Deducing cellular processes from cell-size distributions

(1)

Deducing cellular processes

from cell-size distributions

Junghyo Jo and Vipul Periwal

Laboratory of Biological Modeling

(2)

My group focuses on

Avoid the difficult questions of appetite control and addiction

The gas tank (a.k.a.

adipose tissue)

and the generator

(a.k.a. mitochondria)

(3)

The gas tank

•

Mammals cannot excrete much excess fat

directly. Apparently, there was no evolutionary pressure to include such a process.

•

Mammals can sequester fat in adipose tissue

and use this dense energy source as needed.

•

Insulin is a signal for substrate switching,

shutting off lipolysis and facilitating glucose entry into tissues that require active

(4)

What happens when

the gas tank overflows?

•

Can adipose tissue grow indefinitely?

•

Apart from mechanical stress on joints

and the heart, does stored (=

(5)

Working hypothesis

A dysfunction in fat storage results in an

underutilization of glucose and a loss in

metabolic flexibility, partly from

ineffective insulin signaling. High serum

glucose leads to damage in tissues that

are unable to control glucose entry.

(6)

The dynamical aspects of adipose tissue

growth have been studied but experimental

limitations preclude direct observations.

We use mathematical modeling of

cross-sectional and longitudinal data to

understand the processes by which adipose

tissue grows.

(7)

How can our body regulate energy storage

capacity (fat pads) for a given diet?

•

plasticity of adipose cell number (under weight gain and loss)

•

differences between fat depots

(8)

(9)

Cell-size distributions

0 50 100 150 200 250 0 0.02 Cell diameter (_µm) Relative frequency

(10)

Data

•

Cell-size distributions are probability

distributions of cell number as a function of cell diameter.

•

The overall cell number can be deduced

from the cell-size distribution and the

weight of the fat pad, using the density of adipose tissue.

(11)

Moduli problem

•

Smooth parametrization of cell-size

distributions

•

Kinetics of cell-size distributions

(12)

Energy storage capacity adapts

dynamically to diet

(Guo et al., 2009) 0 5 10 15 20 25 -5 0 5 10 15 20 Fat mass (g) Time (weeks) 19w ND ND HFD 0 5 10 15 20 -5 0 5 10 15 20 Fat mass (g) Time (weeks) 19w ND ND HFD ND

(13)

Energy storage capacity adapts

dynamically to diet

(Guo et al., 2009)

control 7wHF 19wHF

fat pad mass and cell-size distributions

control 7wHF +12wN7wHF 0 5 10 15 20 25 -5 0 5 10 15 20 Fat mass (g) Time (weeks) 19w ND ND HFD 0 5 10 15 20 -5 0 5 10 15 20 Fat mass (g) Time (weeks) 19w ND ND HFD ND

(14)

Only epididymal fat mass decreases

under 19w high-fat diet

** p<0.01 * p<0.05 0 1 2 3 control 7wN 7wHF 19wN 19wHF +12wN7wHF

Inguinal fat mass (g)

** * 0 1 2 3 control 7wN 7wHF 19wN 19wHF +12wN7wHF

Epididymal fat mass (g)

** ** 0 1 2 control 7wN 7wHF 19wN 19wHF +12wN7wHF

Mesenteric fat mass (g)

Retroperitoneal fat mass (g)

**

(15)

Only epididymal fat mass decreases

under 19w high-fat diet

(Streissel et al., 2007) ** p<0.01 * p<0.05 0 1 2 3 control 7wN 7wHF 19wN 19wHF +12wN7wHF

Inguinal fat mass (g)

** * 0 1 2 3 control 7wN 7wHF 19wN 19wHF +12wN7wHF

Epididymal fat mass (g)

Mesenteric fat mass (g)

Retroperitoneal fat mass (g)

**

(16)

Adipose cell-size distributions

epididymal 0 0.01 0.02 0.03 25 50 75 100 125 150 175 200 Relative frequency control 7wN 7wHF 0 0.01 0.02 0.03 25 50 75 100 125 150 175 200 Relative frequency Cell diameter (µm) 19wN 19wHF 7wHF+12wN

(17)

