University of LjubljanaFaculty of Mathematics and Physics

Seminar Ib - 1. letnik, II. stopnja

Detailed structure ofmitochondrion

Author:Jure Lapajne

Advisor:Prof. dr. Primož Ziherl

29. junij 2015

1 Introduction

The first detailed experimental observations of mitochondrion were done using electron microscopy(EM) r1s. It was one of the organelles to receive the most attention. Physicists were especially interestedin the structure of its membranes. For a long time the generally accepted models depicted mitochondrionas an organelle of elliptical shape made of two membranes. The outer protects the inner structuresand separates them from cell matrix. The inner membrane is very curved (it forms cristae (curvesof membrane)) and carries structures necessary for oxidation processes. This early model has foundits way into textbooks r1, 2s. It has persisted until today, although there were many scientists whoexpressed doubts and also conducted different experiments and observations of mitochondria.

Electron microscopy had been for a long time the main method for investigation of mitochondrionstructures. Unfortunately, it did not give definite answers. But research has continued and evolved. Inrecent years another technique has emerged. It is called electron tomography (ET). ET has enableddetailed 3D pictures of the internal structure of mitochondrion r1, 3s. The latter appears to be verycomplex and thus interesting for theoretical and experimental physicists.

2 First studies

In early 1950s Palade and Sjostrand were first to observe internal structure of mitochondrion. Bothobserved that the mitochondrion contain more than one membrane. In the Sjostrand model there arethree membranes - the outer and the inner boundary membranes and a third one inside the innerboundary membrane that forms septa and divides the matrix (content within the inner boundarymembrane) into compartments. Palade only saw two membranes, but in his model (Fig. 1a), the innermembrane curves inwards to form curves/baffles called cristae r1, 3s.

Observation of mitochondria under electronic microscope at different respiration rates revealed thatthere are actually two boundary membranes - inner and outer boundary membrane and those two are notcurved/folded or in any way deformed - they are considered smooth r1s. Within those two membranesthere is another membrane (called the inner membrane) that forms folds and other invaginations calledcristae (Fig. 2c). There are different opinions among scientists, whether this innermost membrane canbe called membrane at all. Since it is so curved and full of different structures, it does not look likea membrane any more. In fact, some publication considers cristae as features inside inner boundarymembrane r2s.

3 Cristae junctions

Observations of mitochondria under EM at different respiration rates yielded another interesting result,showing that cristae are connected to inner boundary membrane by connections called cristae junc-tions (Figs. 2a, 2b) r1s. Unfortunately, EM is not suitable to determine the shape and size of theseconnections.

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Figure 1: (a) The Palade model of mitochondrion, offen called the "baffle" model. It is the mostcommonly depicted model in textbooks. (b) An alternative model. It supplemented Palade modelespecially for mitochondria found in higher animals. It provided explanation for structures observedusing electron tomography. The topology of membrane predicted by these two models are quite different.Perhaps the biggest difference is the absence of tubular structures called cristae junctions, that connectmembrane invaginations to periphery of membrane, in the baffle model. Both models have been provedto be inaccurate r4, 5s.

The interpretation of an image produced by EM is a challenging task. Mitochondria usually formlarge complex reticula. There are overlapping structures and inelastically scattered electrons seen inimages. The latter problem arises because the observed specimens are of a thickness of about 0.25-0.5µm r1, 3, 4s. Fortunately high voltage electrons are available. Required operating voltage ranges from400 kV to 1000 kV r1s. A 3D image is needed to distinguish overlapping structures. Electon tomographyis a technique that enables us to observe 3D images. It is esentially electron microscopy, but specimenis observed from different angles. Further computional image manipulation enables joining informationfrom individual images that correspond to different angles, to create a single 3D image. Lately thetechnique has been improved further. Cryo-electron tomography is a technique where the specimen isobserved at very low temperatures - usually at liquid nitrogen temperatures. The advantage is thatcells or mitochondria do not need to be chemically impaired and as a consequence the structures arepreserved much better than in case of ordinary tomogram r1, 2, 6s.

