In recent years, nonlinear model predictive control (NMPC) has been extensively used for solving automotive motion control and planning tasks.
In order to formulate the NMPC problem, different coordinate systems can be used with different advantages.
We propose and compare formulations for the NMPC related optimization problem, involving a Cartesian and a Frenet coordinate frame (CCF/ FCF) in a single nonlinear program (NLP).
We specify costs and collision avoidance constraints in the more advantageous coordinate frame, derive appropriate formulations and compare different obstacle constraints.
With this approach, we  exploit the simpler formulation of opponent vehicle constraints in the CCF, as well as road aligned costs and constraints related to the FCF.
Comparisons to other approaches in a simulation framework highlight the advantages of the proposed approaches.

§ INTRODUCTION

Trajectory optimization with obstacle avoidance is a major challenge of motion planning and control in autonomous driving and racing.
Trajectories need to be feasible to kinodynamic equations and avoid collisions with objects that are often hard to predict.
Collision avoidance and the related generation of a reference trajectory or collision avoidance as part of the controller, e.g., model predictive control (MPC), is often formulated as a nonlinear discrete time-optimal optimal control problem <cit.>.
Through a carefully chosen nonlinear program (NLP) formulation and by means of dedicated real-time optimization solvers <cit.>, the problem can be solved efficiently.

The transformation of the dynamics into a road aligned coordinate frame (CF), namely the Frenet CF (FCF), has shown many advantages, such as the simplification of references and road boundaries <cit.>.
Nevertheless, the transformed coordinates also come with the disadvantage of transformed geometric obstacle shapes <cit.>, cf. Sec. <ref>.
Typical convex geometric shapes, such as boxes, ellipses or circles, are easier to describe in the Cartesian reference CF and become nonconvex after transformation into the FCF.
The shapes of objects in both frames are shown in Fig. <ref>.

Simulated overtaking of the same maneuver in two CFs, namely the Cartesian (top plot) and Frenet CF (bottom plot).
Planned trajectories with $\Delta t=0.1$s and snapshots every $0.7$s of boundary box alignments.

In nonlinear optimization, "lifting" is a technique where the optimization problem is formulated and solved in a higher dimensional space, which offers advantages regarding convergence rates and region of attraction <cit.>.
The contribution of the presented paper is a way to extend and lift the state space of the vehicle model by including both CFs and thereafter formulate constraints and costs in the more appropriate CF.
We show an increase of overall performance due to the improved description of the obstacle shapes with a variety of deterministic obstacle avoidance formulations in simulation.
Despite the increased state dimensions, even the computation time can be lowered compared to a pure FCF representation.
Additionally, references can be set in any of the two CFs, which allows for flexible combination with planning modules that use either CF, e.g., <cit.>.

Related work:
The effectiveness of NMPC using the FCF related to automotive tasks was shown in numerous works <cit.>.

None of them explicitly considers the shape transformation of objects, which are rather over-approximated with convex shapes in the FCF.
Convex obstacle shapes in the Cartesian CF (CCF) are considered in <cit.> with potential fields, in <cit.> with covering circles and Euclidean distance constraints, in <cit.> with ellipses, in <cit.> with separating hyperplanes and <cit.> with a formulation related to a conjunction of convex planes covering the obstacle.
The most prominent variants are compared within this paper in the Frenet and the lifted formulation.
More importantly, the shape fitting problem with transformed objects in the FCF and an approach with surrogate representations in both CFs was recently considered in a related way in <cit.>.

However, <cit.> focuses on different topics, as it deals with linear MPC and does not consider dedicated obstacle formulations.
Furthermore, it integrates an approximation of the transformation itself into the model, i.e., and approximation of a DAE, whereas in our formulation we provide a reduced index formulation which constitutes an ODE (see Sec. <ref>) or eliminate algebraic variables directly.
Another variant of tracking along a reference path stems from <cit.> and was extended to vehicles in <cit.> and <cit.>.
It uses a method called contouring control, which uses a state on a path-length parameterized reference curve and an additional state for its path position.
Similarly to <cit.>, it approximates the transformation implicitly, which involves the approximation of a bi-level optimization problem for finding the closest point on the reference curve.

This paper proposes novel NMPC formulations that extend the state space to two CFs and allow for an efficient handling of the occurring costs and constraints.
Thereby, the usually convex and simple obstacle shapes in the Cartesian CF can be directly used in the NMPC formulation.
We show in simulation that we outperform the conventional approach of over-approximation <cit.> in terms of computation time and performance.
Furthermore, the obstacle shapes are independent of the states and the road (up to Euclidean transformations), which is not the case in a conventional Frenet representation.
Additionally, we contribute with an extensive comparison of common obstacle avoidance formulations in the proposed formulations.

§ VEHICLE MODELS

In order to formulate our NMPC problem, we use a rear-axis referenced kinematic vehicle model <cit.>.
It comprises three states that are related to the CF which we write as $x^\mathrm{c}$. Particularly, we use $x^\mathrm{c,C}=[x\quad y\quad\varphi]^\top \in \R^3$ for the Cartesian states and $x^\mathrm{c,F}=[s\quad n\quad\alpha]^\top\in \R^3$ for the Frenet states.
We use the Cartesian (earth) position states $x$, $y$ and the heading angle $\varphi$.
Similarly, in the FCF we use position states $s$ and $n$, together with the difference angle $\alpha$, which are all related to a reference curve.
The other states $x^\mathrm{\neg c}=[v\quad\delta]^\top\in\R^2$ are needed for both, CCF and FCF, where $v$ is the absolute value of the velocity at the rear axis and $\delta$ is the steering angle.
For the full CCF model we use the state $x^\mathrm{C}=[x^{\mathrm{c,C}\top} \quad x^{\mathrm{\neg c}\top}]^\top$ and for the FCF model we use the state $x^\mathrm{F}=[x^{\mathrm{c,F}\top} \quad x^{\mathrm{\neg c}\top}]^\top$.
We assume a rear-wheel drive force $F^\mathrm{d}$ as input, which includes the acceleration and braking force.
We simplify that the braking force of the front wheel acts in the direction of the rear-wheel, thus we include it in the force $F^\mathrm{d}$.
This approximation is valid for small steering angles, since the projection of the front wheel force $F^\mathrm{d,front}$ on the rear-wheel reference is given by $F^\mathrm{d,front}\cos(\delta)$.
The most prominent resistance forces for wind $F^\mathrm{wind}(v,\varphi)=c_\mathrm{air}v_\mathrm{rel}(v,\varphi)^2$ and the rolling resistance $F^\mathrm{roll}=c_\mathrm{roll}\mathrm{sign}(v)$ are included, with the total resistance force $F^\mathrm{res}(v,\varphi)=F^\mathrm{wind}(v,\varphi)+c_\mathrm{roll}\mathrm{sign}(v)$.
The air drag depends on the vehicle speed $v$ in relation to the wind speed $v_\mathrm{wind}$ with the air friction parameter $c_\mathrm{air}$.
The rolling resistance is proportional to $\mathrm{sign}(v)$ by the constant $c_\mathrm{roll}$.
We drop the sign function, since we only consider strictly positive speeds.
We model the relative speed related to the air drag, which we assume constant and known, by $v_\mathrm{rel}(v,\varphi)=v+v_\mathrm{wind}\cos(\varphi-\varphi_\mathrm{wind})$, where $\varphi_\mathrm{wind}$ is the angle of the direction the wind asserts force and $\varphi$ is the heading of the vehicle in the CCF.
In most works (e.g., <cit.>) the influence of external wind is ignored, since its influence might be small, unpredictable or absent in experimental indoor setups.
Nevertheless, we include it, since it demonstrates an influence that can be easily modeled in the FCF, but is difficult to model in the FCF.

The input of our model is given by $u=[F^\mathrm{d}\quad r]^\top\in\R^2$, where $r = \frac{\mathrm{d} \delta}{ \mathrm{d} t}$ denotes the steering rate.
The dynamics of the coordinate unrelated states are written as

\begin{equation}
\dot{x}^\mathrm{\neg c}=
% \begin{bmatrix}
% \dot{v}\\
% \dot{\delta}
% \end{bmatrix}=
f^\mathrm{\neg c}(x^\mathrm{\neg c},u,\varphi)=
\begin{bmatrix}
\frac{1}{m}(F^\mathrm{d}+F^\mathrm{wind}(v,\varphi)-F^\mathrm{res}(v))\\
\end{bmatrix},
\end{equation}

where $m$ denotes the vehicle mass.

The lateral acceleration $a_\mathrm{lat}(x^\mathrm{\neg c})$ at the rear wheel axis is given by

\begin{equation}
a_\mathrm{lat}(x^\mathrm{\neg c})=\frac{v^2\tan(\delta)}{l},
\end{equation}

where $l=l_\mathrm{f}+l_\mathrm{r}$ is the total wheelbase length of the vehicle.