Adipose cell-size distributions

inguinal epididymal 0 0.01 0.02 0.03 25 50 75 100 125 150 175 200 Relative frequency control 7wN 7wHF 0 0.01 0.02 0.03 25 50 75 100 125 150 175 200 Relative frequency 0 0.01 0.02 0.03 25 50 75 100 125 150 175 200 Relative frequency Cell diameter (µm) 0 0.01 0.02 0.03 25 50 75 100 125 150 175 200 Relative frequency Cell diameter (µm) 19wN 19wHF 7wHF+12wN

(18)

Frequency Cell diameter Frequency Cell diameter Frequency Cell diameter Frequency Cell diameter Frequency Cell diameter Frequency Cell diameter

Multiple biological processes lead to cell-size distributions recruitment growth death (small cells) shrinkage fluctuation (Jo et al., 2009) death (large cells)

(19)

Frequency Cell diameter Frequency Cell diameter Frequency Cell diameter Frequency Cell diameter Frequency Cell diameter Frequency Cell diameter

Multiple biological processes lead to cell-size distributions ∂n ∂m = βnpδ(s − s0) −v(s) ∂n ∂s recruitment growth death (small cells) shrinkage fluctuation (Jo et al., 2009) death (large cells) +D ∂2n ∂s2 −k(s)φ(m)n

(20)

epididymal

The mathematical model can extrapolate changes of adipose cell-size distributions under 7w HFD

0 2 4 6 8 10 12 25 50 75 100 125 150 175 200 Growth rate ( µ m/g) 0 0.1 0.2 0.3 25 50 75 100 125 150 175 200 Absolute frequency (x10 6 ) Cell diameter (µm) 1.7g (w0) 6.1g (w1) 10.6g (w3) 15.0g (w7) 0 0.3 25 200

(21)

inguinal epididymal

The mathematical model could extrapolate changes of adipose cell-size distributions under 7w HFD

0 2 4 6 8 10 12 25 50 75 100 125 150 175 200 Growth rate ( µ m/g) 0 2 4 6 8 10 12 25 50 75 100 125 150 175 200 Growth rate ( µ m/g) 0 0.1 0.2 0.3 25 50 75 100 125 150 175 200 Absolute frequency (x10 6 ) Cell diameter (µm) 1.7g (w0) 6.1g (w1) 10.6g (w3) 15.0g (w7) 0 0.3 25 200

fat mass (time)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 25 50 75 100 125 150 175 200 Absolute frequency (x10 6 ) Cell diameter (µm) 1.0g (w0) 5.9g (w1) 10.7g (w3) 15.6g (w7) 0 0.7 25 200

(22)

Cell loss of large cells is required to explain the

changes of cell-size distributions under 19w HFD

epididymal 0 20 40 60 80 25 50 75 100 125 150 175 200 0 2 4 6 Growth rate ( µ m/g) Death rate (g -1 ) 0 0.1 0.2 0.3 0.4 25 50 75 100 125 150 175 200 Absolute frequency (x10 6 ) Cell diameter (µm) 15.7g (w7) 17.4g (w9) 19.1g (w13) 20.8g (w19) 0 0.2 25 200

(23)

Cell loss of large cells is required to explain the

changes of cell-size distributions under 19w HFD

inguinal epididymal 0 20 40 60 80 25 50 75 100 125 150 175 200 0 2 4 6 Growth rate ( µ m/g) Death rate (g -1 ) 0 20 40 60 80 25 50 75 100 125 150 175 200 0 2 4 6 Growth rate ( µ m/g) Death rate (g -1 ) 0 0.1 0.2 0.3 0.4 25 50 75 100 125 150 175 200 Absolute frequency (x10 6 ) Cell diameter (µm) 15.7g (w7) 17.4g (w9) 19.1g (w13) 20.8g (w19) 0 0.2 25 200

fat mass (time)

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 25 50 75 100 125 150 175 200 Absolute frequency (x10 6 ) Cell diameter (µm) 15.2g (w7) 17.2g (w9) 19.1g (w11) 21.1g (w19) 0 0.5 25 200

(24)

Size-dependent shrinkage of adipose cells can explain the

changes of cell-size distributions under 7wHFD+12wND.

epididymal 0 5 10 15 25 50 75 100 125 150 175 200 Shrinkage rate ( µ m/g) 0 0.1 0.2 0.3 0.4 25 50 75 100 125 150 175 200 Absolute frequency (x10 6 ) Cell diameter (µm) 14.2g (w7) 12.1g (w7-8) 10.0g (w7-8) 7.9g (w19) 0 0.1 25 200

(25)