The shape and size of cristae junctions were determined by the (cryo) electron tomography tech-niques. The results were consistent among different specimens taken from organisms belonging to thesame species and also among organisms from different species. Crista junctions are of a tubular/trumpetshape (fig. 5). Usually those tubes have circular intersection, but sometimes they are elliptical. Theirdiameter ranges from 24 nm to 30 nm r1, 2, 3, 6s and their length is 20 nm-50 nm r6s. However,the exceptions do exist. Fungus Neurospora crassa represents an extreme deviation. It has slot like

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junctions, which can reach 200 nm r6s in length, though, the average length is much smaller.

4 Cristae shape

As suggested by figs. 2c and 2d, cristae can take different shapes. Their shape depends on the en-vironment the mitochondrion is positioned in and also on chemical balance inside the mitochondrion.Although those states are topologically different, it appears that the cristae shapes are dynamic. De-pending on matrix volume, outer membrane shape, chemical balance in mitochondrion, cristae canundergo fusion or fission to form different shapes. One should also not be confused by the Fig. 5d de-picting tubular shapes morphing into flatter lamellar shapes. Those shapes coexist and bond together.Additionally, two major states of mitochondrion have been identified. They separate themselves by theshape taken by the majority of cristae r2, 6s.

4.1 Ortodox conformation

The ortodox state (Fig. 3) is the most probable state for a mitochondrion in a cell. This conformationalso corresponds to an isolated mitochondrion accompanied by slow respiration rates. Characteristic ofthis state is that the matrix volume (content within inner membrane) is large. The matrix pushes theinner membrane against the outer membrane. In this conformation cristae shapes tend to be tubes.It is possible for several tubular cristae to join and create short lamellar structures r1, 2, 3, 7, 8s.Larger lamellar structures are also possible, but not very common. Possible cristae shapes are nicelyillustrated in Fig. 2d. There is single tubular cristae at the top of mitochondrion (light blue). Theyellow structure below consists of several joined tubular cristae. This one is quite large and consists of5 to 10 tubular cristae.

4.2 Condensed conformation

Condensed conformation (Fig. 3) is another form mitochondrion can posseses. It has been observedin isolated mitochondria, subjected to high respiration rates. This conformation has two significantfeatures. The matrix volume inside inner boundary membrane is not very large and consequentlythe latter moves away from outer membrane - except at sites where the two are connected. Thiscauses intermembrane space to swell. The second feature are cristae shapes. There are many morelamellarly shaped cristae with multiple cristae junctions r5, 1, 2, 8s. Although this conformation hasbeen observed on tomograms, it is not very clear what relevance it has for active mitochondria inorganisms.

The described conformations are not static and can morph from one to another. Thus cristaeare also not static structures. Upon swelling of matrix (transition into ortodox state), cristae startundergoing fission process, while fusion process appears when matrix starts to contract (transition intocondensed state)r2s.

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Figure 2: (a) Electron micrograph obtained by 400 kV electron microscope. Circular structures in-dicated by the arrows, represent examples of crista junction. (b) A section produced by electrontomography. The vertical arrows point to intersections of cristae junctions. The inset at lower leftdepicts an area along the vertical axis of the tomogram. It is zoomed in figure of an aera where theblack line is drawn. Circular objects, pointed by horizontal arrows, are cristae junctions. (c) Com-putionally manipulated 3D tomogram. Outer boundary membrane is depicted by dark transluent blue,inner (boundary) membrane is shown in light transluent blue, while yellow color shows cristae. (d) Atomogram of a mitochondrion with only 4 cristae. Light blue color at the top of mitochondria depictstubular cristae. Such shapes are more common in the orthodox state. Yellow lamellarly shaped cristaeare more common in condensed conformation. Such large cristae forms in a fusion process from multipletubular cristae. There are two other examples of cristae at the bottom of mitochondria. They representsmaller, more common, lamellar cristae r1s.

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Figure 3: Comparison of condensed and orthodox state mitochondrion. Clearly condensed state featuerscristae which look bigger and fatter, while orthodox state features more tubular like cristae. HighADP means high respiration rate, which corresponds to condensed state, while low ADP means lowrespiration rate and corresponds to orthodox state r2, 5s.