§.§ CCF vehicle model

Using simple kinematic relations, the dynamics of the Cartesian states can be written as

\begin{equation}
\label{eq:model_cartesian_only}
\dot{x}^\mathrm{c,C}=
f^\mathrm{c,C}(x^\mathrm{C}, u)=
\begin{bmatrix}
v \cos(\varphi)\\
v \sin(\varphi)\\
\frac{v}{l}\tan(\delta)
\end{bmatrix}.
\end{equation}

The full five-state Cartesian vehicle model is given by

\begin{equation}
\label{eq:model_cartesian_full}
\dot{x}^\mathrm{C}=
\begin{bmatrix}
f^\mathrm{c,C}(x^\mathrm{C}, u)\\
f^\mathrm{\neg c}(x^\mathrm{\neg c},u,\varphi)
\end{bmatrix}.
\end{equation}

§.§ FCF vehicle model

Since in usual vehicle motion control tasks, the vehicle moves mainly close to a reference curve ${\gamma:\R \rightarrow \R^2}$, i.e., the street center line, the transformation into a curvilinear CF is a natural choice.
The reference curve $\gamma(\sigma)=[\gamma_x(\sigma) \quad \gamma_y(\sigma)]^\top$ is parameterized by its path length $\sigma$ and can be fully described by one initial transformation offset $\gamma(\sigma_0)$, an initial orientation $\varphi_0$ and the curvature ${\kappa(\sigma)=\frac{\mathrm{d}\varphi^\gamma}{\mathrm{d}\sigma}}$ along its path.
We use $\varphi^\gamma(\sigma)$ for the tangent angle in each point of the curve.
As part of the Frenet transformation, we project the Cartesian vehicle reference point $p^\mathrm{veh}\in\R^2$ on the closest point in the Euclidean distance of the reference curve with

\begin{equation}
\label{eq:frenet_transformation_argmin}
s^*(p^\mathrm{veh})=\arg \min_{\sigma}\norm{p^\mathrm{veh}-\gamma(\sigma)}^2_2 .
\end{equation}

W.l.o.g., we always set the initial reference point of the transformation to zero.
The position $s$ on the curve is used as longitudinal FCF state.
The vector $(p^\mathrm{veh}-\gamma(s^*))$ is difference of the closest point on the curve to the vehicle.
By using the 90 degree rotation matrix $R^{90}$ and projection to the normal unit vector of the curve $e_n=R^{90}\gamma'(s^*)$, we obtain the Frenet state $n=(p^\mathrm{veh}-\gamma(s^*))^\top e_n$.
The third Frenet state $\alpha$ relates the tangent angle of the curve to the heading of the vehicle with $\alpha=\varphi-\varphi^\gamma(s^*)$.
The relations of the transformation are shown in Fig. <ref>.
We write the transformation of a Cartesian state $x^\mathrm{c,C}=[x\quad y\quad \varphi]^\top$ to a Frenet state $x^\mathrm{c,F}=[s\quad n\quad \alpha]^\top$ by means of the transformation

\begin{align}
s^* \\
(p^\mathrm{veh}-\gamma(s^*))^\top e_n \\
\varphi^\gamma(s^*)-\varphi
\end{bmatrix},
\end{align}

and its inverse by

\begin{align}
\gamma_x(s)-n\sin(\varphi^\gamma(s)) \\
\gamma_y(s)+n\cos(\varphi^\gamma(s)) \\
\varphi^\gamma(s)-\alpha
\end{bmatrix}.
\end{align}

The existence and uniqueness of the transformation is guaranteed under mild assumptions which are discussed in detail in <cit.>.

State relations between Cartesian and FCF

As shown in <cit.>, we obtain the ODE for the kinematic motion in the FCF as

\begin{equation}
\label{eq:model_frenet_only}
\dot{x}^\mathrm{c,F}=
f^\mathrm{c,F}(x^\mathrm{F}, u)=
\begin{bmatrix}
\frac{v \cos(\alpha)}{1-n \kappa(s)}\\
v \sin(\alpha)\\
\frac{v}{l}\tan(\delta) - \frac{\kappa(s) v \cos(\alpha)}{1-n \kappa(s)}
\end{bmatrix}.
\end{equation}

The Cartesian state $\varphi$ is needed in order to formulate the wind disturbance.
It is not available in the FCF, consequently it needs to be computed by evaluating the tangent angle $\varphi^\gamma(s)$ of the current position $s$ on the reference curve $\gamma(\sigma)$.
This can be approximated by a spline function $\hat{\varphi}^\gamma(s)$ that is computed for the road layout.
It yields a spline approximation for the heading angle with $\hat{\varphi}(s,\alpha)=\hat{\varphi}^\gamma(s)+\alpha$.
The full FCF vehicle model is consequently given by the five state model

\begin{equation}
\label{eq:model_frenet_full}
\dot{x}^\mathrm{F}=
\begin{bmatrix}
f^\mathrm{c,F}(x^\mathrm{F}, u)\\
f^\mathrm{\neg c}(x^\mathrm{\neg c},u,\hat{\varphi})
\end{bmatrix}.
\end{equation}

§.§ Model comparison

As indicated in Sec. <ref>, the CF models have different advantages when used in a NMPC formulation, see Tab. <ref> for an overview.
The definition of road boundaries and the reference curve, which are often center-lane-aligned curves are straight forward to define in the FCF, but hard to define in the CCF.
However, the obstacle definition in the FCF is cumbersome due to several reasons.
Despite convex obstacle shapes in the CCF, safety cannot be guaranteed when using SQP to solve the NMPC problem with the Frenet model.

Convex obstacle shapes cannot be guaranteed to be convex, if transformed into the FCF.
This fact can be seen from the following example.
Consider a straight line, which is a convex set, and a circular road. Let the line intersect the road at coordinates $\gamma(\sigma_1)=[x_1\quad y_1]^\top$ and $\gamma(\sigma_2)=[x_1\quad y_1]^\top$. The transformed Frenet states $n_1,n_2$ are zero in either point. At $\sigma_3\in(\sigma_1,\sigma_2)$ the transformed state $n_3\neq 0$, thus the transformed set is not convex.

By considering Lemma <ref> it can be shown that convex obstacles are guaranteed to be a subset of the linearized constraints within an SQP iteration, thus safely over-approximated.

Regard the set $\mathcal{C} = \{x\in \R^{n} \mid g(x) \geq 0\}$  and $\mathcal{C}^{\mathrm{lin}}(x^*) = \{x\in \R^{n} \mid g(x^*) + \nabla g(x^*)^\top (x-x^*) \geq 0\}$. Suppose that the function $g:\R^n \rightarrow \R$ is convex, then $ \mathcal{C} \subseteq \mathcal{C}^{\mathrm{lin}}(x^*)$ for any $x^*$.

Due to convexity, $g(x^*)+\nabla g(x^*)^\top(x-x^*)\leq g(x)$ and therefore, it follows that $\mathcal{C} \subseteq \mathcal{C}^\mathrm{lin}$.

Nonconvex obstacles are not safely over-approximated within SQP algorithms.

Another problem that arises with objects in the FCF is the dependence of the shape on the state.
Consequently, if the obstacle constraints are defined along a discretized time horizon, at each time step $i=0,\ldots,N$, the shape has to be transformed separately, cf. Fig.<ref>.
In typical applications this could lead to $N$ transformations for each obstacle in every NMPC iteration, followed by a convexification (e.g., bounding boxes, convex polygons, covering circles) in order to guarantee safety.
Alternatively, an over-approximation could be used to capture all possible transformed shapes.
However, this would lead to a striking conservatism, especially for long vehicles and low curve radii.

Feature     CCF    FCF

reference definition

boundary constraints

obstacle specification

disturbance specification

Comparison of the two model representations

§.§ Overview of CF lifting formulations

As outlined in Sec. <ref>, it is beneficial to have states of both CFs in the NLP formulation in order to simplify the constraints.
Several different ways of including both CFs are possible and a summary is given in Tab.<ref>.

Formulation     ODE
CF     Obstacle
CF     Cost
CF     $n_x$     $n_z$     Practical
Relevance     Issues

Conventional Frenet			    Frenet     Frenet    Frenet    5     0      yes     nonconvex state-depenent obstacle shapes

usually over-approximated <cit.>

Direct Elimination Frenet 	    Frenet     Cartesian    Frenet    5     0     yes     additional nonlinearities (objective, constraints)

Lifted ODE Frenet 				    Frenet     Cartesian   Frenet     8     0     yes     redundant states

DAE Frenet 				    Frenet     Cartesian   Frenet     5     3     no     bad convergence in our experiments

Conventional Cartesian			    Cartesian     Cartesian    Cartesian     5     0     no     nonconvex boundary constraints <cit.>

Cartesian with Frenet States	    Cartesian     Cartesian    Frenet    $\{5,8\}$     $\{0,3\}$     yes     Difficult bi-level problem. Approximations, e.g. <cit.>

Comparison of CF Formulations in NMPC. Bold typed formulations are compared in this paper.

First, we can choose the main CF ODE and introduce the states related to the other CF as algebraic variables that are determined by the main CF and obtain a differential algebraic equation (DAE) of index 1.
We could either have a CCF based DAE

\begin{align}
\label{eq:dae_cartesian}
\begin{split}
\dot{x}^\mathrm{C}&=f^\mathrm{C}(x^\mathrm{C}, u)\\
\end{split}
\end{align}

or a FCF based DAE

\begin{align}
\label{eq:dae_frenet}
\begin{split}
\dot{x}^\mathrm{F}&=f^\mathrm{F}(x^\mathrm{F},x^\mathrm{c,C}, u)\\
\end{split}
\end{align}

The inverse transformation $\frenetTrans^{-1}$ is computationally cheap, since it just needs explicit function evaluations, whereas the forward transformation $\frenetTrans$ requires solving an NLP as in (<ref>), resulting in a computationally expensive bi-level problem in the final MPC formulation.
Therefore, we choose the FCF of (<ref>) as a basis and exclude CCF formulations (<ref>) from further comparisons.
The DAE of index 1 (Lifted DAE Frenet) is a possible way to formulate the problem and was similarly used in <cit.> for a linearized model.
Another possible formulation (Direct Elimination Frenet) is to directly eliminate the algebraic variables in (<ref>) by using the inverse Frenet transformation in the nonlinear constraint formulation.
If the objective includes Cartesian states with quadratic costs and lifted constraints, the direct elimination would lead to a nonlinear objective and constraints.
Alternatively, we can perform an index reduction of (<ref>), which is obtained by differentiation of the algebraic constraint, leading to

\begin{align}
\dot{x}^\mathrm{c,F}&=f^\mathrm{c,F}(x^\mathrm{c,F}, u)\\
\label{eq:dae_frenet_line1}
0&=\dot{x}^\mathrm{c,C}-\frac{\partial\frenetTrans^{-1}(x^\mathrm{c,F})}{\partial x^\mathrm{c,F}}f^\mathrm{c,F}(x^\mathrm{c,F}, u), \\
&\quad \text{and}\quad x^\mathrm{c,C}(0):=\frenetTrans^{-1}(x^\mathrm{c,F}).
\end{align}

Detailed computation (not presented here) shows the equivalence of (<ref>) to

\begin{align}
\label{eq:dae_ode_frenet_final}
\dot{x}^\mathrm{c,C}&=f^\mathrm{c,C}(x^\mathrm{c,C}, u)
\end{align}

This approach in (<ref>) or (<ref>) (Lifted ODE Frenet) results in redundant states in both CFs, which are coupled through the inputs and the initial state.
We do not consider formulations based on the CCF ODE, since it involves the challenging bi-level problem to obtain FCF states

§ OBSTACLE AVOIDANCE FORMULATIONS

Different formulations for obstacle avoidance constraints are used in NMPC and visualized in Fig.<ref>.
We assume that rectangles represent the real vehicle shapes.