Size-dependent shrinkage of adipose cells can explain the

changes of cell-size distributions under 7wHFD+12wND.

inguinal epididymal 0 5 10 15 25 50 75 100 125 150 175 200 Shrinkage rate ( µ m/g) 0 5 10 15 25 50 75 100 125 150 175 200 Shrinkage rate ( µ m/g) 0 0.1 0.2 0.3 0.4 25 50 75 100 125 150 175 200 Absolute frequency (x10 6 ) Cell diameter (µm) 14.2g (w7) 12.1g (w7-8) 10.0g (w7-8) 7.9g (w19) 0 0.1 25 200

fat mass (time)

0 0.1 0.2 0.3 0.4 0.5 25 50 75 100 125 150 175 200 Absolute frequency (x10 6 ) Cell diameter (µm) 14.2g (w7) 12.1g (w7-8) 10.0g (w7-8) 7.9g (w19) 0 0.2 25 200

(26)

Mathematical modeling can deduce microscopic changes of adipose tissues

Time (weeks)

epididymal inguinal

retroperitoneal

mesenteric

short-term HFD long-term HFD switch to ND

0 10 20

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 7 8 9 10 11 12 13 14 15 16 17 18 19

Total cell number (x10

6 ) 0 1 2 3 4 5 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 7 8 9 10 11 12 13 14 15 16 17 18 19 Fat-tissue mass (g) 50 100 150 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 7 8 9 10 11 12 13 14 15 16 17 18 19

Mean cell size (

µ

(27)

The model can give the detail of cell number changes in terms of recruitment (influx) and loss (outflux)

0 3 6 9 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 7 8 9 10 11 12 13 14 15 16 17 18 19 Death rate (x10 6 /g) Time (weeks) 0 10 20 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 7 8 9 10 11 12 13 14 15 16 17 18 19

Total cell number (x10

6 ) 0 3 6 9 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 7 8 9 10 11 12 13 14 15 16 17 18 19 Recruitment rate (x10 6 /g)

short-term HFD long-term HFD switch to ND

epididymal inguinal

retroperitoneal

(28)

Summary

•

Fat pads continuously recruited new adipose cells

under a high-fat diet.

•

Medium-sized adipose cells (30 to 100 μm

diameter) showed the largest growth/shrinkage rate under weight gain/loss.

•

Cell loss of large adipose cells (above 100 μm

diameter) was observed under a prolonged high-fat diet during 19 weeks.

•

Adipose tissue responds universally to diet changes to regulate energy storage capacity.

(29)

We used an appropriate

modulus to find a systematic

pattern in cross-sectional

mouse data. We expect that

there is systematic temporal

behavior too. Mice are too

small ...

(30)

A longitudinal data set

•

Zucker fatty (fa/fa) genetically obese rats

•

Microbiopsies at multiple time points over

151 days (rat 1) and 163 days (rat 2)

(31)

Data suggests periodicity

-1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 0 50 100 150 200 250 300 day0 day33 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 0 50 100 150 200 250 300 day0 day86 -1.2 -1 -0.8 -0.6 -0.4 -0.2 0 0.2 0.4 0.6 0.8 0 100 200 300 day0 day134

(32)

Problem:

Prove periodicity and

determine the period

(33)

Bayesian analysis

•

Sparse irregularly distributed time points

•

Model period = period-bin-size X

number-of-bins

•

Additional variable defining a model: phase of

first time-point

•

Consider models with periods between 30 and

100 days

(34)

Cost function

•

A model assigns period-bin numbers to data time-points.

•

For each cell-size-bin and each period-bin, what is the probability of the cell number counts at

different time points assigned to this period-bin belonging to the same log-normal distribution?

•

For each cell-size-bin and each period-bin, two

parameters for the lognormal distribution (median and width).

(35)

Data limitations

•

Limited number of time points implies requirements on testable periodic models.

•

(0 0 0 1 1 1) is not a model that tests periodicity

because there are no non-contiguous time points that are assigned to the same period-bin. However, (0 1 0 0 1 1) and (0 1 0 1 0 1) are bin assignments that test

(36)

Parallel-tempering MC

•

Parallel tempering Monte Carlo to compute the

model likelihood for each model

•

Ten equally spaced inverse temperatures

•

For each cell-size-bin and each temperature, run

a Monte Carlo (10^4 steps for equilibration, 10^3 steps for integration)

•

The mean likelihood for all the temperatures is

the model likelihood, balancing model complexity against goodness of fit.