5 Causes of cristae and cristae junction shapes

Membrane topology depends on multiple parameters including biochemical factors (reactions, certainprotein properties), but there also are physical factors. Curvature energy, intristic curvature of mem-branes caused by protein composition, dynamics of membranes, physical interactions with outer mem-brane (volume constraints, bridges between outer and inner membrane) can all be assigned to morephysical properties.

It has been proposed that cristae junctions are thermodynamically stable structures r1, 7, 9s. Thefirst evidence for this claim comes from observation of distribution of cristae radii. Distribution of aform:

dP

dx“ e´p

axm`bxq (1)

fits the observed distribution of radii better than gaussian distribution. This suggests that radius ofcristae junctions are not just random.

The second evidence comes from observations of protein and lipid movement within the membrane.The movement is rapid, almost free-like. This suggest that molecules are able to take thermodinamicallymost favorable positions.

Although thermodynamically stable, cristae and cristae junctions are not static strucures. Theyform, reshape and disappear. The actual physical reasons for this may be found in changing innermembrane surface/volume, matrix volume and intermembrane space volume r1s. At this point physicalreasons almost can not be separated from chemical reasons. Respiration rate, pH gradient,.. caninterfere with protein structure and thus also affect membrane shape.

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Figure 4: Distribution of cristae junctions. Dotted line represents measurements from different to-mograms, while solid line shows Boltzmann distribution using model which will be described in nextsection r1s.

6 Mathematical modeling of membrane shapes

As we have seen, there are some distinct shapes present in membranes of a mitochondrion. In thissection I will focus mostly on description of structures of inner membrane, especially the shapes ofcristae and cristae junctions.

6.1 Cristae junction shape

The fundamental formula, which describes membrane energy as a function of its shape is

E1 “ k1

ż

dApc1 ` c2

2´ c0q

2 ` k2

ż

c1c2dA` σ

ż

dA`∆pV. (2)

c1 and c2 are the principal curvatures and c0 is the spontaneous curvature of membrane. The secondintegral involves the gaussian curvature and is zero for membrane of constant topology - as in this case.The bending constants k1 and k2 depend on composition of membrane. The third term describes surfacetension, while the fourth term accounts for osmotic pressure difference across the inner membrane -i.e. osmotic pressure difference between matrix within inner boundary membrane and intermembranespace r9, 10, 11s.

Minimization of energy means finding surface shape which minimizes the total energy. Based uponobservations of cristae junction shapes the following parameterization xpu, vq, ypu, vq and zpu, vq hasbeen proposed r11s:

x “ rpR

r` 1´ cospuqq cospvq, (3)

x “ rpR

r` 1´ cospuqq sinpvq, (4)

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Figure 5: Shape of cristae junction. The drawing shows parameterization described by equations (3)-(5). It is in fact torus parameterization, but boundary conditions ensure that the shape resembles atrumpet. By definition, the radius R is where surface of junction is vertical and radius R`r is locationof horizontal surface.

x “ rp1´ sinpuqq. (5)

However, this parameterization in fact describes torus like shape and boundary conditions are necessary.Parameterization is obeyed until vertical and horizonatal surfaces are reached (Fig. 4). Horizontalsurface is reached at x2 ` y2 “ pr `Rq2 and vertical surface is reached when x2 ` y2 “ R2. Boundaryconditions correspond to parameters u and v belonging to interval r0, π{2s and r0, 2πs respectively.

Using this parameterization, one can obtain energy of crista junction:

E “ k1

ż π{2

0du

ż 2π

0dv

ˆ

´2 cospuq ´R1 ´ 1

2pcospuq ´R1 ´ 1q´ c10

˙2`

R1 ` 1´ cospuq˘

, (6)

where c10 “ c0r and R1 “ Rr . Integration yields

E “ 2k1π

˜

pR1 ` 1q2?R12 ` 2R1

tan´1

˜?R12 ` 2R1

R1

¸

´

ˆ

c102` 1

˙ˆ

2´ πc102

ˆ

R1 ` 1´2

π

˙˙

¸

(7)

The energy only depends on R1 “ R{r. The ratio caracterises the shape of crista. Thus energy doesnot depend directly on size of junction but rather on the shape of junction.