Schematic drawing of obstacle constraints. (a: ellipse CCF, b: covering circles CCF, c: separating hyperplanes CCF, d: ellipse FCF, e: covering circles FCF, f: separating hyperplanes FCF)

Often simple geometric covering shapes (circles <cit.> or ellipses <cit.>) and related distance functions are used.
Alternatively, covering polygons and restrictions on edges or vertices (hyperplanes) are formulated, <cit.>.
Furthermore, the boundaries can be deflected in order to integrate the obstacle into the boundary constraints <cit.>.
The later approach is not within the scope of this work, due to the generally different formulations that for instance, include a combinatorial planner for choosing the passing side <cit.>.
We compare the formulation of obstacle avoidance constraints by an
ellipse <cit.>, covering circles <cit.> and separating hyperplanes <cit.>.
We also implemented a formulation introduced in <cit.>, which we refer to as set-vertices-exclusion, but which poorly converged in our experiments.
We assume a rectangular shape of the vehicles with the rear/front chassis length $l_\mathrm{ch}=l_\mathrm{r,ch}+l_\mathrm{f,ch}$ related to the vehicle CG and chassis width $w_\mathrm{ch}$.
When we considering the over-approximated area to evaluate which constraint formulations is superior, the separating hyperplane do not increase the obstacle size, whereas circles and ellipses add additional unoccupied area.

§.§ Obstacle approximation by an ellipse

The distance between a circle and an ellipse can be computed explicitly, while computing the distance between two ellipses is more involved.
Thus, we cover the ego car with a circle.
The main axes $a,b$ of an ellipse covering a rectangle are computed by ${a=\frac{1}{\sqrt{2}}(l_\mathrm{f,ch}+l_\mathrm{r,ch})}$ and $b=\frac{1}{\sqrt{2}}w_\mathrm{ch}$.
Increasing this by the ego radius $r^\mathrm{ego}$ leads to the extended ellipse matrix $D=\mathrm{diag}(\left[a+r^\mathrm{ego},b+r^\mathrm{ego}\right])$.
With the rotation matrix $R(x^\mathrm{c,opp})\in \R^{2\times2}$ and a translation vector $t(x^\mathrm{c,opp}) \in \R^{2}$ related to the orientation and position of the obstacle vehicle, we can formulate the collision avoidance constraint with the matrix $\Sigma(x^\mathrm{c,opp})=R(x^\mathrm{c,opp}) D R(x^\mathrm{c,opp})^\top$ via the feasible set

\begin{align}
\label{eq:collision_ellipse}
\begin{split}
\free^\mathrm{ell}(x^\mathrm{c,opp}) =
\Bigl\{&x^\mathrm{c} \in \R^3\Big|
\norm{x^\mathrm{c}-t(x^\mathrm{c,opp})}^2_{\Sigma^{-1}(x^\mathrm{c,opp})}
\geq 1\Bigr\} .
\end{split}
\end{align}

§.§ Obstacle approximation by covering circles

We use the union of a set of circles to contain the set of the vehicle shape <cit.>.
For $l_\mathrm{ch}\geq w_\mathrm{ch}$, the number of covering circles $n_\mathrm{circ}$ must be larger than $\lceil\frac{l_\mathrm{ch}}{w_\mathrm{ch}} \rceil$ and have a radius $r^{\{\mathrm{ego,opp}\}}$ of $w_\mathrm{ch} \frac{1}{\sqrt{2}}$.
For each combination of the $n_\mathrm{circ,ego}$ and $n_\mathrm{circ,opp}$ circles a distance constraint must be satisfied, leading to $n_\mathrm{circ,ego} n_\mathrm{circ,opp}$ inequality constraints.
The covering circle center points are computed according to <cit.>, which gives us a function $p^i:\R^3\rightarrow \R^2$ for the circle center $i$ that computes the center positions $p_i=p^i(x^\mathrm{c,C})$ from the states $x^\mathrm{c,C}$.
With $\Delta r= r^\mathrm{ego} + r^\mathrm{opp}$ and $x:=x^\mathrm{c,C}$, we write the free set as

\begin{align}
\label{eq:collision_circles}
\begin{split}
\free^\mathrm{circ}(x^\mathrm{opp}) &=
\biggl\{x \in \R^3\big|
\norm{p^i(x)-p^j(x^\mathrm{opp})}_2 \geq \Delta r, \\
% &\forall i \in [0, \ldots ,n_\mathrm{circ, ego}-1],\\
% &\forall j \in [0,\ldots ,n_\mathrm{circ, opp}-1]
% TODO: use one line less. Dont use [] for indices, this are not set bracekts, better i = ...
&\textrm{for}~ 1 \leq i \leq n_\mathrm{circ, ego}, 1 \leq j \leq n_\mathrm{circ, opp} \biggr\}
\end{split}
\end{align}

§.§ Obstacle approximation by separating hyperplanes

When formulating collision avoidance with separating hyperplanes, we optimize for a feasible solution of the parameters $\theta\in \R^3$ of a hyperplane $h^\theta(p)$.
The parameterized hyperplane needs to separate all (four) vertices $p_i^{\{\mathrm{ego,opp}\}}(x^\mathrm{c\{ego,opp\}})$ of either vehicle's bounding box.
We write the feasible region using the related hyperplane existence problem with points ${\bar{p}^{\{.\}\top}=[p^{\{.\}\top} \quad 1]}$ as

\begin{align}
\begin{split}
\label{eq:hyperplane_theta}
&\free^\mathrm{hp}(x^\mathrm{opp})=\Bigl\{x \in \R^3, \theta \in \R^3\Big|\text{ }
\theta_1^2 + \theta_2^2=1,\\
&\text{ }\theta^\top\bar{p}_i^{\mathrm{ego}}(x)\leq 0,\text{ }
\theta^\top\bar{p}_i^{\mathrm{opp}}(x^\mathrm{opp})\geq 0,\text{ }
\forall i=0,\ldots,3
\Bigr\}.
\end{split}
\end{align}

With the constraint $\theta_1^2 + \theta_2^2=1$ for the hyperplane parameters we avoid a degenerate solution.

§ NMPC FORMULATION

The aim of the NMPC is to plan a feasible trajectory of a vehicle to drive on a road with bounded curvature on a reference lane parallel to the center line and with a desired reference speed.
Furthermore, the NMPC must avoid static and dynamic obstacles.
As motivated in Sec. <ref>, we use two variants of an FCF-based ODE to obtain Cartesian states, i.e. the direct elimination and lifted ODE formulation and compare it to the conventional formulation with over-approximation, such as shown in <cit.>.
First, we define the costs and constraints.

§.§.§ General costs & constraints

Some constraints are unrelated to the CF, such as the lower and upper bounds for states $x^\mathrm{\neg c}$ and inputs $u$.
For control costs  $u^\top R u$, we use the positive semi-definite weight matrix $R\in \R^{2\times2}$.

§.§.§ FCF related costs & constraints

State costs are related to FCF states, since there is no practical advantage of including CCF state costs.
A cost related to a desired reference path parallel to the road center line is accounted for by a square penalty of the deviation of the Frenet lateral coordinate $n$ to its reference $n_{\mathrm{ref}}$.
For a reference speed $v_\mathrm{ref}$, a square penalty with positive weight $w_s$, as well as a penalty on precomputed longitudinal reference positions $s_{\mathrm{ref},i}=\hat{s}_{0}+i \Delta t v_\mathrm{ref}$ is used, with the measured state $\hat{s}_{0}$ and sampling time $\Delta t$.
Since we assume a road with constant width, boundary constraints simplify in the FCF to box constraints for an upper bound $\ub{n}$ and a lower bound $\lb{n}$.

§.§.§ CCF related costs & constraints

We use the collision avoidance formulations, which could be one out of $\mathcal{O}=\{\mathrm{ell},\mathrm{circ},\mathrm{hp}\}$ in the CCF, thus have the constraint ${x^\mathrm{c,C} \in \free^{\{\mathrm{ell},\mathrm{circ},\mathrm{hp}\}}}$.
FCF costs are defined via the positive weight matrix $Q=\text{diag}(q)$ with the weight vector $q\in\R^5$ and the reference states in $x^\mathrm{F}_\mathrm{ref}$.
We use a terminal cost $Q_N=\text{diag}(q_N)$ with the weight vector $q_N\in\R^5$ after $N$ discrete time steps.