(37)

Results

•

Periodic with period approximately 53 +/- 3

days, independently for both rats

•

Different bin numbers selected for different

animals for most likely model: Rat 1 -> 5 bins of 10 days each, Rat 2 -> 7 bins of 8 days each

(38)

Log Likelihood of Period (Rat 1) -400 -350 -300 -250 -200 -150 -100 -50 0 0 30 60 90 120 Period (days) Log likelihood

(39)

Log Likelihood of Period (Rat 2) -400 -350 -300 -250 -200 -150 -100 -50 0 0 30 60 90 120 Period (days) Log likelihood

(40)

0 0.5 1 1.5 2 2.5 3 3.5 4 cell-sizes22.905524.318325.818327.410829.101630.896632.802334.825636.973739.254341.675644.246246.975349.872852.949156.21559.682463.363767.272171.421575.826980.50485.469690.741596.3385102.281 108.59115.288122.399129.948137.964146.474155.508165.1 175.284186.096197.574209.761222.699 0 0.5 1 1.5 2 2.5 3 3.5 4 cell-sizes22.905524.318325.818327.410829.101630.896632.802334.825636.973739.254341.675644.246246.975349.872852.949156.21559.682463.363767.272171.421575.826980.50485.469690.741596.3385102.281108.59115.288122.399129.948137.964146.474155.508165.1175.284186.096197.574209.761222.699 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5 cell-sizes22.905524.318325.818327.410829.101630.896632.802334.825636.973739.254341.675644.246246.975349.872852.949156.21559.682463.363767.272171.421575.8269 80.50485.469690.741596.3385102.281 108.59115.288122.399129.948137.964146.474155.508165.1 175.284186.096197.574209.761222.699 0 0.5 1 1.5 2 2.5 3 3.5 cell-sizes22.905524.318325.818327.410829.101630.896632.802334.825636.973739.254341.675644.246246.975349.872852.949156.21559.682463.363767.272171.421575.826980.50485.469690.741596.3385102.281108.59115.288122.399129.948137.964146.474155.508165.1175.284186.096197.574209.761222.699 0 0.5 1 1.5 2 2.5 3 3.5 cell-sizes22.905524.318325.818327.410829.101630.896632.802334.825636.973739.254341.675644.246246.975349.872852.949156.21559.682463.363767.272171.421575.826980.50485.469690.741596.3385102.281108.59115.288122.399129.948137.964146.474155.508165.1175.284186.096197.574209.761222.699

Rat 1 cycle

(41)

Rat 2 cell-size distributions

0 0.5 1 1.5 2 2.5 3 3.5 0 50 100 150 200 250

cell diameter (microns)

% bin 0 pred bin 1 pred bin 2 pred bin 3 pred bin 4 pred bin 5 pred bin 6 pred

(42)

Towards dynamics

•

Model as a compression of data

•

Model as a limited predictor

•

Compressing the model - connecting

(43)

What could lead to

periodicity?

•

Maximal lipid uptake occurs when there are

a large number of cells in the convective size range.

•

If lipid is available but there is limited

uptake capacity available, then new recruitment is needed.

(44)

Size distribution of pancreatic islets

5 mm MIP-GFP mouse

• basic development (islet neogenesis, proliferation potential, and islet fission)

• physiological and pathological conditions

(aging, pregnancy, diabetes, obesity, and insulinoma) 0 0.1 0.2 0.3 0 50 100 150 200 Relative frequency Effective diameter (µm) Postnatal day 7

(45)

0 0.01 0.02 0 50 100 150 Relative frequency Cell diameter (µm) size distribution of

adipose cells size distribution of pancreatic islets

0 0.1 0.2 0.3 0 50 100 150 200 Relative frequency Effective diameter (µm) Postnatal day 7

(46)

Acknowledgment

Fellow: _{Junghyo Jo}

Collaborators:

• Samuel W. Cushman (Diabetes Branch)

Shawn Mullen Teresa Liu

• Kevin D. Hall (Lab. of Biological Modeling)

Juen Guo

• Oksana Gavrilova (Mouse Metabolism Core Lab.)

Stephanie Pack William Jou

• Anne E. Sumner (Clinical Endocrinology Branch)

• Manami Hara (The University of Chicago)

German Kilimnik Abraham Kim