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The next step is to calculate distribution of cristae as a function of R’. Distribution can be calculatedusing partition function:

P pR1q “e´

EpR1,c0q

kT

ş8

0 dxe´Epx,c0q

kT

. (8)

Taking limits R Ñ 8 and R Ñ 0, one can connect equation 8 with equation 1. Further, usingexperimental data different parameters can be calculated. Measurements of R from different speciesfit the distribution described by equation 1 nicely. Measurements of R’, however, do not match thedistribution at all. Further analysis shows that there are differencies in size of r among the species.Measurements of R1 have to be divided into multiple groups to account for differences in r and afterthat each group can be fitted described model. Fitting the model shows, that there are differencesmainly in spontaneous curvatures c0 of membranes of different species, while k1 stays approximatelythe same. The differences in c0 likely come from different protein structures of different species r11s.

6.2 Cristae shapes

Two cristae shapes have been observed - tubular and lamellar shapes. The model tries to explain both.It is based on energy minimization principle. Again, the energy is calculated by the Helfrich membraneenergy presented already in previous subsection, however, it is convenient to also include a term dueto external force ´fl.

In general, cristae vary in size and shape. A generalised model desribing cristae thus includs severalgeneralisations and simplifications. A simple model of crista consists of (Fig. 6):

1. Body of a cristae; a circullar lamellar membrane of radius R and thickness 2r.

2. Membrane wrapping the body of crista. It is a half-torus shaped membrane. For simplicity, it istaken as half-cylinder. Its radius is r and length L. Larger radius R is negledcted since in mostcases R " r.

3. N tubular membranes growing out of body of cristae. They connect the cristae to the innerboundary membrane. They are of radius r and length L.

Places where described shapes morph together are in this model neglected. For simplicity cristajunctions are also omited. Integration is carried out by dividing integral (13) into the three describedparts. Following the integration, one can find the shape of cristae by minimization of energy withrespect to different parameters. For now, the quantitative results of this model are not very important,since C0, σ and κ are not very well known. Still, some estimations can be made. Typical value of κfor biological membrane is 4 ¨ 10´20 J r7, 10s. Further, one can obtain average values of r, R, L and Nfrom observation. This enables us to calculate surface tension constant σ, tensil force f and pressuredifference ∆p.

Obtained tensil force of around 20 pN seems very resonable. Common tensil force exerted by proteinsin average cell is in vicinity of 5 pN. Calculated pressure difference is negative and is ranging form ´10kPa to ´5 hPa r10s. Although the range is large, negative pressure difference fits the observations.

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Figure 6: Simple model of crista consists of cilcular lamellar body, wrapper membrane and tubulesgrowing out of the body connecting to inner boundary membrane r10s.

Calculated values for surface tension range from 0.023 pN/nm to 0.28 pN/nm, which is below thecritical surface tension, for which an average biological membrane ruptures r10s.

One can also compare parameters N , R and L of the model with experimental results. In order todo that, good approximations for surface tension and pressure difference need to be made. Appropriatenumbers would be ∆p “ ´34 hPa and σ “ 0.058 pN/nm, while κ is the same as before. Applyingdescribed values to a mitochondrion found in HeLa cell results in some interesting comparisons (Fig.7,8), r10s.

6.2.1 Tubules or lamellae

The coexistence of tubules and lamellae can be explained as two-phase equilibrium model. Simpleexamples of such equilibrium states are helical ribbons under a stretching force. Some parts unfoldand the ribbon is separated into straight and curved parts. Something similar happens to polymerssubjected to a stretching force - some parts stay coiled whereas others stretch and straighten. Pullingtethers from vesicles also shows similar behavior. At a certain point when a tether is pulled from thevesicle, the force becomes independent of extension. As shown in Fig. 8b, this happens also in the caseof tubules. It turns out that lengthening of tube (increasing L) results in a slight decrease in the sizeof lamellar part and in an increase in the size of tube radius r. At a smaller l, the radius of tube r issmaller, whereas the radius of the lamella is larger. A crista with a small N and large l mostly consistsof tubular parts, whereas a crista with small N and small l is made up of a small lamellar part andsmall tubular part. A large crista with large lamellar part is possible only at large numbers N r7s.