§.§.§ Direct elimination NMPC formulation

With the direct formulation, we can directly use the inverse transformation ${x^\mathrm{c,C}=\frenetTrans^{-1}(x^\mathrm{c,F})}$ to eliminate the Cartesian states in the constraint formulation.
Thus we have less states, but "more" nonlinear constraints.
We discretize the continuous trajectory with ${N-1}$ control intervals and use direct multiple shooting <cit.> with one step of an RK4 integration function $\Phi^\mathrm{F}(x^\mathrm{F},u, \Delta t)$ for the ODE in (<ref>) and the NLP formulation

x^F_0, …, x^F_N,

_0, …, _N-1

θ_1, …, θ_n_opp

∑_k=0^N-1 _k_R^2 + x^F_k-x^F_ref,k_Q^2 + x^F_N-x^F_ref,N_Q_N^2

x^F_0= x̅^F_0,

x^F_i+1= Φ^F(x^F_i,u_i, Δt), i=0,…,N-1,
u≤_i≤u ,i=0,…,N-1,
x^F ≤x^F_i ≤x^F, i=0,…,N,

x^c,C ≤^-1(x^c,F) ≤x^c,C, i=0,…,N,

a^lat ≤a^F_lat(x_i) ≤a^lat,i=0,…,N,
v_N ≤v̅_N,
^-1(x^c,F)∈(x^c,opp,j_i, θ_j),i=0,…,N-1,

Decision variables $\theta_1,\ldots,\theta_{n_\mathrm{opp}}$, where ${\theta_j = [\theta_j^0,\ldots,\theta_j^{N}]\in\R^{3\times N}}$ are only used for the separating hyperplanes formulation.

§.§.§ Lifted ODE NMPC formulation

In this formulation we use the extended state  $x^\mathrm{d}=[x^{\mathrm{F}\top} \quad x^{\mathrm{c,C}\top }]^\top$
and the extended ODE (<ref>).
The additional states increase the size of the state-space to eight in our case, where three states stem from either CF and additional two states are CF independent states.
In this formulation we use the RK4 integration function $\Phi^\mathrm{d}(x^\mathrm{d},u, \Delta t)$ of dynamics (<ref>).
We can write the final NLP for the lifted ODE formulation as

x^d_0, …, x^d_N,

_0, …, _N-1

θ_1, …, θ_n_opp

∑_k=0^N-1 _k_R^2 + x^F_k-x^F_ref,k_Q^2 + x^F_N-x^F_ref,N_Q_N^2

x^d_0= x̅^d_0,
x^d_i+1= Φ^d(x^d_i,u_i, Δt), i=0,…,N-1,
u≤_i≤u ,i=0,…,N-1,
x^d ≤x^d_i ≤x^d, i=0,…,N,
a^lat ≤a_lat(x^d_i) ≤a^lat,i=0,…,N,
v_N ≤v̅_N,
x^c,C_i∈(x^c,opp,j_i, θ_j),i=0,…,N-1,

§ NUMERICAL EXPERIMENTS

In order to evaluate the performance of the proposed approach, we simulate two randomized scenarios that constitute three non-ego vehicles in front of the ego vehicle with a lower cruise speed.
The scenario is simulated for 20 seconds, where usually three overtakes are possible.
In total, 500 full simulation runs are evaluated for each scenario type.
We record the solution times of the NMPC and the final driven distance after the simulation ended, which we take as a performance indicator.
We use different types of obstacles, particularly long ones in the dimensions of a truck (truck-sized), as well as short ones resembling normal cars (car-sized).
We make several simplifications in order to avoid performance influences of sources unrelated to our formulation.
Firstly, there is no model-plant mismatch, i.e., the simulation framework and the NLP use the same kinematic vehicle model and discretization.
Secondly, the ego NMPC has access to the planned trajectories of the other vehicles in order to avoid an influence of prediction errors.
Finally, we model non-ego participants to be non-interactive.
They aim at driving along a reference line parallel to the center line.
The simulations were run on an Alienware m-15 Notebook with an Intel Core i7-8550 CPU (1.8 GHz).
The parameters for the environment and the NMPC are shown in Tab. <ref> and Tab. <ref>, respectively.

Module     Name     Variable     Value

Road     road bounds$^2$     $\lb{n},\ub{n}$    $\pm 8.5$

curvature$^1$     $\kappa$     $[\shortminus0.05, 0.05]$

wind speed     $v_\mathrm{wind}$     $20$

wind direction     $\varphi_\mathrm{wind}$     $0$

Ego vehicle     length wheelbase     $l_\mathrm{r}, l_\mathrm{f}$     $1.7$

length chassis     $l_\mathrm{r,ch}, l_\mathrm{f,ch}$    $2$

width chassis    $w_\mathrm{ch}$    $1.9$

mass     m     1160

lateral acc. bound     $\lb{a}_\mathrm{lat},\ub{a}_\mathrm{lat}$     $\pm 5$

input bounds     $\lb{u}, \ub{u}$     $\pm[10\mathrm{kN}, 0.39]$

velocity bound     $\ub{v}$     $40$

steering angle bound     $\lb{\delta},\ub{\delta}$     $\pm0.3 $

starting position$^1$    $x^\mathrm{c,F}_0$    $[0,\shortminus5,0]$-

$ [0,5,0]$

reference speed     $v_\mathrm{ref}$    $40$

Opp. vehicles     length wheelbase$^2$     $l_\mathrm{r}, l_\mathrm{f}$     $10$

length chassis$^2$     $l_\mathrm{r,ch}, l_\mathrm{f,ch}$    $13$

width chassis$^2$     $w_\mathrm{ch}$    $4$

mass$^2$     m     $3000$

input bounds$^2$     $\ub{u}$     $[30\mathrm{kN}, 0.39]$

input bounds$^2$     $\lb{u}$     $[\shortminus45\mathrm{kN}, \shortminus0.39]$

starting position$^1$ $i$    $s^i_0$    $50(i+1)$

$n^i_0$     $ [\shortminus 5,5]$

reference speed     $v_\mathrm{ref}$    $15$

Environment parameters.$^1$ Randomized with uniform distribution.$^2$ Parameters only differ in long vehicle scenario. The parameters are equal for all vehicles, if not noted explicitly. We use SI units, if not stated explicitly.

We use the NLP solver  <cit.> with  <cit.>, RTI iterations and a partial condensing horizon of $\frac{N}{2}$.

Name     Variable     Value

nodes / disc. time     $N$/ $\Delta t$      $40$/ $0.1$

terminal velocity     $\ub{v}_N$     $15$

state weights     $q$     $[1, 500, 10^3, 10^3, 10^4]\Delta t$

terminal state weights      $q_N$     $[10, 90, 100, 10, 10]$

control weights      $R$     diag($[10^{-3}, 2\cdot 10^6])\Delta t$

Parameters for MPC in SI units.

We use obstacle constraint formulations of Sec. <ref>.
Besides the different obstacle dimensions, the proposed NMPC formulations convetional, direct elimination and lifted ODE were evaluated with the different obstacle formulations of Sec. <ref>.
We use the ellipse ("EL"), the $n$ covering circles ("C$n$") and the separating hyperplane ("HP") formulation.
In Fig <ref> the computation times and the maximum achieved progress of the randomized scenarios are shown.
Clearly, final progress after overtaking in the truck-sized scenario is increased in the proposed formulation significantly due to the more accurate representation of the obstacle shape.
For car-sized vehicles the extended states do not yield a prominent advantage, since here the Frenet transformation does not deform the obstacles vastly.
The maximum progress is nearly equal for both proposed approaches, since the obstacle constraint formulations based on Cartesian states is equal.
A striking difference between the two proposed formulations can be seen in the computation times, shown detailed in Tab. <ref>.
While the lifted ODE formulation even decreases the average computation time for nearly all obstacle formulations, the direct elimination formulation increases the computation time by around 30%.
Remarkably, the ellipsoidal obstacle formulation in the proposed lifted ODE formulation outperforms all other obstacle formulations in both, the computation time, as well as the performance measured in the average progress after overtaking, which highlights the advantage of the formulation.
Contrary to our expectations, the separating hyperplane formulation shows weaker performance in computation time and average progress.
In theory, separating hyperplanes should be more accurate in capturing the obstacle shape, nevertheless, due to the disadvantageous linearizations within the SQP iterations, the shape is not captured well.
This might be mainly due to the nonconvex and nonlinear constraint in (<ref>).

Box-plot comparison of the MPC solution timings for each real-time iteration and the final progress after 20 seconds for different obstacle formulations for truck- and car-sized vehicles.

6cComputation times (ms) for truck-sized obstacles

Conventional     2lDirect Elimination      2cLifted ODE

EL    $       1.5\pm        0.4$     $       1.9\pm        0.2$    $      28.9\%$     $       1.4\pm        0.3$     $      -6.6\%$

C5    $       7.2\pm        1.9$     $       7.6\pm        1.7$    $       5.5\%$     $       7.2\pm        1.8$     $      -0.0\%$

C7    $      14.0\pm        3.2$     $      14.0\pm        2.8$    $      -0.1\%$     $      13.9\pm        2.9$     $      -0.4\%$

HP    $       7.5\pm        1.5$     $       7.5\pm        1.5$    $      -0.1\%$     $       7.4\pm        1.7$     $      -1.6\%$

6ccar-sized obstacles

EL    $       1.5\pm        0.5$     $       2.0\pm        0.4$    $      29.6\%$    $       1.4\pm        0.4$     $      -5.7\%$

C1    $       1.4\pm        0.4$     $       1.9\pm        0.4$    $      34.0\%$    $       1.4\pm        0.4$     $      -3.5\%$

C3    $       3.6\pm        1.1$     $       4.0\pm        1.0$    $      12.4\%$    $       3.6\pm        1.1$     $       0.6\%$

HP    $       8.0\pm        2.3$     $       7.9\pm        1.9$    $      -0.6\%$    $       7.7\pm        2.0$     $      -4.0\%$

Mean and standard deviation of computation times for different scenarios, obstacle formulations and lifting formulations. Additionally, the difference in percent to the conventional formulation is given.

