viscosity of the sap, or the
sugar concentration inside.
So to demonstrate that, I show we made like a little artificial plant
here.
I have a number of channels.
This top one just has water.
This has 5% sugar.
This has 10%.
That's the Coca-Cola again.
And yes, we tried to run Coca-Cola through this device to calibrate
it.
And 20% sugar, 40% sugar.
Now these are all flowing at the same speed.
Let me see if we can play this.
Oh no, it's not working.
It worked a little bit earlier.
Anyway, they're all flowing at the same speed.
But what you would have seen in this video is that this dye diffuses,
so it spreads out faster in pure water than it does in highly
concentrated sugar solutions.
So because the liquid is more dense, it's more difficult for the dye
molecules to do diffusion.
So we can quantify how much more, and we can use that to measure the
concentration inside.
So this can be done on some plants.
And I show here some data that we collected on the sugar
concentration.
So again, the question we're trying to answer is, is it a good idea to
supercharge photosynthesis, to supercharge flow and transport?
And these are some 50 species, and this is your concentration.
So down here at 10% we have the Coca-Cola level, down here.
And this up here at 60%, that's maple syrup level.
So we have broad range in between.
And I'm just numbering the species here.
I sorted them according to concentration.
So I'll go into detail with what some of them are in a moment.
So there are two groups.
There's one called active phloem loaders.
So active loaders can actively pump, accumulate sugars inside the
phloem.
So it's the green group shown up here.
So they span the range from, let's see, 50% sugar inside.
That's really concentrated.
It's extremely viscous.
It's almost as viscous as maple syrup.
So from roughly like 20% up to 50%.
And the passive loaders, they just rely on diffusion inside the leaf.
So they cannot go as high.
They go to maybe 20 at most.
And the question is, why do we have this particular distribution?
And what is going on with these extremely high concentration species
up here?
So if we think of this sugar flow again as a pipe flow, we know that
the volumetric flow depends on geometry, radius, length of pipe,
pressure difference, and then it's inversely proportional to
viscosity.
Now the sugar mass flow, so how much energy I can actually transport,
that's the flow rate multiplied by the concentration.
So this whole factor here multiplied by concentration.
And you see there's in the final expression for the flow down here, I
have a term which only depends on geometry.
Then I have something which depends on concentration divided by
viscosity.
Now the viscosity of solutions of small molecules basically don't
depend on which molecule it is.
So down here I show for some different types of sugar.
Sucrose, glucose, fructose, which are all...
I mean sucrose is the most abundant in plants, but these all occur.
Also I have small ions.
Essentially it's the same curve if you plot it as a function of just
mass fraction.
Concentration in terms of mass fraction.
Viscosity divided by this.
So essentially what I need to do is I need to take concentration
divided by this curve.
And it looks like this.
So at around 20 something percent, 23 percent, I get the most for my
transport.
So if I increase the concentration higher than this, a constant
pressure drop, I will actually transport less.
And what you find is that most of these species, they sit sort of
within this envelope.
There's something seriously going wrong with these guys out here.
They're really underperforming in the sense that they have very high
sugar concentration, but they could potentially transport more if
their sap was more diluted.
So does anyone have an idea what these plants might be?
Is it the cucumber?
No, not the cucumber.
Maple?
Don't let me get into maple syrup, that will be a different time.
Sorry?
Aloe plants?
That's a good suggestion, not to my knowledge.
They're not particularly concentrated.
Yes?
Also a good suggestion?
Yes?
So the answer is that these guys are actually crops.
So they are not natural.
They have been selected, engineered by humans to do as much
photosynthesis, to accumulate as many seeds, as large seeds as
possible.
And one consequence of that selection has actually been that we have
supercharged the transport system already now too much.
So if we invest in even more photosynthesis for these guys, they'll
just move even further out here.
And essentially the viscosity of the sap will just kill the transport
process.
Which is, I think, an interesting point.
Okay.
So I want to finish off with just showing one slide on a conference
that we're organizing in Copenhagen in just a few weeks.
In case some of you, I sparked interest in some of you, you can have
two days of nothing but the flowing.
What could possibly be better?
We have an exciting program, a good lineup of speakers.
Otherwise, I think I'll thank you for your attention.
I'll be happy to take questions.