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(a) (b)

Figure 7: (a) Shows comparison between predicted number of tubular connections from crista bodyto inner boundary membrane from described model and experimental number of those connections.Comparison has been made for 8 cristae found inside selected mitochondrion in HeLa cell. (b) The samecomparison for parameter L. As can be seen the model predictions are not far away from experimentalresults. Perhaps further model improvements and better values for membrane constants could improveresults r10s.

(a) (b)

Figure 8: (a) Comparison between predicted length of tubular connections from crista body to innerboundary membrane from described model and experimentally measured length of those connections.Comparison has been made for 8 cristae found inside selected mitochondrion from HeLa cell. (b)Parametric plot of force f versus length l “ L`R as a functions of tube length L at different numberof tube N . Data used in modeling are data calculated from this model; ∆p “ 34 hPa and σ “ 0.058pN/nm. Upper branches correspond to smaller L and are not observed in experiments, while plateaucorresponds to larger L r10, 7s.

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7 Conclusion

In this paper we presented the most important findings about structure of the mitochondrion. Thereare still some unknows and more measurements need to be done to determine the physical properties ofmembranes. On theoretical side I presented a model describing crista junction and a model descibingcrista. Especially the latter seems very simplified and could be improved. The inclusion of chemicalpotential, temperature dependance (enthropy) would also be welcome. It is also not entirely explainedwhat happens when mitochondrion transforms from orthodox to condensed state and vice-versa.

References

[1] T.G. Frey, C.W. Renken, and G.A. Perkins, Insight into mitochondrial structure and function fromelectron tomography, Biochemica et Biophysica Acta 1555, 196 (2002).

[2] Carmen A. Mannella Structure and dynamics of the mitochondrial inner membrane cristae Biochim-ica et Biophysica Acta 1763, 542 (2006).

[3] T. G. Frey, C. A. Manella, The internal structure of mitochondria, Trends in Biochemical Sciences25, 319 (2000) .

[4] T.G. Frey and G.A. Perkins, Recent structural insight into mitochondria gained by microscopy,Micron 31, 97 (2000).

[5] D. Logan, The mitochondrial compartment, Journal of Experimental Botany 57, 1225 (2000).

[6] Carmen A. Mannella The relevance of mitochondrial membrane topology to mitochondrial functionBiochimica et Biophysica Acta 1762, 140 (2006)

[7] Ghochani, M. (2010). Experimental and theoretical study of mitochondrial inner membrane con-formation: electron microscope tomography and thermodynamics (Master’s thesis). Retrieved fromhttp://oatd.org/oatd/record?record=handle%5C%3A10211.10%5C%2F298

[8] M. Zick, R. Rabl, A. S. Reichert, Cristae formation—linking ultrastructure and function of mito-chondria, Biochimica et Biophysica Acta - Molecular Cell Research 1793, 5 (2009).

[9] A. Ponnuswamy, J. Nulton, J. M. Mahaffy, P. Salamon, T. G. Frey, and A. R. C. Baljon, Modellingtubular shapes in the inner mitochondrial membrane, Physical biology 2, 73 (2005).

[10] M. Ghochani, J. D. Nulton, P. Salamon, T. G. Frey, A. Rabinovitch, and A. R. C. Baljon TensileForces and Shape Entropy Explain Observed Crista Structure in Mitochondria Biophysical Journal99, 3244 (2010).

[11] C. Renken, G. Siragusa, G. Perkins, L. Washington, J. Nulton, P. Salamon, and T.G. Frey, Athermodynamic model describing the nature of the crista junction; a structural motif in the mito-chondrion, Journal of Structural Biology 138, 137 (2000).

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