A rendered resulting simulation in both CFs can be found on the website <https://rudolfreiter.github.io/obstacle_avoidance/>

§ CONCLUSIONS

We have presented two novel FCF-based formulations of MPC for vehicles that include states of the CCF in order to gain numerical advantages.
Simulated evaluations and several wide-spread obstacle constraint formulations show that the proposed approaches are capable of representing the obstacle shapes more suitably and that with the lifted ODE formulation even the computation time was decreased.
Furthermore, our evaluations show that an ellipsoidal obstacle representation outperforms all other obstacle formulations in computation time.
In conclusion, we state that the combination of the ellipsoidal obstacle constraint formulation and the lifted ODE formulation yields superior results in all categories.

§ ACKNOWLEDGMENT

This research was supported by DFG via Research Unit FOR 2401 and project 424107692 and by the EU via ELO-X 953348.


# Cholesterol impairment contributes to neuroserpin aggregation

Costanza Giampietro  FIRC Institute of Molecular Oncology, 20139 Milan, Italy
these authors contributed equally to this work  Maria Chiara Lionetti  Center
for Complexity and Biosystems, Department of Biosciences, University of
Milano, via Celoria 26, 20133 Milano, Italy  these authors contributed equally
to this work  Giulio Costantini  Center for Complexity and Biosystems,
Department of Physics, University of Milano, via Celoria 16, 20133 Milano,
Italy  Federico Mutti  Center for Complexity and Biosystems, Department of
Biosciences, University of Milano, via Celoria 26, 20133 Milano, Italy
Stefano Zapperi  Center for Complexity and Biosystems, Department of Physics,
University of Milano, via Celoria 16, 20133 Milano, Italy  CNR - Consiglio
Nazionale delle Ricerche, Istituto di Chimica della Materia Condensata e di
Tecnologie per l’Energia, Via R. Cozzi 53, 20125 Milano, Italy  ISI
Foundation, Via Alassio 11C, Torino, Italy  Department of Applied Physics,
Aalto University, P.O. Box 14100, FIN-00076, Aalto, Finland  Caterina A.M. La
Porta  Center for Complexity and Biosystems, Department of Biosciences,
University of Milano, via Celoria 26, 20133 Milano, Italy  Corresponding
author<EMAIL_ADDRESS>
###### Abstract

Intraneural accumulation of misfolded proteins is a common feature of several
neurodegenerative pathologies including Alzheimer’s and Parkinson’s diseases,
and Familial Encephalopathy with Neuroserpin Inclusion Bodies (FENIB). FENIB
is a rare disease due to a point mutation in neuroserpin which accelerates
protein aggregation in the endoplasmic reticulum (ER). Here we show that
cholesterol depletion induced either by prolonged exposure to statins or by
inhibiting the sterol regulatory binding-element protein (SREBP) pathway also
enhances aggregation of neuroserpin proteins. These findings can be explained
considering a computational model of protein aggregation under non-equilibrium
conditions, where a decrease in the rate of protein clearance improves
aggregation. Decreasing cholesterol in cell membranes affects their
biophysical properties, including their ability to form the vesicles needed
for protein clearance, as we illustrate by a simple mathematical model. Taken
together, these results suggest that cholesterol reduction induces neuroserpin
aggregation, even in absence of specific neuroserpin mutations. The new
mechanism we uncover could be relevant also for other neurodegenerative
diseases associated with protein aggregation.

## Introduction

Conformational diseases, such as Alzheimer’s and Parkinson’s diseases,
spongiform encephalopathies and serpinopathies, are an increasingly common
class of neurological disorders characterized by the aggregation of aberrant
conformations of proteins. Familial Encephalopathy with Neuroserpin Inclusion
Bodies (FENIB) [1, 2] is an autosomal dominant inclusion body dementia
characterized by protein aggregation within the endoplasmic reticulum (ER) [1,
2]. FENIB is associated with mutations in neuroserpin [3], a protein belonging
to the superfamily of serpins that plays an important role in brain
development, neuronal survival, and synaptic plasticity [4]. Indeed, mutants
of neuroserpin in FENIB patients show accelerated rates of polymerization
compared with wild type proteins, both at the protein level in vitro [5, 6]
and in cell models [1, 2]. Moreover, neuroserpin is the most important
inhibitor of the tissue plasminogen activator (tPA) in the brain which is
known to be increased in Alzheimer’s disease [7, 8, 9].

Lipid metabolism has been shown to play a role in FENIB, but also in other
neurodegenerative diseases, from Alzheimer’s to Parkinson’s disease [10, 11,
12, 13]. In FENIB the inhibition of hydroxymethyl glutaryl-CoA reductase
(HMGCR) has a critical role in the clearance of mutant neuroserpin from the ER
[13]. Numerous studies have also reported that the modification of cholesterol
content can affect amyloid precursor protein (APP) processing, which is needed
for neuronal activity [11]. In this connection, it has been shown that the
E693D (Osaka) mutation in APP promotes intracellular accumulation of
Amyloid$\beta$ (A$\beta$), thus disturbing amyloid-mediated cholesterol efflux
from the cell [14]. Cholesterol has also been shown to influence APP
processing and A$\beta$ generation by modulating A$\beta$\- and
Amyloid-$\gamma$-secretase activities [15, 16, 17, 18]. In turn, APP cleavage
products regulate cholesterol homeostasis [19].

In a recent paper, our group has shown by 3D numerical simulations and mean-
field calculations that protein aggregation undergoes a non-equilibrium phase-
transition controlled by the rates of protein synthesis and degradation,
suggesting a crucial role of intracellular trafficking, particularly from the
ER [20]. According to this study, a decrease in the rate of protein clearance
from the ER can lead to rapid and irreversible protein aggregation in the ER
itself [20], leading to the onset of neurodegenerative diseases.

Cholesterol plays a crucial role in regulating the properties of phospholipid
membranes, affecting their fluidity and rigidity [21, 22, 23], the function
and dynamics of membrane proteins [24, 25], and thus vesicular trafficking
within the cell [26, 27, 28]. Hence, by regulating the biosynthesis of
cholesterol it is possible to affect the form and function of all the
membranes within the cell, including the ER. In particular, experiments show
that lowering the level of cholesterol delays ER-to-Golgi transport [29, 28].

Cholesterol homeostasis is controlled by a family of transcription factors,
known as sterol regulatory element binding proteins (SREBPs) [30]. In cells
with sufficient cholesterol supply, SREBPs are transmembrane proteins retained
in the ER, associated with SREBP-cleavage-activating protein (SCAP), a
cholesterol sensor [31]. Upon cellular cholesterol depletion, SREBP leaves the
ER to reach the Golgi, where cleavage by site-1 protease (S1P) releases the
amino-terminal half of SREBP, which can be further cleaved within its
membrane-spanning helix by site-2 metalloproteinase (S2P) [32]. The mature
processed form of SREBP is released in the cytosol and can translocate into
the nucleus where it modulates the expression of several genes controlling
cholesterol and fatty acid homeostasis [33], including HMGCR, HMG-CoA synthase
(HMGCS), low density lipoprotein receptor (LDLR) and SREBP1/2 itself. SREBP1
activates fatty acid synthesis and SREBP2 cholesterol synthesis and uptake
[34].

In this paper, we combine experiments and computational analysis to show that
lowering the level of cholesterol either by using statins or by inhibiting
SREBP1 and SREBP2 pathways with the small molecule betulin has a significant
impact on the aggregation of non-mutated neuroserpin within the cell. Statins
are competitive inhibitors of HMGCR, a rate limiting enzyme for cholesterol
levels. They are widely used as plasma cholesterol lowering drugs in
dyslipidemic patients. In spite of the clear effect of statins in lowering
morbidity and mortality, in particular in cardiovascular events, many recent
studies suggest that prolonged use of statins in Alzheimer’s disease patients
leads to mixed outcomes and could contribute to statin-adverse effects [35,
36, 37, 38, 39]. Whereas current guidelines encourage an aggressive use of
statins to achieve long-term cholesterol lowering in order to decrease
cardiovascular events, the resulting effect on cell membranes is unknown.
Here, we focus our experimental analysis on non-mutated (wild type)
neuroserpin and directly observe a dramatic increase in aggregates when cells
are chonically exposed to statins. To confirm the direct involvement in
aggregation of lower cholesterol levels, we use betulin, a pentacyclic
triterpene commonly isolated from the birch bark, that specifically inhibits
the maturation of SREBPs by enhancing interaction between SCAP and INSIG thus
promoting SREBPs retention in the ER [40]. Also in this case, we observe lower
cholesterol and more neuroserpin aggregation. Our findings are consistent with
a cholesterol induced decrease in the protein clearance rate, leading to
faster aggregation in our model [20]. To elucidate the biophysical mechanism
underlying this observation, we study a mathematical model of coated vesicle
release from lipid membranes, illustrating the crucial role of cholesterol for
vesicle formation and therefore the impact of its unbalance on intracellular
trafficking. Taken together, our results suggest that long-term treatment with
statins may affect intracellular trafficking in a way to enhance protein
aggregation.

## Results

### Prolonged treatment with statins enhances neuroserpin aggregation in HeLa
cells

Simvastatin (SIM), pravastatin (PRA) and rosuvastatin are all known to be
inhibitors of cholesterol synthesis by acting on HMGCR. PRA is, however,
reported to be less effective both in patients and in vitro due to its
hydrophylicity [41]. We test the effect of long-term (i.e. up to 8 days)
treatment with statins (5$\mu$M SIM or 10$\mu$M PRA or 0.1$\mu$M rosuvastatin
or 1$\mu$M rosuvastatin) on the capability of neuroserpin to aggregate in HeLa
cells. Fig. 1a shows that, in native gel, the treatment with SIM is able to
induce multiple bands of neuroserpin, suggesting the presence of aggregates.
We obtain similar results for the more powerful drug rosuvastatin, as reported
in Fig. 1b [42]. In contrast, the exposure to PRA does not affect neuroserpin
aggregation, as shown in Fig. 1a. To directly confirm the presence of
neuroserpin aggregates after prolonged treatment with statins, we use the
Duolink in Situ staining. Using this technique allows the detection of a
fluorescent spot if at least two monomers of neuroserpin protein are in close
proximity, such as when they form a dimer. We mix (1:1) HeLa with HeLa-GFP
cells to confirm that possible aggregation is independent from the specific
cell line used. Moreover, in HeLa—GFP cells the cytoplasm is green. We then
treat the cells with statins for up to 8 days. Long-time treatment (8 days)
with 5$\mu$M SIM or 0.1$\mu$M rosuvastatin induces neuroserpin aggregation
while shorter incubation times with 5$\mu$M SIM or 8 days of treatment with
10$\mu$M PRA do not induce any effect, as shown in Fig. 2. Moreover, 8 days of
treatment with 0.1$\mu$M rosuvastatin gives a stronger effect with respect to
5$\mu$M SIM Fig. 2 and Fig. 3.

### Effect of the treatment with SIM, rosuvastatin or PRA on cholesterol and
HMGCR levels in HeLa cells

Filipin is commonly used to detect cholesterol, but it is not very stable,
rapidly photobleaches and results in acute toxicity for living cells[43]. To
avoid these problems, we use a fluorescently labeled theonellamide (TNM-AMCA).
TNM-AMCA was originally isolated from a marine sponge, and has been reported
to bind in a specific manner to the 3$\beta$-hydroxyl group of cholesterol
without acute toxicity [44, 45, 46]. 5$\mu$M SIM, 0.1$\mu$M rosuvastatin or
20$\mu$M betulin affect the level of cholesterol after 8 days of treatment
(Fig. 4 and Fig. 3), while PRA is less effective and does not change
significantly the level of cholesterol[41] (Fig. 4). Finally we investigate by
western blot whether the treatment with statins affects the expression level
of the key enzyme of cholesterol synthesis, HMGCR. As shown in Fig. 1c, no
significant changes in the level of expression of this enzyme occurs after
long-time treatment either with SIM or PRA.

### Effect of the treatment with SIM, rosuvastatin or PRA on the level of
expression of SREBP1 and SREBP2

SREBPs are transmembrane proteins expressed at the level of ER, associated
with SCAP which is a cholesterol sensor [31]. Intracellular trafficking of
SREBPs is sensitive to cholesterol depletion: when the level of cholesterol is
reduced, SREBP leaves the ER to reach the Golgi where it is cleaved. Then, the
mature processed form of SREBP is released in the cytosol and translocates
into the nucleus where it modulates the expression of several genes
controlling cholesterol and fatty acid homeostasis[33] and SREBP1/2 itself.
Furthermore, it is known that SREBP1 regulates fatty acid synthesis and SREBP2
cholesterol synthesis and uptake [34]. We thus investigate by
immunofluorescence the effect of statins on the level of expression of SREBP1
and SREBP2. Fig. 5 shows that SREBP1 is not affected by the either SIM or PRA
treatments and SREBP2 increases significantly after 8 days of treatment with
SIM. Rosuvastatin, in contrast, affects mainly SREBP1 (Fig. 6a).

### Effect of the treatment with betulin on the level of expression of
cholesterol, neuroserpin aggregation, SREBP1 and SREBP2

An alternative route to interfere with the cholesterol level and study its
effect on neuroserpin aggregation is to inhibit SREBPs activity using betulin,
a small molecule that is known to inhibit the maturation of SREBP by inducing
interaction of SCAP and Insig [47, 40]. Betulin induces a decrease of
cholesterol after 8 days of treatment (Fig. 3) [40] and an accumulation of the
precursor of SREBPs (pre-SREBP), in agreement with Ref. [48] [49] (Fig. 6b).
Under these conditions, we show that the treatment for 8 days with 20$\mu$M
betulin induces neuroserpin aggregation (Fig. 1 and Fig. 3a).

### Computational simulations reveal aggregation induced by the impairment of
intracellular trafficking

In a previous paper our group developed a model three dimensional to explain
the critical role for protein aggregation of the non-equilibrium conditions in
the ER[20]. Here we implement the model in order to understand how the
impairment of intracellular trafficking might induce protein aggregation. The
model performs coarse—grained simulations of diffusing monomers and polymers
in a closed system, representing the ER, under a constant rate of protein
production $k_{\rm in}$. Proteins exit the ER with a rate $k_{\rm out}$ when
they are close to the boundaries simulating vesicular trafficking. Hence, the
parameter $k_{\rm out}$ encapsulates the efficiency of intracellular
trafficking which can be affected by the level of cholesterol in the cell.
Fig. 7a shows that reducing $k_{\rm out}$, which should simulate an impairment
in the intracellular trafficking leads to an increase in protein aggregation,
thus recapitulating in silico the experimental observations.

### Mathematical model illustrates mechanical regulation of vesicle formation

The mechanism by which the level of cholesterol determines the value of
$k_{\rm out}$ is related to its role in affecting the biophysical properties
of membranes [21, 22, 23, 24, 25] and thus their ability to form the vesicles
needed for intracellular trafficking [26, 27, 28]. Vesicle formation in the ER
is assisted by the Coat protein complex II (COPII) that is recruited on the
surface of the ER membrane [50]. The COPII complex self-assembles into a cage
structure that encapsulate the lipid membrane forming a vesicle [51]. The
success of the process crucially depends on the binding strength of the COPII
to the membrane and on its bending stiffness, as we can illustrate by a simple
mathematical model. We considers an elastic membrane coupled to a curvature
induced coat, representing the action of a COPII cage in the ER. The model,
described in details in the methods section includes two key dimensionless
parameters: i) the ratio between the bending stiffness of the membrane $K_{m}$
and the one of the coating scaffold $K_{s}$, and ii) the ratio between the
scaffold bending energy and its interaction energy with the membrane
$\epsilon_{LJ}$. Simulations show that vesicle can only form is $K_{m}/K_{s}$
is small and $\epsilon_{LJ}/K_{s}$ is sufficiently large. For larger values of
$K_{m}/K_{s}$, the coating scaffold is not able to bend the membrane into a
vesicle while if $\epsilon_{LJ}/K_{s}$ is small the coat detaches from the
membrane. This behaviour is summarized in the phase diagram reported in Fig.
7b. A reduction in the level of cholesterol in the ER membrane affects both
$K_{m}$ [21, 22, 23] and $\epsilon_{LJ}$ [29], thus displacing the system in
different regions of the phase diagram. In particular, experiments show that
cholesterol depletion leads to a reduced recruitment of the Sec23p component
of the COPII complex on the ER membrane, delaying ER-to-Golgi transport [29].

## Discussion

Long-term lowering serum low density lipoprotein cholesterol with statin drugs
is used extensively and has proven very effective to reduce the incidence of
cardiovascular events. De-novo cholesterol which is the target of statin
therapy is found in all membranes and lipid based bodies, contributing to
vesicle formation and migration, as well as other membrane functions [52]. We
can therefore expect that regulating the biosynthesis of cholesterol could
change the form and function of every membrane within the cell. For this
reason, statin therapies could also cause potential harm [53]. The role of ER
in the aggregation of misfolded proteins, such as neuroserpin, and therefore
as a crucial determinant of cellular toxicity, is important in many
neurodegenerative diseases. Our group recently demonstrated the presence of a
phase transition towards rapid protein aggregation due to a breakdown of
homeostasis in intracellular organelles, such as the ER, controlled by the
rates of protein synthesis and clearance, $k_{\rm in}$ and $k_{\rm out}$,
respectively [20]. The open question stated in the conclusions of our earlier
paper was related to the identification of possible biological processes and
factors that would tune these key parameters into an aggregation-prone
pathological phase [20]. In the present paper, we combine experiments and
computational models to investigate this critical issue.

We focus our attention on cholesterol depletion by statins as a potential
inhibitor protein clearance from the ER. We investigate three different
statins in a chonic treatement (8 days): SIM, rosuvastatin and PRA.
Rosuvastatin has been shown to be more effective than SIM [42], while PRA is
less effective when compared with SIM [41]. Here we show indeed that long-term
treatment with SIM or rosuvastatin decreases the level of cholesterol,
increases the level of expression of SREBP, compensating for cholesterol
depletion, and finally enhances neuroserpin aggregation even in the absence of
specific mutations. In contrast, PRA is less effective than the others two
statins and therefore does not affect neither the level of cholesterol nor
neuroserpin aggregation [41].

To ensure that the aggregation we observe for neuroserpin is really due to the
lowering of cholesterol in the cell rather to a direct interaction between
statins and neuroserpin, we interfere with the cholesterol level indirectly by
targeting SREBP through betulin, whose action ultimately leads to cholesterol
depletion [40]. We found that betulin decreases the level of cholesterol and
accumulates pre-SREBPs [40, 49, 48]. Also in this case, we find an enhancement
of neuroserpin aggregation after long term treatment (8 days).

It is well known that N-glycosylation plays determinant roles in protein
folding and trafficking, and N-glycosylated proteins are especially important
in regulating extracellular activities. Recent papers show that aberrant
N-glycosilation happens frequently in relation to human diseases, including
Alzheimer’s[54] and FENIB, as pointed out in a recent paper [55]. Furthermore,
a systematic and quantitative analysis of surface proteins was carried out in
HepG2 liver cells treated with statin showing that many glycosilation sites
are downregulated compared to untreated cells [56]. On the other hand, statins
inhibit the pathway of dilichol, which plays a critical role in protein
N-glycosilation and acts as a membrane anchor for the formation of a precursor
oligosaccharide [57]. Therefore, the long-treatment with statins could lead to
important impairment of intracellular trafficking, effectively decreasing the
clearance rate $k_{\rm out}$ [20]. This is shown directly by numerical
simulations of a protein aggregation model where reducing $k_{\rm out}$ leads
to faster aggregation.

The role of cholesterol on the biophysical properties of lipid membranes has
been the subject of extensive experimental investigation. It is known that the
level of cholesterol affects the membrane bending stiffness $K_{m}$ [21, 22,
23] the dynamics of membrane proteins [24, 25] and has thus an effect on
vesicle formation and trafficking [26, 27, 28]. In particular, experiments
show that the attachment of coating proteins, such as COPII, to membranes is
dependendent on cholesterol [29]. Here, we develop simple mathematical model
that can incorporate into a single framework all this experimental
observations and help predict the effect of statins on the intracellular
trafficking. While the model is too simple to provide a quantitative
explanation of vesicle formation in dependence on the cholesterol level, the
phase diagram illustrates the possibility that small changes of key membrane
biophysical parameters such as the bending stiffness or the binding affinity
of coat proteins, all known to be affected by cholesterol, result in a net
impairment of vesicle formation.

Recent results in the literature show an increased level of neuroserpin in
Alzheimer’s disease patients, the critical role of tPA and the association
between neuroserpin and A$\beta$ plaques in Alzheimer’s brain tissues [58].
Our paper shows that neurosperpin aggregation can be modulated by
environmental factors affecting intracellular trafficking, even in the absence
of deleterious mutations that are known to induce aggregation [5, 6, 1, 2].
The aggregation mechanism we propose is fairly general, depending only on the
modification of kinetic rates for protein clearance and it could therefore be
relevant also for other neurodenerative pathologies. For instance, it would be
extremely interesting to clarify the possible role of statins in A$\beta$
aggregation and more generally on the development of Alzheimer’s disease.

## Methods

### Cell lines and treatments

Human Hela cells (ATCC-CRL-2) or HeLa-GFP cells (AKR-213 Cell Biolabs Inc.)
were cultured in 80%DMEM, 10% FBS with the addition of 1% antibiotics and 1%
L-glutamine at 37∘C / 5% CO2. SIM (S-6196, Sigma) needs to be activated by
opening of the lactone ring before use in cell culture. We used the protocol
described by [59]. Briefly, 40mg/ml of SIM were dissolved in 100% ethanol with
subsequent addition of 0.1N NaOH [59]. The solution was heated at 50∘C for 2h
and then neutralized with HCl to pH 7.2. The resulting solution was bought to
a final volume (1ml) with distilled water and aliquots were stored at -80∘C
for no more then 3 months. Rosuvastatin (SML1264, Sigma) was dissolved in DMSO
stock solution (20mM) and stored in aliquots at -20∘C. Pravastatin (P4498
Sigma) was dissolved in distilled water and stored in aliquots at -80∘C.
Betulin (B9757, Sigma) was dissolved in 100% etanol (500$\mu$M) and stored in
aliquots at -20∘C. The medium has been replaced every 4 days with fresh medium
containing statins or betulin or fresh medium for untreated cells.

### Immunofluorescence

Subconfluent cells were fixed with 3.7% paraformaldeide, incubated with
0.1%Triton X-100 in PBS for 15min at room temperature, then with 3%BSA in PBS
for 30min at room temperature and finally with anti-SREBP-1 (1:100, ab28481
Abcam. 3.7% PFA fixed cells were incubated with 1% BSA/10% goat serum/0.3M
glycine/0.1% tween in PBS for 1h at room temperature and incubated with anti-
rabbit (1:250, Alexa Fluor 488, ab 150077) for 1h at room temperature.
1$\mu$g/ml Actinstain-555 phalloidin (Cytoskeleton Inc) is used to stain actin
for 1h at room temperature. The nuclei were counterstained with DAPI and the
slides mounted with Pro-long anti fade reagent (Life technologies). The images
were acquired with a Leica TCS NT confocal microscope.

### Duolink assay

Subconfluent cells were fixed with 4% paraformaldeide and incubated with
0.1%Triton X-100 in PBS for 15min at room temperature, then with 5%BSA in PBS
for 1h at room temperature and finally incubated with anti-neuroserpin
(ab32901 AbCam) coniugated PLA probes (1:50) overnight at 4∘C. We detected
neuroserpin aggregates with Duolink In Situ staining. The Duolink In Situ
staining are based on in situ PLA, which is a proximity ligation assay
technology. A pair of oligonucleotide labeled secondary antibodies (PLA
probes) generates a signal only when the two PLA probes have bound in close
proximity (at a distance of less than 40nm), either to the same primary
antibody or to two primary antibodies bound to the sample in close proximity.
The signal from each detected pair of PLA probes is visualized as an
individual fluorescent spot. Duolink In Situ Probemaker PLUS enables quick and
convenient conjugation of the PLA PLUS oligonucleotide arm directly to the
primary antibody (DUO92009 and DUO92010 for PLUS and MINUS probe). The cross
reacted proteins were detected according to the manufacturer’s instruction
(sigma.com/duolin, section 7.3 Detection protocol). The nuclei were mounted
with Duolink in situ mounting medium containing DAPI (DUO82040, Sigma). The
images were acquired with a Leica TCS NT confocal microscope. These PLA
signals were quantified as described in the section spot detection.

### Cholesterol staining

Cells were fixed with 4% paraformaldehyde (PFA) in PB 0.1M for 10 minutes at
room temperature (RT). Then, they were incubated with 50$\mu$g/ml Digitonin
for 5 minutes and then for 1 hour with 4% Bovine Serum Albumin (BSA) in PBS.
The cells were then incubated with TNM-AMCA (1$\mu$M, gently gifted by
Shinichi Nishimura and Minoru Yoshida, RIKEN Center for Sustainable Resource
Science, Saitama, Japan) for 1 hour. All the specimens were mounted with
Fluoromount Aqueous Mounting Medium without DAPI (Sigma-F4680) and the images
were acquired with Leica SP2 laser scanning confocal microscope.

### Western Blot

Confluent cells were lysed by boiling in a modified Laemmli sample buffer (2%
SDS, 20% glycerol, and 125 mM Tris-HCl, pH6.8). The protein concentration was
measured by BCA Protein Assay Kit (Thermo Scientific). Equal amount of
proteins were loaded on gel and separated by SDS-PAGE and transferred to a
PVDF membrane (Whatman). After blocking, primary and HRP-linked secondary
antibodies, specific bindings were detected by chemiluminescence system (GE
Healthcare). Goat polyclonal antibody anti-neuroserpin (1:500, ab32901, AbCAM)
or rabbit anti-HMGCR (1:1000, MAB5374, Millipore) or anti-SREPB1 (1:500,
ab28481, AbCam) or anti-SREBP2 (1:100, ab30682, AbCam) were used overnight at
4∘C. Mouse anti-vinculin (1:10000, V9264, Sigma) or anti-$alpha$ Tubulin
antibody (1:5000, Sigma) for 1h at room temperature was used as housekeeping.
A secondary antibody anti goat-HRP (1:5000, ECL Blotting reagents (GE
Healthcare RPN2109) / SuperSignal™West Femto Maximum Sensitivity Substrate,
Thermo scientific) was used for 1h at room temperature to detect
chemiluminiscence.

For non denaturating gels, cells were resuspended in 50mM Tris-HCl, pH7.4
containing 5mM EDTA, 1% Triton-X100 and Protease inhibitors cocktail
(1ml/100ml lysate, P8340, Sigma), passed for 4 times through an insulin
syringe and maintained in ice for 20min. After a brief centrifugation, the
protein concentration was measured by BCA Protein Assay Kit (Thermo
Scientific). 100$\mu$g protein were loaded on 10% non denaturating gel,
transferred on PVDF membrane as described above. The PVDF membrane was
reversible stained with Ponceau S solution (P7170, Sigma) to verify the
correct loading of the samples.

### Spot detection

Image analysis of Duolink In Situ staining was performed using the bioimage
informatics platform Icy [60]. To detect spots, we first use the Thresholder
plugin to eliminate the random noise in the images. We used the automated mode
so that the algorithm automatically calculates the optimal threshold value. We
then applied the Spot Detector plugin [61] that is based on an UnDecimated
Wavelet Transform tool designed to detect spots. Then, we used the default
values, enabling the scale=2 setup with a threshold equal to 100 and a size
filter to accept only spots with a size bigger than three pixels. We also
extracted the number of cell nuclei and compute the ratio between the number
of spots and the number of cells.

### Statistical Analysis

Statistical significance analysis is performed using the Kolmogorov-Smirnov
test.

### Polymer aggregation model

Simulations are performed using the protein aggregation model described in
Ref. [20] where individual protein molecules sit on a three dimensional cubic
lattice. Monomers diffuse with rate $k_{\rm D}$ and attach to neighboring
monomers or polymer endpoints with rate $k_{\rm H}$. Polymers move
collectively by reptation with a length-dependent rate $k_{\rm R}/i^{2}$,
where $i$ is the number of monomers in the polymer, and locally by end
rotations, with rate $k_{\rm E}$ and kink motions with rate $k_{\rm K}$. A
polymer can attach to another polymer with rate $k_{\rm H}$ if their endpoints
meet, and can fragment by breaking an internal bond with rate $k_{\rm f}$.
Inspired by experimental results on neuroserpin polymerization [62], we allow
for polymerization after at least one of the monomers has been activated with
rate $k_{\rm A}$. Active monomers can also become latent with rate $k_{\rm L}$
and after that they do not aggregate. We consider a system of size ($L\times
L\times H$) with $L=100L_{0}$ and $H=25L_{0}$, where $L_{0}$ is the linear
size of a monomer, with periodic boundary conditions along x and y and closed
boundary conditions along z. Monomers enter the system from both closed
boundaries with rate $k_{\rm in}$ and monomers and polymers can exit from the
same boundaries with rate $k_{\rm out}/i^{3}$. Numerical simulations are
performed using Gillespie Montecarlo algorithm [63]. We measure time in units
of $1/k_{\rm in}$ and set $k_{\rm H}=k_{\rm E}=k_{\rm R}=k_{\rm K}=k_{\rm
A}=k_{\rm in}$, $k_{\rm L}=0$, $k_{\rm D}=10^{2}k_{\rm in}$, $k_{\rm
f}=10^{-3}k_{\rm in}$ and vary $k_{\rm out}$.

### Vesicle formation model

We consider a simple model with three basic interaction energies: two bending
energies related, respectively, to the elasticity of the membrane and external
cage, and a potential representing their mutual interaction. The membrane is
modeled by a polygon with $N_{m}$ nodes connected by inextensible bonds and
resisting bending through an angular spring of energy

$E_{a}=K(\theta-\theta_{0})^{2},$ (1)

where the parameter $\theta_{0}=\theta_{m}=0$ is the equilibrium value of the
angle between two bonds, and $K=K_{m}$ is the bending stiffness of the
membrane. The coating scaffold is modeled, instead, as an open polygon with
$N_{s}$ nodes, rigid bonds and a bending energy given by Eq.1 with $K=K_{s}$
and $\theta_{0}=\theta_{s}\geq 0$. In order to simulate a realistic structure,
we chose $N_{m}/N_{s}=10$, corresponding to the ratio between the typical
lipid distance in a biological membrane and the characteristic length of the
COPII proteins in a cage [51]. For the membrane-scaffold interaction we employ
a Lennard-Jones potential where the depth of the potential well is equal to
$\epsilon_{LJ}$. Starting from a pseudo-circular equilibrium configuration,
where the scaffold does not perturb the membrane, we simulate the vesicle
formation by changing gradually the equilibrium angle of the scaffold
$\theta_{s}$. The simulations are performed using the LAMMPS molecular
dynamics simulator package [64] with a constant NVE integration and the
addition of a viscous damping force to reduce oscillations.

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## Acknowledgements:

We thank S. Nishimura and M. Yoshida, RIKEN Center for Sustainable Resource
Science, Saitama, Japan who kindly provided the cholesterol probe TNM-AMCA; GC
and SZ are supported by the ERC advanced grant SIZEFFECTS. SZ acknowledges
support from the Academy of Finland FiDiPro progam, project 13282993. CAMLP
thanks the visiting fellow program of Aalto Science Institute.

## Author contributions:

CG carried out the native elecrophoresis, MCL and FM carried out
immunofluorescences and western blots experiments, GC performed numerical
simulations and image analysis, SZ designed the model and assisted in the
writing of the paper. CAMLP designed the project, performed and supervised
experiments, and wrote the paper.

## Additional information

Competing financial interests The authors declare no competing financial
interests.

Figure 1:  Aggregation of neuroserpin with statins or betulin a) 100$\mu$g
protein of HeLa cells or treated with 10$\mu$M PRA or 5$\mu$M SIM or 0.1$\mu$M
rosuvastatin or 1$\mu$M rosuvastatin or 20$\mu$M betulin were loaded on 10%
native gel and transferred on PVDF membrane. The sheet was incubated with
anti-neuroserpin (ab32901, Abcam, 1:500) overnight at 4∘C and then with the
secondary antibody anti goat-HRP (1:5000, ECL Blotting reagents (GE Healthcare
RPN2109) / SuperSignal West Femto Maximum Sensitivity Substrate, Thermo
scientific) for 1h at room temperature. Ponceau S solution (P7170, Sigma) was
used to verify the correct loading of the samples. b) Effect of statins or
betulin treatments on the level of expression of neuroserpin or HMGCR.
10$\mu$g total protein were loaded on 10% polyacrilamide gel, transferred on
PVDF and incubated with anti-neuroserpin (ab32901, AbCAM, 1:500) or with anti-
HMGCR (1:1000, MAB5374, Millipore) overnight at 4∘C). Vinculin (1:10000,
Sigma) was used as housekeeping.  Figure 2:  Duolink In situ Staining of
neuroserpin in HeLa untretaed or treated with SIM or PRA. a) Hela-GFP and Hela
cells were mixed 1:1 and treated with 5$\mu$M SIM or 10$\mu$M PRA up to 8
days, fixed in 3.7% paraformaldeide and incubated with the primary antibody
anti-neuroserpin (Ab32901, Abcam) coniugated PLA probes (1:50), overnight at
4∘C. The presence of aggregates are visualized using Duolink in Situ staining
according to the manufacturer’s instructions. Nuclei are stained with DAPI.
Images are acquired by Leica SP2 laser scanning confocal microscope. Each red
spot represents a neuroserpin aggregate. b) The number of spots per cell for
each condition reported in panel a is quantified as described in the methods
section. Figure 3:  Effect of rosuvastatin and betulin on neuroserpin
aggregation and cholesterol level. a) Hela-GFP and Hela cells were mixed 1:1,
treated with 0.1$\mu$M rosuvastatin or 20$\mu$M betulin for 8days, fixed in
3.7% paraformaldeide and incubated with the primary antibody anti-neuroserpin
(Ab32901, Abcam) coniugated PLA probes (1:50), overnight at 4∘C. The presence
of aggregates are visualized using Duolink in Situ staining according to the
manufacturer’s instructions. Nuclei are stained with DAPI. Images are acquired
by Leica SP2 laser scanning confocal microscope. Each red spot represents a
neuroserpin aggregate. The number of spots per cell for each condition in the
images showed in this panel is quantified as described in the methods section.
b) Untreated or treated cells with 0.1$\mu$M rosuvastatin or 20$\mu$M betulin
(8 days) were fixed with 3.7% paraformaldeide and then incubated with TNM-AMCA
(1$\mu$M) to stain cholesterol for 1h at room temperature. Images were
acquired by Leica SP2 laser scanning confocal microscope. Figure 4:  Detection
of cholesterol with TNM-AMCA in untreated and treated HeLa cells after statins
treatments. Untreated or 5$\mu$M SIM or 10$\mu$M PRA treated HeLa cells for
the time reported in the figure, were fixed with 3.7% paraformaldeide and then
incubated with TNM-AMCA (1$\mu$M) to stain cholesterol for 1h at room
temperature. Images were acquired by Leica SP2 laser scanning confocal
microscope. Figure 5:  Level of expression of SREBP1 and SREBP2 after SIM or
PRA treatment in Hela cells. Subconfluent cells are fixed with
3.7%paraformaldeide after 8 days treatment with statins and incubated with
anti-SREBP1 (1:100, ab28481) or anti SREBP-2 (1$\mu$g/ml ab30682, Abcam)
overnight at 4∘C. Next, the cells are incubated with anti-rabbit Alexa Fluo488
for 1h at room temperature. Actin is stained with 1$\mu$g/ml Actistain-555
phalloidin for 1h at room temperature. Nuclei are stained with DAPI. Untreated
cells are also shown for reference. Images are acquired by Leica SP2 laser
scanning confocal microscope. Figure 6:  a) Level of expression of SREBP1 and
SREBP2 after rosuvastatin or betulin treatment in Hela cells. Subconfluent
cells a fixed with 3.7%paraforldeide after 8 days of rosovastatin (0.1$\mu$ M)
or betulin (20$\mu$M) treatment and incubated with anti-SREBP1 ( 1:100,
ab28481) or anti SREBP-2 (1$\mu$g/ml ab30682, Abcam) overnight at 4∘ C. Then,
the cells are incubated with anti-rabbit AlexaFluo488 for 1h at room
temperature. Nuclei are stained with DAPI. Untreated cells are also shown for
reference. Images are acquired by Leica SP2 laser scanning confocal
microscope. b) Western blot of SREBP1 and SREBP2 in Hela cells untreated and
after 8 days of treatment with (20$\mu$M betulin. 10$\mu$g total protein were
loaded on 10% polyacrilamide gel, transferred on PVDF and incubated or anti-
SREPB1 (1:500, ab28481, AbCam) or anti-SREBP2 (1:100, ab30682, AbCam) were
used overnight at 4∘C. Mouse anti-$alpha$ Tubulin antibody (1:5000, Sigma) for
1h at room temperature was used as housekeeping. A secondary antibody anti
goat-HRP (1:5000, ECL Blotting reagents (GE Healthcare RPN2109) /
SuperSignal™West Femto Maximum Sensitivity Substrate, Thermo scientific) was
used for 1h at room temperature to detect chemiluminiscence. The bands
reported are the precursor of SREBP1 and SREBP2, pre-SREBP Figure 7:
Numerical simulations show protein aggregation induced by impairment of
trafficking. a) The main figure shows the average number of aggregates
observed at a fixed time as a function of $k_{\rm out}/k_{\rm in}$, the ratio
between rate of exit from the ER by means of vesicular transport and the rate
of protein production. A reduced value of $k_{\rm out}$ leads to the formation
of aggregates. The inset shows the time dependence of the average number of
aggregates for different values of $k_{\rm out}$. b) The phase diagram of the
vesicle formation model. The stiffness ratio between the membrane and the
external scaffold $K_{m}/K_{s}$ as a function of the intensity of the mutual
interaction with respect to the scaffold rigidity $\epsilon_{LJ}/K_{s}$. For
soft membranes there is almost always vesicle formation (yellow zone) or the
breaking of the coat-membrane interface, if binding is not strong enough
(white zone). Increasing the membrane stiffness leads to a new phase appears
where the coat-membrane interface is stable but vesicle can not form (grey
zone). Typical configurations in the different phases are reported on the
right-hand side of the panel.